Undergraduate Research Positions
NSERC USRAs and WLI URAs
The Mathematics Department is pleased to offer summer research positions for academically highachieving undergraduate students. The NSERC Undergraduate Student Research Awards (USRAs) and Work Learn International Undergraduate Research Awards (WLI URAs) give students the opportunity to work with a faculty supervisor, acquiring valuable research experience in an academic setting. This opportunity is also useful for enhancing students’ resumes, for those interested in enrolling in a graduate program or for helping to decide a future career route.
Jump to Research Projects
Position Details
The normal duration of these research position is 16 weeks of fulltime work in the summer. Applicants must be willing to be on campus every weekday during the summer, and not be taking summer courses. At the end of the research position, students write a short report on their activities.
The stipend for these awards is yet to be confirmed. In 2023 summer the minimum paid was $8,931.83
Eligibility
 Applicants must have a strong academic standing, an average of over 80% in Mathematics Courses and a minimum of 70% overall.
 NSERC USRAs  If you are a Canadian citizen or permanent resident, check UBC NSERC website for eligibility details. We welcome applications from students at other eligible universities.
 WLI URAs  If you are an international student, check the Work Learn International website Work Learn International website for eligibility details. You must be attending UBC Vancouver.
 Students who are offered a position but do not meet the criteria above will not be able to accept a position.
 Students intending to graduate in 2024 will be given a lower priority.
 You must be residing in British Columbia during the length of the project (usually May  August).
How to ApplyThe application deadline for 2024 summer positions is Thursday, January 25, 2024.

Science Undergraduate Research Experience
There is no formal application for the Science Undergraduate Research Experience (SURE) Awards in the Department of Mathematics. Applicants to the NSERC USRA or WLIURA who are not offered a position may be considered for the SURE Award, provided that they have also satisfied the SURE eligibility requirements. You must be attending UBC Vancouver in the Faculty of Science and be residing in British Columbia. You can read more about the SURE awards here: https://science.ubc.ca/students/blog/researchopportunitiesscienceundergraduates2024
Questions?
Contact the Resource Coordinator, Allen Yang.
You may also reach out to the faculty supervisor(s) for each project for more information: UBC Mathematics Faculty Directory
Research Projects
2024
* We will investigate the mixing behavior of Markov chains satisfying that the graph supporting the transitions of the chain is "treelike". This problem is very well understood when the Markov chain is reversible. The aim of the project is to extend some of the results to the nonreversible case.
* The LUCE random permutation model is a general class of random permutations which has appeared in several settings but its properties are not well understood. We aim to generalize some of the classical results on random permutations to this class of models.
* Pattern avoiding permutations are an active area of research with many recent results. We wish to study several Markov chains on the set of pattern avoiding permutations, and to study their mixing properties and large scale behaviour.
The nematode C. elegans is a model organism commonly used to study animal development. In this project, the student will participate in developing a mathematical and computational model of the process by which leftright asymmetry first appears in C. elegans embryos. The ideal student will have some familiarity with nonlinear differential equations, some basic mechanics (equations of motion including friction, torque, vectors etc.), comfort in handling data (from the Sugioka Lab in Zoology), strong coding skills. There is also a possibility for the student to work on an image processing algorithm involving the tracking of structures in 3D timelapse images of C. elegans embryos (in addition or instead of the above described project). In addition to regular oneonone meetings with me, the student will attend my regular research group meeting, learn about other ongoing modelling efforts and get mentoring from more senior researchers in the group (grad students/postdocs).
The growth of plants depends on the spatially organized differentiation of cells in the growing root. In this project, the student will participate in developing a mathematical and computational model of the process by which hormone signalling, cell elongation, and cell division together lead to the organization of the root tip into three functionally distinct zones which form a kind of traveling wave as the root grows. The ideal student will have some familiarity with nonlinear differential equations and strong coding skills. In addition to regular oneonone meetings with me, the student will attend my regular research group meeting, learn about other ongoing modelling efforts and get mentoring from more senior researchers in the group (grad students/postdocs).
My group is offering 1 or 2 USRA/WLI positions for undergraduate students, to participate in collaborative projects with other faculties at UBC in computer science, biochemistry and the Institute for Resources, Environment and Sustainability (IRES). Projects include development of data driven green house gas models for farming systems, development of workflow and database for cryoEM, cell shape analysis, dashboard for data visualization. Students joining the lab will also have the opportunity to participate in our group meetings and seminars to learn about other current projects and more advanced topics in machine learning, mathematical biology and computational geometry (more info and publications available at https://kdaoduc.com). For enquiries, please send an email with transcripts and cv to kdd@math.ubc.ca
Organoids are clusters of cells engineered to mimic properties of tissue organization and function. These invitro cellular systems hold great promise for drug discovery, medical therapies, and fundamental biology. The group of Peter Zandstra (Biomedical Engineering, UBC) is developing organoids using human pluripotent stem cells (hPSCs), with the aim of controlling size, shape, and cellular organization in these structures. This USRA project will develop early mathematical and computational models to support the experiments and theoretical development in the Zandstra lab. We will model the response of gene regulatory systems of these cells to chemical stimuli (e.g. Bone Morphogenetic Protein, BMP), to cellcell contact, to cell population size, etc. The mathematical modeling will consist of several steps. (1) Investigating models for the internal generegulatory systems of the cells using ordinary differential equation models (some already developed by the above team, some to be developed). (2) Assembling computational models of cells with internal generegulatory dynamics in the free software platform Morpheus (TU, Dresden) that allows us to visualize the dynamics of cell shapes, cellcell interactions, and cellchemical interactions. This platform will serve as an "insilico" representation of the Zandstra experiments, where math modeling, hypothesis testing, and parameter tuning can be easily and cheaply conducted. (3) Modeling the microbead experiments, where hPSC aggregates are exposed locally to BMP4laden beads. (4) Determine how organoid development differs in global vs local BMP exposure. The ultimate goal is to determine conditions for the development of elongated organoids with several cell layers observed in the lab
Minimization problems arise naturally in many branches of mathematics and science. For example, problems in navigation involve finding paths of least length (geodesics) on the earth's surface. Minimal surfaces, which are twodimensional analogs of geodesics, are minimizers (or simply critical points) of the area function, and arise naturally in material science; for example in fluid interface problems and elasticity problems. A simple physical example of a minimal surface is the soap film that forms after dipping a wire frame into a soap solution. By the laws of surface tension this soap film has the property that it is stable, that is it becomes larger under slight deformations. The theory of minimal surfaces has had striking applications, for example to general relativity and low dimensional topology.
This project will involve theoretical, numerical, and graphical problems with minimal surfaces and eigenvalue problems, tailored to the interests and background of the research assistant.
How can mathematics help design new materials? If you want to build a structure out of Lego, you put the pieces together onebyone, however, if you want to build new materials out of nanoscale or microscale particles, this strategy doesn’t work, because such particles are constantly moving around due to thermal fluctuations in their environment — like a box of Lego that is being constantly and vigorously shaken. Nevertheless, by designing particles appropriately (shapes, sizes, who they stick to, etc), they can be made to spontaneously assemble into a desired structure — the Lego structure forms itself, even as it is being shaken. Such spontaneous assembly happens all the time in biology, where proteins fold, viruses assemble, etc, and we’d like to understand the principles behind such selfassembly so we can use them to design better materials. This project will work towards this goal by developing algorithms to solve inverse problems in self assembly. We’ll ask: how should particles be designed (shape, size, who they stick to, etc), so they spontaneously assembly into a given target structure under thermal fluctuations? The project will explore a simplified model of particle assembly that comes from statistical physics, which considers sticky squares on a lattice, and will study ODE reactiondiffusion models to predict the yield of different types of target structures. It will then use optimization and automatic differentiation tools to try and optimize these yields. If time and interest permits, we can also study particlebased simulations, and develop tools to estimate and optimize yields by adapting automatic differentiation to stochastic particle trajectories.
Prerequisites: good knowledge of ODEs, linear algebra, probability, programming (or a strong interest in learning this quickly). Familiarity with some aspects of physics, especially statistical physics, is helpful but not necessary, but a willingness to think in an interdisciplinary manner is critical.
Optimal transport theory considers phenomena arising when mass distributions are matched in a cost effective way. Starting from being one of the main problems in linear programming, it now has emerged as a central topic in several areas of mathematics such as partial differential equations, probability, and geometry. It also gives a quantitative way to analyze data and has become a prominent tool in data science/machine learning/AI. In this summer project, we will explore both theoretical and computational aspects of optimal transport, aiming at making new contribution. Some possible directions are as follows:
• Constrained data fitting via optimal transport: how can we fit a curve/manifold into a distribution in an optimal way?
• Statistical inference problems: given a sample dataset can we effectively figure out the original distribution?
• Gradient flows: what is a natural way a distribution evolves over time?
References: Book: Computational Optimal Transport https://optimaltransport.github.io/book/
Also check out https://kantorovich.org and https://kantorovich.org/event/otgradientflows/
Qualifications: We will hire a student who has strong background in mathematical analysis (e.g. Math 321 or higher) and coding (e.g. Python). Open mindedness and willingness to tackle difficult problems are crucial.
This project seeks to recruit an enthusiastic student to work on the mathematical and computational modelling of cell clustering and dispersion in the presence of a chemotractant signals in the environment. The aim is to carry out detailed literature review on cell clustering and directed cell migration, learn new mathematical models of geometric surface partial differential equations and how these can be coupled with hooke's elastic models for cell and nuclear deformation. The models will then be solved efficiently and robustly by the use of finite element methods on domains that change in time. The ultimate goal is to have a numerical algorithm that characterises cell clustering and dispersion in 2D and 3D under the influence of a chemotactic signal. Algorithm development and implementation will be carried out in FeNiCs, a finite element open source software package.
This project will require background in both quantum mechanics and differential geometry. We will start with the concept of an anomaly in quantum theory. The first objective will be to understand simple toy models of this phenomenon. After understanding the basic examples, we will move on to the “anomaly cancellation equation” which is an equation studied in the field of differential geometry as a nonlinear partial differential equation on complex manifolds. Though this equation is studied purely as a PDE problem in mathematics, the goal of this project is to clarify its physical origins.
The concept of size of a set is ubiquitous in many areas of quantitative sciences. Measure, dimensionality and thickness are mathematical concepts that aim to quantify the size of a set. Many distinct notions of dimension arise in analysis, combinatorics, geometric measure theory and the study of fractals, leading to indepth investigations of their properties. Yet many natural questions about them remain open.
The project consists of two parts. The first half will build some background in several different notions of size, such as topological, Minkowski, Hausdorff, Fourier dimensions, porosity, gaps and Newhouse thickness, and establish wellknown relations between them. The second part will focus on computing some of these quantities for sets where they are still unknown. Applications to different mathematical areas will be a feature.
Secondyear undergraduate real analysis and elementary measure theory are prerequisites for this project.
New measurement technologies like singlecell RNA sequencing are bringing 'big data' to biology. One of the most exciting prospects associated with this new trove of data is the possibility of studying temporal processes, such as differentiation and development. leveraging tools from probability and optimal transport. We aim to develop a mathematical theory to answer questions like How does a stem cell transform into a muscle cell, a skin cell, or a neuron? How can we reprogram a skin cell into a neuron? We model a developing population of cells with a curve in the space of probability distributions on a highdimensional gene expression space. We design algorithms to recover these curves from samples at various timepoints and we collaborate closely with experimentalists to test these ideas on real data.
Summer students in my group will receive rigorous training in optimization, probability, statistics, and mathematical programming. During the rotation period, we will work through this tutorial on applying optimal transport to stem cell reprogramming: https://broadinstitute.github.io/wot/
If time permits, will then explore various extensions on either the theoretical or applied side depending on student interest.
A proof assistant is a is a software tool designed to help mathematicians and other researchers in verifying that proofs are correct, and thus also with proof development. It provides a formal framework for expressing mathematical statements and reasoning about their correctness.
We will first learn how to interact with the Lean proof assistant, that is how to program definitions, statements, and proofs in the Lean language. We will then formalize and thus verify mathematical results.
Possibile goals for the mathematics to formalize include Lindenstrauss's measure rigidity result, my work in algebraic number theory,and undergraduate topics like Galois theory and Fourier analysis. The selection will be made in cooperation with the student(s).
This project aims to experiment with dynamic presentations in mathematics, where static exposition has typically been the norm. As an example, Francis’ Topological Picturebook offers an enticing point of entry to constructions and theorems in topology; but what if these could be animated? We will draw on material from lowdimensional topology and from algebra to learn some mathematics, and then think of new ways to present these ideas using computer graphics. A keen interest in learning mathematics outside the standard undergraduate curriculum, as well as advanced computing skills, will be essential to the project.
Past Projects
2023
We will investigate the large scale structure and asymptotic properties of certain random permutations. The starting point will be recent work on blockwise simple permutations, and several recent papers on permutations with forbidden patterns. we would try to extend some of the results from hard constraints to soft constraints, i.e. weighted random permutations.
A problem in extremal combinatorics asks how many (discrete) objects are possible subject to some condition. We consider simple matrices which are (0,1)matrices with no repeated columns (thus an encoding of a set system). We say a simple matrix A has F as a configuration if a submatrix B of A is equal to a row and column permutation of F. Assume F is given. Much has been determined about the maximum number of columns and mrowed simple matrix can have subject the condition of no configuration F. There is an attractive conjecture (A. and Sali 2005) to pursue with some challenging open problems. A variety of open problems exist including two 4rowed cases that, if we could prove quadratic bounds, would establish the conjecture for all 4rowed F.
I have plans to launch a new YouTube channel whose goal will be showcasing the value and methods of mathematical modelling in everyday contexts. The videos will target a range of audiences from high school students through grad level but will mostly be aimed at people who have seen some second and third year math (linear algebra, ODEs, PDEs, stochastic processes etc.). For now, I am looking for a student with a strong background in these areas of mathematics and some experience with modelling (e.g. MATH 360, 361, 462, or similar courses in other departments) to help with background research and preparation for the videos. If the student is interested in video production and editing, we could also start that process. Some video ideas include: The mathematics of colour vision, The basics of epidemic modelling, Why Translink has to work to avoid bus clumping, Installing a drip irrigation system, Population dynamics of muggles and wizards. The student will gain experience building and analyzing models and designing and producing "viral math videos”.
As part of a an ongoing collaboration with the Richman Lab in Dentistry, we are developing models and data analysis tools to understand the initiation and ongoing replacement of teeth in reptiles. This project is focused on developing an image analysis pipeline and user interface that will allow us to accelerate the work currently being done manually. A previous USRA student made quite a bit of progress but there is still refinement of the algorithm and improvements to the UI required. Strong programming skills are required (the existing code is in Matlab) and familiarity with basic image processing, both traditional and machine learning, would be useful. The student will gain experience developing and coding an image processing algorithm and producing a GUI for use by a nontechnical audience; they will also gain interdisciplinary experience, working closely with experimentalists and modellers.
Throughout their lives, reptiles continually lose and replace teeth in an organized manner, in contrast with the single tooth replacement of humans and other mammals. Kulesa et al. (1996) proposed a partial differential equation (PDE) model to explain how the locations at which teeth form in young alligators are determined in the growing jaw. More recently, we proposed a model (a system of ODEs) to explain the spatiotemporal patterns in tooth replacement seen in the adult jaw of leopard geckos (Brink et al. 2023). The goal of this project is to integrate the two models into a single model for tooth replacement throughout the life of the organism. The ideal student for this project would have an interest in developmental biology, knowledge of systems of ODEs, some familiarity with PDEs, and experience with numerical solutions/scientific computing. This project is part of an ongoing collaboration with the Richman Lab in the Faculty of Dentistry.
Refs:
Kulesa et al. 1996: https://doi.org/10.1006/jtbi.1996.0103
Brink et al. 2023: https://doi.org/10.1101/2022.11.17.513312
The roundworm Caenorhabditis elegans is a model organism that biologists have used to explore many aspects of animal development. One fascinating aspect is the appearance of chirality in the early embryo when it has only 4 cells. Motivated by recent and ongoing observations in the laboratory, we will develop a computer simulation to test a hypothesis for how two of the cells, ABa and ABp, rotate clockwise (viewed from the above) when they divide, thus introducing chirality to the 6cell embryo.
The ideal USRA student will have strong coding skills, good intuition about the mechanics of cellcell adhesion, and some biological knowledge of the cell division process.
Bubble formation and consequent gas emission are common in natural geological materials such as flooded soils and terrestrial sediments via biodegradation, but also in manmade ponds such as oil sands tailings ponds. Recent data suggests that in 2020 around 7 megatonnes of methane and carbon dioxide were released from oil sands tailings ponds in Canada. The tailings ponds consist of the FFT and MFT layers, which include water, sand, anaerobic microorganisms and naphtha. In these layers, degradation of the naphtha by the microorganisms leads to the production of methane and carbon dioxide, both potential causes of GHG emissions. The complex fluids group at UBC has been studying this process in a lab setting, using lab experiments, models and computations, from a fluid mechanics perspective. The MFT/FFT layers can be categorized as timedependent yield stress fluids. These materials behave like a solid when the applied stresses are below a threshold, i.e. the yield stress, while above the threshold they deform and flow like a liquid. This suggests the ability of these layers to retain gas bubbles. Our interest is in the entrapment and release of bubbles, understanding the physical processes and eventually how the fluid rheology might be used as a means to entrap the bubbles and potentially control GHG emission from the ponds.
To date, we studied the onset of motion of a single bubble as well as the effect of interactions of bubbles and eventually the stability of bubble clouds by smallscale lab experiments. We are interested in extending our experiments to study bubble entrapment/entrainment and stability in multilayered fluid systems as well as bubbles’ propagation and stability in materials with nonuniform rheology where networks of low viscous layers control the bubble motion and propagation. We plan to use a largescale pressurized vertical flow loop equipped with an imaging system to study gasliquid flows of interest. We are seeking 12 students to work on the project. The students will assist in all operations related to the experiment: fluid preparation, rheometry measurements of the fluids, running experiments and image processing. The students will learn the physical background of the experiments and may help in the design of new components and implementation of changes to the apparatus.
Atomic reconstruction techniques for cryoelectron microscopy (cryoEM) have neglected the lipid micelle of membrane proteins, which are an important class for biomedical research because of their disease relevance. While 3D protein structure can be challenging to model and infer from 2D cryoEM images, lipid micelles are more tractable due to their simpler geometry (e.g. ellipsoids with cylindrical cavities). This project aims to leverage this geometric representation with the theory of image formation in CryoEM to find a theoretical formulation of images generated by micelles, with an efficient numerical implementation in a high performance physics simulator. The forward model of image formation in cryoEM is well defined and involves 3D rotations, integration along the imaging axis, and the multiplication of a filter in Fourier space which accounts for the microscope lensing effects, that results in a complex integral to solve and estimate, for simple geometric shapes. Furthermore, the student will work on using the results of this geometric study to develop inference methods of the geometrical parameters from 2D images in synthetic and experimental datasets.
Oil and gas wellbore leakage is a primary concern for the public due to risks of groundwater contamination and greenhouse gas (GHG) emissions. The wellbore travels through sensitive geological zones containing brine and hydrocarbons. Casing is placed in the wellbore to protect against collapse, and cement is used to replace the water between the rock and casing to maintain stability and protect against leakage. Proper displacement of water by cement is vital to ensure a proper bond is made between the cement and pipe as wells can remain active for decades. When the wells are no longer active and abandonment occurs, a long cement plug is pumped into the casing to hydraulically seal the well.
As a result of poor cementing practice there many thousands of wells in Western Canada that now leak, contributing significantly to total industrial emissions of methane. Therefore this research aims to understand displacement and exchange flows using viscoplastic fluids. A key feature of a viscoplastic fluids is its yield stress: the material flows only if the imposed stress exceeds the yield stress. This raises question regarding how yield stress impacts displacement flow, or how a yield stress fluid can hinder buoyancy flow of a lighter fluid.
We are seeking 2 students to work on two subprojects: (i) Displacement flow of a viscoplastic fluid; (ii) Exchange flow between a viscoplastic fluid and Newtonian fluid. Both these project will hopefully lead to better industry practice and help Canada meet its GHG target reductions. The students will assist in all operations related with the experiments: fluid preparation, rheometry measurements of the fluids, running experiments and image processing. The students will learn the physical background to the experiments and may help in design of new components and implement changes to the current apparatus.
This position would perhaps suit a student in Engineering, Engineering Physics, Physics or Chemistry with hands on laboratory experience who wishes to gain research experience in fluid mechanics and rheology. Interested students should contact Scott Charabin (scott.charabin@ubc.ca) with resume and transcript. The intention is that the successful students be funded under the NSERC USRA program or international student equivalent..
This project will investigate properties of topological structures called ``skyrmions" which are known experimentally to appear in certain (``chiral") magnetic materials, and are considered to have technological potential. Mathematically, chiral skyrmions arise as solutions to very interesting (``micromagnetic") PDE models. The project will involve some combination of literature review, numerical computation, asymptotic computations and rigorous analysis.
Many small particles like viruses, bacteria, or DNAcoated colloids, move by socalled “ligandreceptor” contacts, where sticky “legs” on the particle bind and unbind to sticky patches on another surface. Averaged over many binding and unbinding events, the particle generally diffuses, meaning its meansquared displacement increases linearly with time. At low temperatures however, some systems start to undergo subdiffusion — their meansquared displacement increases as t^alpha where alpha is less than 1 — which means the particles move about and explore their accessible space much more slowly. The reasons for this are not fully understood. This project will explore the origins of subdiffusion, by building and running a numerical simulation, and possibly by studying a theoretical model.
Prerequisites: excellent knowledge of linear algebra, PDEs, ODEs, and probability theory; some familiarity with Markov chains or stochastic differential equations is also a plus. Interest in physics, particularly statistical mechanics, is also a plus. Good coding skills necessary in one language (eg Matlab, Python, C++, etc), or a willingness to quickly develop such skills. Possibility for 2 students, to study another aspect of such ligandreceptor particle motion.
Please see https://arxiv.org/abs/2110.03112 for more information about the project.
Where should a filter be placed in a fluid, to remove as much of the chemical it is filtering as possible? This question has numerous applications. For example, there is growing interest in seeding oysters in water basins to remove some of the pollutants (such as nitrogen) in the basins, yet, the locations of the oysters affects how much pollutant is removed. One can formulate this problem as an optimization problem, to maximize the amount of pollutant removed, over all possible filter locations. However, since the amount of pollutant removed depends on the evolution of pollutant concentration over the whole water basin, and since the concentration solves a partial differential equation (PDE), this is a “PDEconstrained optimization problem”.
This project will involve learning about PDEconstrained optimization and techniques to solve them, and then implementing these techniques to solve for the optimal location of filters in a simple model of oysters in a river, developing increasingly more complex models if time permits.
Prerequisites: Excellent knowledge of linear algebra and PDEs. Some familiarity with numerically solving PDEs. Familiarity with fluid dynamics and/or optimization a plus. Good coding skills, or a willingness to quickly learn them.
My research is focused on studying the shape, motility, and behaviour of single mammalian cells, as well as cell groups such as clusters and sheets ("epithelia"). I connect the internal states of proteins or genes in the cell to properties of its adhesion, motion, cell division, etc. My projects build up differential equation models of genetic or protein circuits in cells, and investigate the effect of those states on cellcell interaction in collective cell behaviour. Using mathematics and computational methods, I study aspects of early embryonic development, as well as processes that lead to metastasis in cancer. The USRA project would be part of this effort. I will teach the USRA student how to use the multiscale simulation software Morpheus for this purpose. The student should have some experience with ordinary differential equations. Previous exposure to modelling is a plus, but not essential. The student will be expected to attend weekly research group meetings, weekly USRA seminars, aside from regular meetings with me.
I am writing a graduate textbook on mathematical and computational modeling in cell biology. A project, suitable for an advanced undergraduate is to assist in proofreading, fixing and preparing some figures, finding citations, and providing feedback. The student will learn several new mathematical techniques, including ordinary and partial differential equation models, use of multiscale simulations, and applications to cell biology. The work will include a number of possible research miniprojects to support development of examples for this text.
This project seeks to formulate using first principles, a reactiondiffusion formalism based on current experimental datasets, to study the spatiotemporal dynamics of epigenetic interactions in stem cells. The model formulation will assume that the reactionkinetic parts can be obtained using an integrated approach, based on the principles of reactiondiffusion theory for pattern formation. Here activation and inhibition will be based on biological knowledge that will be quantified using statistical machine learning approaches. Once the model has been formulated, computational methods will be developed to solve such systems (e.g. based on finite differences or finite elements). Finally, using the computational toolkit, model predictions of the dynamical trajectories of the biological epigenetic components and elements, will provide insights into the dynamical characteristics for searching these components and elements within experimental data. The modelling approach itself, could also serve to provide a theoretical and computational formalism for the foundations of our understanding of the biophysical mechanisms for the evident order observed in the developmental gene expression programs.
This project seeks to study the mathematical analysis of reactiondiffusion systems with linear crossdiffusion using linear and weakly nonlinear analysis. The aim is to extend current results established, currently only for the case of linear stability analysis, the conditions for Turing diffusiondriven, to study bifurcation pathways that allow for Hopf and Transcritical bifurcations. Establishing the longtime behaviour of the solutions through numerical bifurcation analytical packages such as PDE2PATH or other alternatives will result in new knowledge for such models. Applications of reactiondiffusion systems are prevalent in nature, from pattern formation to would healing, cancer invasion, chemical reactions and many other areas.
We aim to study presistance on fractals with Sierpinski carpet as the main example. By a result of Bourdon and Kleiner (Combinatorial modulus, the Combinatorial Loewner Property, and Coxeter groups, Groups Geom. Dyn., 7(1):39–107, 2013), we known the existence of presistance exponents. We will investigate if this exponent as a function of p has convexity and monotonicity properties. Knowledge of real variables (at the level of MATH 320) is a prerequisite. Some programming knowledge (of Python) is desired as we will try some numerical experiments.
The AmitsurLevitsky Theorem (proved in 1950) asserts that a certain multiplinear polynomial S_k (X_1, ... , X_k) (called the standard polynomial) in k noncommuting variables vanishes when on any ktuple of n x n matrices whenever k is at least 2n. The DixonPressman conjecture (2018) predicts what happens when we substitute square matrices of larger size into S_k. For even k this conjecture was proved in https://link.springer.com/article/10.1007/s1185602223498
(The preprint version of this paper can be found at https://arxiv.org/abs/2010.04679 .)
For odd k the DixonPressman conjecture remains open. The project will involve learning the background material in linear and abstract algebra (with a bit of algebraic geometry mixed in), then conducting computer experiments with matrices over finite fields in support of the DixonPressman conjecture. For any given n and k the conjecture can be proved in this way, though the computations get harder as n and k increase. The goal is to verify the conjecture in as many cases as possible and to look for patterns which will enable a theoretical proof. Course prerequisites for this project are Math 223, 322 and 323. I will expect familiarity with symmetric groups and finite fields as well as a high degree of facility with computer algebra computations.
Segmenting foreground and background objects in images is wellstudied for color images. Approaches include traditional ways, such as thresholdbased segmentation on green screen, regionbased segmentation methods, and deep learning methods. However, none of these apply well to the monochrome video recordings that neuroscientists obtain in their experiments. Neuroscientists require to use monochrome infrared (IR) cameras as colored light can change animal behavior and invalidate experiments.
In this project, we will attempt to replace the green screen used in the colored light setting with blackwhite textures, such as stripes or checker patterns. These can be printed to appear only in the IR spectrum and hence would be compatible with existing laboratory animal recording principles. A former student created a synthetic training dataset with varying textures and real animal recordings. This project will focus on building a deep learning method to do the segmentation and entails experimenting with different texture types to find an optimal texture pattern that best segments the laboratory animals, in particular the fur and libs of rodents.
Logconcave graphical models have recently been proposed as a general nonparametric version of Gaussian graphical models. However, a fast algorithm for estimation is still missing. In this project we will combine the state of the art algorithm for logconcave density estimation with graphical models in order to produce a fast algorithm. This project will require reading up on graphical models and optimization, and then implementing a new package for estimation in logconcave graphical models.
New measurement technologies like singlecell RNA sequencing are bringing 'big data' to biology. One of the most exciting prospects associated with this new trove of data is the possibility of studying temporal processes, such as differentiation and development. leveraging tools from probability and optimal transport. We aim to develop a mathematical theory to answer questions like How does a stem cell transform into a muscle cell, a skin cell, or a neuron? How can we reprogram a skin cell into a neuron? We model a developing population of cells with a curve in the space of probability distributions on a highdimensional gene expression space. We design algorithms to recover these curves from samples at various timepoints and we collaborate closely with experimentalists to test these ideas on real data.
Summer students in my group will receive rigorous training in optimization, probability, statistics, and mathematical programming. During the rotation period, we will work through this tutorial on applying optimal transport to stem cell reprogramming: https://broadinstitute.github.io/wot/
If time permits, will then explore various extensions on either the theoretical or applied side depending on student interest.
Previously, the existence and stability of standing waves for one dimensional nonlinear Schroedinger equations (NLS) with 234 triple power nonlinearities was studied analytically and numerically by Liu, Zwiers and myself (https://doi.org/10.1016/j.na.2021.112409). This project aims to extend it in two directions: (a) consider general triple power nonlinearities, not just powers 2, 3, and 4; (b) Consider periodic waves for 234 triple power nonlinearity. A focus will be the coexistence of different types of periodic waves for the same nonlinearity with same period and same mass. We hope to understand which one is the constrained energy minimizer, and how to compute the other one numerically. Their stability property will be also considered. The ideal student should have taken MATH 215/255 and is comfortable with MATLAB coding.
Let G be a group, which we presume is infinite. Fix an integer n. We consider all the (complex) representations of G of dimension n: these are the ways in which the abstract group G can act as linear symmetries of the complex vector space C^n. It is reasonable to identify two representations if they differ by change of coordinates in C^n. Such equivalence classes assemble to form a geometric structure: the character variety. We concentrate on the classes of those representations where there is no nontrivial subspace of C^n that is invariant under all the elements of G (the socalled irreducible representations). These constitute the majority of the character variety, and are easier to work with.
Character varieties are used extensively in lowdimensional topology: e.g., they can be used to distinguish between inequivalent knots. Motivated by these applications, we consider the circumstance where the group G is itself symmetrical, potentially reflecting a symmetry in an underlying knot. Any symmetry of G should translate to a symmetry of the associated character variety X. In some cases a nontrivial symmetry of G may result in a trivial symmetry of X, and in these cases we may enrich X with further structure that detects the symmetry of G.
There is an absence of worked examples in this area. This project proposes taking a group G arising from lowdimensional topology and carrying some symmetry, then giving an explicit presentation for the character variety X (or at least the part corresponding to the irreducible representations) and investigating the symmetry it carries.
Excellent knowledge of linear algebra (Math 221 or 223) and good knowledge of group theory (Math 322) and rings and modules (323) are necessities for this project. Galois theory (Math 422) will also be helpful. Knowledge of topology is not required.
Ramsey theory is a branch of extremal combinatorics that considers the following type of question: how many edges can a graph on n vertices have, before it is forced to contain a copy of a given subgraph? This project investigates a question of this type, where the vertices of the graph represent lines in space, and two vertices are adjacent if the corresponding lines intersect. The setting is motivated by recent developments in incidence geometry.
2022
Problems in extremal combinatorics asks how many (discrete) objects are possible subject to some condition. We consider simple matrices which are (0,1)matrices with no repeated columns. We say a simple matrix A has F as a configuration if a submatrix B of A is equal to a row and column permutation of F. Much has been determined about the maximum number of columns an mrowed simple (0,1)matrix can have subject to condition it has no configuration F for some given F. There is an attractive conjecture (A. and Sali 2005) to pursue with many challenging open problems. We might pursue a special 8rowed choices for F that when forbidden in an mrowed simple matrix A should yield a cubic bound (cubic in m) on the number of columns of A. This would generalize a bound of Keevash et al. Moreover it would extend work with Dinh from 2020.
Problems in extremal combinatorics asks how many (discrete) objects are possible subject to some condition. We consider simple matrices which are (0,1)matrices with no repeated columns. We say a simple matrix A has F as a configuration if a submatrix B of A is equal to a row and column permutation of F. Much has been determined about the maximum number of columns an mrowed simple (0,1)matrix can have subject to condition it has no configuration F for some given F. This maximum is denoted forb(m,F). In certain cases the available constructions (avoiding F) suggest an exact bound. With Nikov, a number of exact results were obtain for krowed F that contain K_k, the krowed simple matrix with all 2^k possible columns. This relates to VCdimension. A number of related problems will be explored.
This is an opportunity to do some undergraduate research project in the realm of number theory under the joint supervision of Dr. Schraven and Prof. Bachmann. The goal is for you to experience research in mathematics
In mathematics we have several ways of quantify whether something is ”big” or ”small”. In topology that might be that a set is dense or in real analysis that a set has small Lebesgue measure (for an interval [a,b] the Lebesgue measure gives you ba, i.e. the length of the interval. This can be done for much wilder sets). In our setting we have a subset A of Z^d and we want to understand what proportion of Z^d is in the set A. For example for A being the even integers, we intuitively understand that half of the integers are in A. This proportion (if one can make sense of it) is called the natural density of A.
In general it is quite hard to compute the natural density of a given set. However, under some conditions this can be done using padic numbers. Intuitively, we get the padic numbers if we tweak the metric on the real numbers a bit (padic numbers are fun and we will learn about them in the beginning of the project). If you are familiar with probability theory, you can also think of the natural density as some kind of probability measure on the lattice Zd. Naturally we can ask whether we can push this analogy and define a notion of expected value. This is indeed possible and using similar ideas as in the density case one can get some effective tool to compute this expected value again in terms of padic numbers. In this talk https://www.youtube.com/watch?v= dtE2Zv7bryE&t=1041s you can hear me talk about this.
As some of my colleagues would say ”be wise, generalize”. If we can do expected values for Zd, can we do the same thing for more general rings? Certain things are already known. For example we can do the expected value over the ring of integers of number fields (see for example [3]) and the density argument carries over to function fields (see [1]). The goal of this project is to investigate the notion of expected value for function fields.
A quiver is a graph whose edges are directed (they are arrows). A quiver representation is linear algebra data associated to a quiver: a vector space for each vertex and a linear map for each arrow. You will study moduli spaces which parameterize quiver representations up to isomorphism. Such moduli spaces, while built entirely from combinatorial and linear algebra data, often can be used to describe moduli spaces of geometric objects. In this project, you will study this idea in various cases. In particular, you will explore how to use quiver representations to construct certain geometric moduli spaces, namely spaces parameterizing certain holomorphic bundles on the complex projective line. We will see that these ideas provide an algebrogeometric approach to a famous theorem in algebraic topology: Bott periodicity. Prerequisites for this project include a strong knowledge of linear algebra and group theory, and preferably some knowledge in a geometric subject (e.g. algebraic geometry, topology, or differential geometry).
Many different mechanisms have been proposed to describe the development of type II diabetes. This project would assess these proposed mechanisms by modifying an established differential equation model to account for some of these mechanisms and comparing results against qualitative expectations for diabetic progression. The student will learn some modelling techniques through modifying and interpreting the Topp model for glucose regulation, how to compute steady states of systems of ordinary differential equations, and how to simulate the time dynamics of these systems. This student will need to be familiar with differential equations and ideally have some coding experience.
Some references:
 Phatak, S.R. et al., 2021. Sweetening sixteen: Beyond the ominous octet. Journal of Diabetology, 12(1), p.1.
 Topp, B. et al., 2000. A model of βcell mass, insulin, and glucose kinetics: pathways to diabetes. Journal of theoretical biology, 206(4), pp.605619.
Mycobacteria represent pathogens exhibiting exceptional tolerance to stresses such as host immunity and successive antibiotic treatments. To investigate the molecular mechanisms driving their phenotypic heterogeneity and enhanced tolerance, longterm timelapse atomic force microscopy (LTTLAFM) is a powerful technique that allows to precisely and dynamically image and probe mycobacteria at the necessary spatial or temporal resolution. In particular, LTTLAFFM offers a multiparametric view of bacterial surfaces and bulk cell features at the nanoscale, opening new avenues for devising selections and screens to identify molecular determinants controlling various cellular processes (e.g. growth, development, death). However, the analysis of and combined interpretation of the multiple unique datasets harvested by LTTLAFM remains limited by the lack of automation and quantitative tools. The goal of this project is to develop methods for systemization of data collection and processing, which includes cell segmentation, medial axis and height profile extraction, and tracking of division events and location over time series trajectories, using a variety of algorithms and libraries available in Python. In addition, the student will also specifically focus on the properties of nanovesicules at the cell surface, evidenced by experimental data obtained by Dr. Eskandatian. Upon characterizing their shapes, distributions, and dynamics as a Poisson point process, the student will study how vesiculogenesis is deregulated under various stress conditions, and represent a potential hallmark of stress and death in mycobacteria.
The oil sands industry is both a significant contributor to the Canadian economy and is widely regarded as a cause of adverse environmental effects, e.g. it has been estimated to account for 10% of GHG emission in Canada. Recent studies show that anaerobic microorganisms contribute to the degradation of Naphtha hydrocarbons and naphthenic acids in the FFT and MFT layers of oil sands tailings ponds, producing methane and CO2, both potential causes of GHG emissions. The FFT and MFT layers are colloidal suspensions, which behave like viscoplastic fluids with timedependent rheology: changing both with age and depth in the pond. The key feature of a viscoplastic fluid is its yield stress: the material flows only if the imposed stress exceeds the yield stress. This raises questions regarding the stability of bubbles, which are trapped in a yield stress fluid.
Previously, we performed a series of experiments with a model yield stress fluid, Carbopol gel, to study the onset of motion of bubbles. We would like to continue this project with some other laboratory yield stress fluids, i.e. Laponite and Kaolinite suspension, which are considered as better models for tailings material. Besides, we'd like to extend our study to the bubbles migration in a yield stress fluid including networks of angled 'damaged' layers within which the yield stress is destroyed. This might model the effect of nonuniform rheology of the tailings material or/and the presence of water chimneys on the stability and migration of bubbles in the tailing ponds.
We are seeking a student to work on two subprojects: (i) the growth and stability of bubbles in several different viscoplastic fluids; (ii) bubbles migration towards and along damaged networks in a yield stress fluid. This might lead to fundamental understanding of how the rheology of the fluid, shear history of the fluid, and interaction between the stress field around the bubbles affect the onset of motion and bubbles propagation.
The students will assist in all operations related with the experiment: fluid preparation, characterizing the rheology of the fluids, performing experiments and image processing. The students will learn the physical background to the experiments and may help in design of new components and implement changes to the current apparatus.
The student will work under the daytoday supervision of Dr. Masoud Daneshi in complex fluids lab, LSK, UBC.
Ideally the candidate would have lab experience e.g. in an engineering discipline or physics. We prefer those who has experience in instrumentation, Labview programming and imaging and are familiar with Matlab and SolidWorks.
I am writing a graduate textbook on mathematical and computational modeling in cell biology. I will have a project, suitable for an advanced undergraduate, that would assist in proofreading, fixing and preparing some figures, finding citations, and providing feedback on clarity of the material. The student will learn several new mathematical techniques, including ordinary and partial differential equation models, use of multiscale simulations, and applications to cell biology. This USRA position would be an excellent preparation for the graduate course Math 563, for which this text is being written. The work will include a number of possible research miniprojects to support development of examples for this text.
Organoids are small tissues grown artificially in experiments designed to study how cells of various types interact, and how they respond to drugs. (For example, brain organoids may have several kinds of neurons and support cells.) My projects build up differential equation models of genetic circuits in cells, and investigate the effect of those genetic states on cellcell interaction in organoids. I will teach the USRA student how to use the multi scale simulation software Morpheus for this purpose. The student should have some experience with ordinary differential equations. Previous exposure to modelling is a plus, but not essential. The student will be expected to attend research group meetings weekly, aside from regular meetings with me.
Stem cells are cells that have no specialization, and that can consequently produce any of the final specialized cell types needed in the body (nerves, brain, kidney, etc.). To do so, stem cells divide to produce daughter cells that specialize (“differentiate”) as well as more stem cells (“self renewal”). However, the number of times stem cells can divide and selfrenew over a lifetime is not known. This project focuses on stem cells that form our blood system. The adult human body has between 50,000 to 200,000 such stem cells$^{2}$. It has been a popular belief that what limits the number of stemcell divisions (the Hayflick limit) is gradual wearingaway of pieces of genetic material (“telomeres”) that take place at each cell division. It is estimated that roughly 50100 DNA base pairs (at the end of a telomere) are lost at each cell division$^{3}$, leading to ageing and eventually ending the selfrenewal of the stem cells.
This USRA project will combine mathematical modeling supervised by Prof Leah Keshet, with new biological data and insights provided by Prof Peter Lansdorp. What is the role of math? For example, Erwin Schrödinger predicted in 1944 “We infer by an easy computation that on average as few as 5060 successive divisions suffice to produce the number of cells in a grown man (very roughly a hundred or a thousand x $10^{12}$ or say, ten times that number, taking into account the exchange of cells during a lifetime. Thus, a body cell of mine is, on average, only the 50th or 60th ‘descendant’ of the egg that I was”$^{1}$. More sophisticated math (differential equations) has been applied to predict the dynamics of stem cells, based on loss of telomeric DNA affecting stem cell turnover$^{5,6}$. However, there is now new data on the decline in telomere length in nucleated blood cells over a lifetime$^{4}$ as well as a hierarchical model of progressively more numerous and more rapidly dividing cells$^{7}$. The USRA student will help to advance this interdisciplinary area, combining modeling, simulations, and biological data.
 Schrödinger E. What is life? : the physical aspect of the living cell ; with, Mind and matter ; & Autobiographical sketches. Cambridge ; New York: Cambridge University Press; 1992.
 LeeSix H, Obro NF, Shepherd MS, et al. Population dynamics of normal human blood inferred from somatic mutations. Nature 2018;561:4738.
 Vaziri H, Dragowska W, Allsopp RC, Thomas TE, Harley CB, Lansdorp PM. Evidence for a mitotic clock in human hematopoietic stem cells: loss of telomeric DNA with age. Proc Natl Acad Sci U S A 1994;91:985760.
 Lansdorp P. Telomeres, aging and cancer, the big picture. Blood 2021; in press
 EdelsteinKeshet L, Israel A, Lansdorp P. Modelling perspectives on aging: can mathematics help us stay young? J Theor Biol 2001;213:50925.
 Werner B, Beier F, Hummel S, et al. Reconstructing the in vivo dynamics of hematopoietic stem cells from telomere length distributions. Elife 2015;4.
 Derenyi I, Szollosi GJ. Hierarchical tissue organization as a general mechanism to limit the accumulation of somatic mutations. Nat Commun 2017;8:14545.
The road colouring theorem is a famous result in graph theory regarding synchronization of colourings of finite directed graphs G with constant outdegree. There is a generalization of this theorem, known as the O(G) conjecture. This has been established for various classes of special cases but is unknown in general. In this project, the student would review the relevant literature, write computer programs to generate important structures corresponding to various graphs G, and explore possible routes to a solution of the O(G) conjecture. The student must have programming experience and have completed a course in linear algebra, such as Math 221.
CalabiYau manifolds appear in many different branches of mathematics, including algebraic geometry, differential geometry, nonlinear PDE, and theoretical physics. It is predicted in string theory that these objects come in mirror pairs. Historically, the first example of this phenomenon was discovered by physicists Candelas, de la Ossa, Greene and Parkes. This project will explore some of the calculations from the groundbreaking paper of COGP and fit them into a broader modern mathematical framework.
Structural equation models are a way to mathematically represent causal relationships among random variables. A linear structural equation model consists of a directed graph, one random variable at each of the vertices of the graph, and linear equations that express each random variable as a noisy linear combination of its parents. Such a model can be learned by considering the second and third order moments of the set of random variables. In this project, we will study the polynomial relationships that hold among these moments, in order to help design better algorithms for causal inference.
Tensor eigenvectors behave quite differently from matrix eigenvectors. In this project we will work on computing all of the eigenvectors for several families of tensors.
New measurement technologies like singlecell RNA sequencing are bringing ‘big data’ to biology. My group develops mathematical tools for analyzing timecourses of highdimensional gene expression data, leveraging tools from probability and optimal transport. We aim to develop a mathematical theory to answer questions like: How does a stem cell transform into a muscle cell, a skin cell, or a neuron? How can we reprogram a skin cell into a neuron?
We model a developing population of cells with a curve in the space of probability distributions on a highdimensional gene expression space. We design algorithms to recover these curves from samples at various time points and we collaborate closely with experimentalists to test these ideas on real data. We have recently applied these ideas to shed light on cellular reprogramming (to learn more, see here: https://broadinstitute.github.io/wot )
We are looking for students with some background in optimization, probability, and mathematical programming. We are looking to hire several students to work on separate aspects of this largescale project.
The time dependent solution of a large class of FokkerPlanck equations for the distribution functions of electrons and/or reactive species can be obtained numerically with an efficient pseudospectral method defined with nonclassical basis polynomials. The solutions are expressed in terms of the eigenfunctions and eigenvalues of the linear FokkerPlanck equation. The FokkerPlanck eigenvalue problem is isospectral with the Schroedinger equation so that the pseudospactral methods developed can be applied to both eigenvalue problems. There are large number of quantum problems of current interest.
There are several well defined projects for 2 undergraduate summer students. A publication on the FokkerPlanck equation with a previous undergraduate student can be found at the following link; https://journals.aps.org/pre/abstract/10.1103/PhysRevE.102.062103
A publication on the Schroedinger equation with a previous undergraduate student can be found at the following link; https://www.sciencedirect.com/science/article/pii/S2210271X20303595
The pseudospectral methods have been described in the book "Spectral Methods in Chemistry and Physics", Springer 2015  http://www.springer.com/gp/book/9789401794534
It is anticipated that the summer research will result in a publication for each student.
Fix a field F of characteristic zero, and let the symmetric group $S_n$ act on the polynomial ring $F[x_1,\ldots,x_n]$ by permuting the variables. It is wellknown that the ring of symmetric polynomials over F (the subring of $S_n$invariants) is itself a polynoimal ring, either in the \emph{elementary symmetric functions} $s_k = \sum_{A\in\binom{[n]}{k}} \prod_{i\in A} A_i$ ($1\leq k\leq n$), or alternatively in the \emph{power sum polynoimals} $t_k = \sum_{i=1}^{n} x_i^k$ (same range of $k$).
A similar result holds when we replace the polynomial ring with the set of Laurent polynomials $F[\{x_i\}^\pm\}]$ (for technical reasons this is not a ring, though that can be fixed). The Laurent polynomials admit an action by rescaling the variables, and in particular have a "subring" consisting of the elmeents of degree zero (e.g. the symmetric polynomial $\sum_{i\neq j} x_i/x_j$). We will investigate generating sets for Laurent polynomials of homogenous degree zero and for higherdimensional generalizations.
We will investigate the padic character of recent physics papers on "padic string theory", specifically whether harmonic analysis on trees rather than harmonic analysis on padic groups is sufficient for the computations.
This project is concerned with twocomponent competing system on lattice having both local (perimeter) and nonlocal interactions ($H^{s}$). First one derives the most general form of $\Gamma$convergence energy functional which involves fractional $s$perimeter and Riesz potential interaction $\frac{1}{xy^{n2t}}$ where $s$ and $t$ are both in the range of $(0,1)$. Then we want to find (numerically and variationally) the most stable lattice shape for this system on a twodimensional or threedimensional lattice. Basic knowledge of multivariable calculus, functional analysis, PDE and computer programming are needed.
2021
Arrays of boxes filled with positive integers subject to certain rules are called Young tableaux. They generate functions known as Schur functions, which date from Cauchy in 1815, and arise today in many areas including combinatorics, algebra, quantum physics and algebraic geometry. In this project we will investigate a natural generalization of skew Schur functions with the aim of discovering their algebraic and combinatorial properties. We may also investigate connections to other areas. A strong background (A+) in abstract algebra such as Math 322 or Math 323 is required, and an aptitude for combinatorics and programming skills will be an asset.
A problem in extremal combinatorics asks how many (discrete) objects are possible subject to some condition. We consider simple matrices which are (0,1)matrices with no repeated columns. We say a simple matrix A has F as a configuration if a submatrix B of A is equal to a row and column permutation of F. Much has been determined about the maximum number of columns an mrowed simple (0,1)matrix can have subject to condition it has no configuration F for some given F. There is an attractive conjecture (A. and Sali 2005) to pursue with some challenging open problems. We might pursue some 4rowed choices for F that when forbidden in an mrowed simple matrix A should yield a quadratic bound (quadratic in m) on the number of columns of A. The proofs will no doubt use a multiset approach (A. and Lu). There are many variations of this problem including considering specially structured families of forbidden configurations. We will have plenty of problems to work on. An interested student could contact undergrads Zachary Pellegrin or Kim Dinh who worked with me summers 2019 and 2020 respectively.
In this project, the student will implement, analyze, and validate a computational model for the development of dentition in reptiles. The model will be a quantification of a hypthesis first proposed in the mid 70s by J.W Osborn. We will collaborate with the Richman Lab in the Faculty of Dentistry at UBC who have been quantifying the patterning of tooth eruption across the jaw in leopard geckos. The student will carry out simulations of both deterministic and stochastic versions of the model and use nonlinear regression and maximum likelihood estimation to fit parameters using the Richman Lab’s data.
The group of Khanh Dao Duc at the Math department of the University of British Columbia (Vancouver, Canada) is seeking to recruit a USRA student, to participate in the development of mathematical and computational methods for investigating biological structures from cryogenicelectron microscopy (cryoEM) data. For the following project, Dr. David Dynerman (Chan Zuckerberg Biohub, San Francisco, USA) will be cosupervisor. The group is also working in close collaboration with other research groups and scientists, including Professor Nina Miolane (ECE Department, UC Santa Barbara) and Dr. Frédéric Poitevin (LCLS, Stanford Linear Accelerator Center) so this position will also give the opportunity to learn more about the strong coupling between instrumentation in electron microscopy, advanced mathematical concepts and machine learning algorithms.
As the level of detail in 3D models produced by cryogenicelectron microscope (cryoEM) continues to increase, one barrier faced by all commonly used software is assuming that experimental images are linear projections of the protein’s 3D density. This linearity assumption starts to break down at high resolution due to an electron optical effect called Ewald Sphere Curvature. We recently developed some basic theory on how to account for this curvature and directly recover 3D structures from 2D cryoEM images without relying on the standard linear assumption. Humorously, our approach to correct this nonlinear optical effect involves constructing and solving a large, but tractable, linear system. Our goal is to now run numerical simulations to prove that the method improves 3D model resolution. This project would involve inventing an efficient algorithm to construct this linear system as well as applying standard numerical analysis techniques to solve it. Once this prototype implementation is complete, we plan to analyze its robustness against noise and other perturbations as a next step to bringing these ideas into common use in the cryoEM field.
Biologists have discovered remarkable patterns of cell migration in many essential biological processes, including development of embryos and wound healing. For example, the socalled neural crest cells migrate in streams along the spine of the embryos of chicks, frogs and zebrafish. Recently, our group built a model to explain cell polarization and migration in terms of the chemical signaling inside the cells. This has been used to study the cellcell coordination in collective migration of neural crest cells, with interesting results. However, our simulations so far have been in two dimensions only, on a planar surface. The USRA project for summer 2021 extends the existing model to 3D to simulate how a cell migrates on a textured substrate, and on a 3D scaffold of collagen fibers. The project will require programming in Python and Rust, with training and help from a graduate student.
Minimization problems arise naturally in many branches of mathematics and science. For example, problems in navigation involve finding paths of least length (geodesics) on the earth's surface. Minimal surfaces, which are twodimensional analogs of geodesics, are minimizers (or simply critical points) of the area function, and arise naturally in material science; for example in fluid interface problems and elasticity problems. A simple physical example of a minimal surface is the soap film that forms after dipping a wire frame into a soap solution. By the laws of surface tension this soap film has the property that it is stable, that is it becomes larger under slight deformations. The theory of minimal surfaces has had striking applications, for example to general relativity and low dimensional topology. This project will study connections between an eigenvalue problem on surfaces with boundary and minimal surfaces in the unit ball.
Canada has the thirdlargest crude oil reserve in the world behind Venezuela and Saudi Arabia. Canadian oil reserve is believed to include approximately 170 billion barrels of oil or 11% of total global oil reserves. The oil sands industry is both a significant contributor to the Canadian economy and is widely regarded as a cause of adverse environmental effects, e.g. it has been estimated to account for 10% of GHG emission in Canada.
Recent studies show that anaerobic microorganisms contribute to the degradation of Naphtha hydrocarbons and naphthenic acids in the FFT and MFT layers of oil sands tailings ponds, producing methane and CO2, both potential causes of GHG emissions. The FFT and MFT layers are colloidal suspensions, which behave like viscoplastic fluids with timedependent rheology: changing both with age and depth in the pond. The key feature of a viscoplastic fluid is its yield stress: the material flows only if the imposed stress exceeds the yield stress. This raises questions regarding the stability of bubbles, which are trapped in a yield stress fluid, that we try to answer in this project through a series of targeted experiments.
We are seeking 2 students to work on two subprojects: (i) the growth and stability of bubble clouds in a viscoplastic fluid; (ii) the growth and stability of a single bubble or multiple bubbles introduced in a perfectly degassed fluid. This might lead to fundamental understanding of how the rheology of the fluid and interaction between the stress field around the bubbles affect the onset of motion. The students will assist in all operations related with the experiment: fluid preparation, rheometry measurements of the fluids, running experiments and image processing. The students will learn the physical background to the experiments and may help in design of new components and implement changes to the current apparatus.
To model the shape, motility, and behaviour of biological cells, a variety of computational platforms are commonly used. This project will be based on the cellular Potts model approach, whose origin is based on the physics of foams and soap bubbles. The student will be taught and then expected to use the opensource software (Morpheus) to simulate single and multiple interacting cells. The research will consist of the following (1) Deriving conditions for simple cell behaviour based on the underlying mathematics of the CPM. (Briefly: to use calculus to find minima of an energy function called the Hamiltonian for simple geometries.) (2) To test the analytic predictions using simulations and (3) To assist in developing simulations for interactions of (real) cells in several biological casestudies.
Many systems that evolve in time, for instance the firing of neurons, appear to be governed by a combination of dynamical (deterministic) and stochastic (random) causes. This project will involve the study of neuron models (see ref. [1]), for example the MorrisLecar neuron model, using mathematical analysis together with numerical simulation (with Matlab, or Python, etc.). The models we consider will include ODEs, SDEs (stochastic differential equations: ODEs influenced by random noise), and possibly PDEs and SPDEs, depending on background and interest.
References:
[1]. P.E. Greenwood, L.M. Ward, Stochastic Neuron Models, Springer 2016 (free online)
Inference problems involving 3dimensional point data (i.e., data where each observation is comprised of a collection of points in 3deuclidian space) are abundant in data science with many application both in science and engineering. For example, the learning of force models from atomic configurations and the classification of 3dimensional objects based on their geometric shapes can be formulated as a regression and classification task on 3D point data, respectively.
In order to perform inference on such 3D point data efficiently certain symmetries induced by the Euclidian symmetry group and the type of predicted quantity should be directly built into the respective inference model. For example, an inference model used for the prediction of interatomic forces should be invariant under translation of the input configurations and covariant with respect to rotation of input configurations.
A wealth of techniques has been developed in recent years within the applied mathematics and material science community for the learning of force fields. The aim of this project is to use and extend some of these techniques in the form of new algorithmic implementations for use in a broader scope of applications (e.g., the above mentioned classification of geometrical shape data.)
The student should be interested both in theoretical aspects of the project as well as computational aspects pertaining to the implementation of the developed algorithmic approaches. Implementation of algorithms will likely require coding in PyTorch (a python API for deep learning) and/or Julia.
Geometry optimisation in molecular modelling typically involves the computation of stable equilibria, i.e. minima of a potential energy surface, or more generally transition states (saddle points) or transition paths. Minimizing computational cost is of course a general goal of numerical optimisation but the extremely high computational cost of accurate molecular potential energy models makes this even more critical. This project would initially focus on a mutually agreed upon mathematical aspect of geometry optimisation, for example the design of accelerated dynamical systems to compute saddle points, or the development of a molecular multigrid algorithm. In a next step, the mathematical theory would be implemented in a prototype code. Finally, in collaboration with a modelling group, the new algorithms could be integrated into a general purpose molecular modelling software and tested on realworld modelling scenarios. The project would likely involve aspects of numerical optimisation, linear algebra, and ODEs. For further details and references see http://www.math.ubc.ca/~ortner/research/optimisation/
In compressed sensing and it's generalizations, one captures linear measurements of a signal (i.e., an ntuple of real numbers) which has some known structure, i.e., the signal is known to belong to a particular subset of ntuples of real numbers. The ability to recover the signal from these measurements depends on the invertibility of the linear measurements restricted to this subset, moreover stable recovery is implied if the linear measurements act as a near isometry restricted to this subset. In a recent paper, we gave a way of determining a number of random linear measurements sufficient to achieve this near isometry depending upon the signal structure and the tail bounds of the random measurements. In this USRA, we wish to show that the results of that paper are essentially unimprovable. This project involves an indepth understanding of subGaussian random variables and highdimensional geometry; ideally it results in a publication. We believe that the load is most appropriate for two USRA students.
The Nullstellensatz is a foundational theorem in algebraic geometry proved by David Hilbert around 1890. The classical version (often covered in Math 423) is for polynomials with coefficients in an algebraically closed field. There are also variants for polynomials with real coefficients and more generally, with coefficients in a pclosed field. This project is inspired by the recent preprint
https://arxiv.org/abs/1911.10595
which proves a version of the Nullstellensatz for polynomials with quaternion coefficients. Here are the specific activities I have in mind:
1. learning existing versions of the Nullstellensatz for polynomials with commutative coefficients,
2. learning background material on pclosed fields and central simple algebras,
3. trying to generalize the above preprint.
Prerequisites: Strong background and interest in abstract algebra, preferably Math 423.
Tensor networks are diagrams (or graphs) that allow one to describe a complicated decomposition of a given tensor (multidimensional array). They are used in quantum physics since they represent good approximations of steady states of different quantum systems.
Since the edges in tensor networks generally signify physical proximity between quantum particles, it makes sense to consider tensor networks given by hypergraphs in dimensions higher than 1. In this project we are going to try constructing a meaningful version of the tensor network called MERA (Multiscaled Entanglement Renormalization Ansatz) in dimensions 2 and up, which uses hypergraphs and satisfies all the properties that 1 dimensional MERA does.
With the emergence of big data, information more and more often comes in the shape of a multidimensional array (or tensor). The importance of finding a decomposition of such an object is at least twofold. First, it often uncovers additional information about the incoming data. For example, in the famous Netflix prize problem, finding the decomposition allows us to find different user types, and predict movie preferences. Second, and sometimes more important, the decomposition allows for efficient storage of the tensor.
A nonnegative tensor decomposition is a decomposition in which each of the factors is nonnegative. Such decompositions appear in many applications, including in hidden variable models, such as the model in the Netflix problem. Despite its usefulness, nonnegative tensor decomposition is an NPhard problem.
In this project we are going to study the space of tensors of given nonnegative rank. It has been shown that tensors of nonnegative rank at most 2 correspond to probability distributions that are totally positive, a type of distribution that signifies strong dependence between random variables. A similar result has been shown for 2 X 2 X 2 tensors of nonnengative rank at most 3. Here, we will study larger tensors of nonnegative rank at most 3, as well as higher nonnegative ranks.
In this project we are going to study decompositions of tensors according to an orthogonal tensor network.
We will design algorithms for finding such decompositions, and study the properties of tensors that can decompose according to an orthogonal tensor network..
New measurement technologies like singlecell RNA sequencing are bringing ‘big data’ to biology. My group develops mathematical tools for analyzing timecourses of highdimensional gene expression data, leveraging tools from probability and optimal transport. We aim to develop a mathematical theory to answer questions like: How does a stem cell transform into a muscle cell, a skin cell, or a neuron? How can we reprogram a skin cell into a neuron?
We model a developing population of cells with a curve in the space of probability distributions on a highdimensional gene expression space. We design algorithms to recover these curves from samples at various time points and we collaborate closely with experimentalists to test these ideas on real data. We have recently applied these ideas to shed light on cellular reprogramming (to learn more, see here: https://broadinstitute.github.io/wot )
We are looking for students with some background in optimization, probability, and mathematical programming. We are looking to hire several students to work on separate aspects of this largescale project.
In biology, quorum sensing is an intercellular form of communication that bacteria use to coordinate group behavior such as biofilm production and the production of antibiotics and virulence factors. It is also responsibe for the mechanism underlying the onset of luminescence production in cultures of the marine bacterium Vibrio fischeri. Luminescence and, more generally, quorum sensing are important for V. fischeri to form a mutualistic symbiosis with a small Hawaiian squid, Euprymna scolopes. During the day the bacterial cells grow to a high density in the light organ of the squid, which then triggers by sunset a luminescence behavior that camouflages the squid at night from its predators by eliminating its shadow in thin pools of ocean water.
In this USRA project, the candidate will work together with members of my research group to analyze and perform computer simulations of various nonlinear ODE systems characterizing quorum sensing behavior in 3D domains. These model ODE systems have been recently derived from an asymptotic reduction of more elaborate PDE/ODE systems of quorum sensing. The ideal candidate will have strong calculus skills, a good background in ODEs, some exposure to PDEs, strong computational skills in a scientific programming language such as MATLAB, as well as a willingness to learn some of the biological issues from reading some journal articles with members of my research group. Mathematically, the student will learn techniques in nonlinear ODE dynamics, some bifurcation theory, and be exposed to an interesting interdisciplinary application.
For two recent references from my group on this general topic see:
https://www.math.ubc.ca/~ward/papers/lux_new.pdf
https://www.math.ubc.ca/~ward/papers/qpde_revise.pdf
An elliptic curve over the rationals can be expressed via an equation in variables x and y that is cubic in x and quadratic in y. They play an important role in many areas of number theory and have been the subject of research for over a century. To each elliptic curve is associated a primeindexed complex sequence that has interesting statistical properties, and a supersingular prime for an elliptic curve is then a prime at which the corresponding value in the sequence vanishes.
We can ask  how often do such primes arise? Their occurrence for a large family of elliptic curves has been conjectured by Lang and Trotter. We will investigate variations of this question further through computation and the study of heuristic models.
This project involves studying the background of elliptic curves, understanding the history of the problem, and working with some analytical machinery as well as the program SAGE (no prior experience needed) to gain further insight into the distributions of supersingular primes. A student would benefit from a background in algebra and complex analysis.
We want to investigate reactiondiffusion systems with nonlocal diffusions. Nonlocal diffusions are ubiquitous in nature. In this project we will study several prototype fractional GiererMeinhardt, Schnakenberg, GrayScott, Brusselator systems. One difficulty is that these are quite nonlocal problems and even the definition requires singular integrals. We will study the effects of different fractional indices, different boundary conditions, combinations of nonlinearities on the existence and stability of localized patterns.
2020
The phenomenon of Anderson localization for random Schrödinger operators is well established in low dimensions or strong disorder. One of its facets is the fact that eigenvectors for eigenvalues at the bottom of the spectrum are well localized in space. However, the determination of the localization centers of these eigenvectors is a difficult problem. The use of an effective potential has recently been advertised and shown to be numerically very effective. This project will explore analytical aspects of the relation between localization and the effective potential with an eye towards a new proof of localization via the effective potential.
The classical gambler's ruin problem can be thought of as a random walk stopped upon exiting an interval. Since simple random walk on Euclidean lattice approximates Brownian motion on the Euclidean space, one might naturally expect that simple random walk killed upon exiting a set approximates Brownian motion killed upon exiting a continuous domain. In this project, we consider simple random walks killed (or stopped) upon exiting a domain. The goal of this project is to obtain quantitative estimates that compare transition probabilities of random walk and Brownian motions killed upon exiting (a large class of) domains. The student will learn tools from analysis (Whitney decomposition, singular integrals) and probability (Doob transform, heat kernel bounds).
The translation of mRNA into protein is a fundamental, yet complex biological process, mediated by ribosomes. To explain what can affect its efficiency, it is crucial to unravel the interactions between the ribosomes and other molecular complexes, but also to take into account other factors at a larger scale. The main goal of this proposal is to draw a global picture of the role played by ribosomes in translational systems across different scales, encompassing molecular, cellular and evolutionary aspects. More specifically, we shall investigate 1) the biophysical properties and evolution of the ribosome exit tunnel, 2) different modes of translation, transport and remodeling of the ribosome in response to specific spatial cellular organizations and 3) the limiting factors that drive protein translation at the system level. These completing approaches will both elucidate the functional impact of the ribosome structure, and conversely, determine how spatial or resources management impose evolutionary and design constraints at the molecular level.
We will analyze the biophysical properties of the tunnel from cryo EM data, and elucidate the interplay between the electrostatics and geometry of the exit tunnel. These biophysical properties will also be studied in the context of evolution, with cryo EM structures of the ribosomes available for many species. Studying the evolution of the ribosome and its tunnel will require inferring ancestral shapes that can potentially explain different modes of translation.
At the mesoscopic scale, we will focus on two important modes of translation: First, the translation of membrane protein genes, which involves transport to the endoplasmic reticulum (ER). Interestingly, this gives rise to geometric patterns of polyribosomes on the ER membrane. To explain these patterns, we will study a new biophysical model of translation, and compare the patterns with imaging data. The second local mode of translation to investigate occurs in dendritic regions, located far from the cell nucleus. By combining theoretical analysis of modes of transport, with differential expression data, we will build statistical tools to distinguish, for different genes, their mode of transport, and infer the associated local translation dynamics. At the system scale, we are interested in the metabolic cost of translation. In vitro timeseries measurements of protein levels plateau, with multiple potential limiting factors. Upon fitting a mathematical model to a various set of experimental conditions, we will disentangle the contribution of these factors. This modeling approach will serve as a first step for high throughput measurement of translation rates. In vivo, living systems also need to manage the ribosomal population. In particular, "ribophagy" pathway can not only decrease the ribosomal pool, but also allows the recycling of ribosomal components. To assess the robustness and optimality of this pathway, we will use optimal control.
The study of homological invariants of knots has benefited from the existence of knot tables (see). This gives a collection from which to draw examples, compute, and gain intuition about the invariants in question. This has been the source of new conjectures and, ultimately, structure related to these novel invariants. Knots are sometimes studied by decomposing them into simpler pieces called Conway tangles. In the case of knot Fleor homology and Khovanov homology, we now have a good idea of the form these invariants take when evaluated on a Conway tangle. However, Conway tangles are more difficult to tabulate than knots, and as a result there is not a table of Conway tangles that researchers can work with. This project will take a step in this direction: we’ll restrict to Conway tangles of a particular form, and systematically build a table of these special examples. With this in hand, appealing to existing software, we will compute the aforementioned invariants. This project will require some algebraic background and an interest in topology. Familiarity with Python and or C++ will be very useful.
A problem in extremal combinatorics asks how many (discrete) objects are possible subject to some condition. We consider simple matrices which are (0,1)matrices with no repeated columns. We say a simple matrix A has F as a configuration if a submatrix B of A is equal to a row and column permutation of F. Much has been determined about the maximum number of columns an mrowed simple (0,1)matrix can have subject to condition it has no configuration F for some given F configuration. There is an attractive conjecture (A. and Sali) to pursue with some challenging open problems. The bounds we obtain are sometimes exact, sometimes asymptotic (Theta(f(m))) and sometimes the coefficient of the leading term in the bound is exact. A variation is to consider a specially structured family of forbidden configurations. Under the supervision of Richard Anstee, we will have plenty of problems to work on.
Experimental biologists have acquired lots of data on the distribution and mobility of B cell receptors (BCR) on the surface of B cells of the immune system. The goal of this project will be to build and parameterize a suite of stochastic mathematical models that describe the mobility and organization of BCR. We will design the models based on reasonable biophysical assumptions and with careful reference to existing models in the literature. The models will be implemented as stochastic computer simulations in Matlab or Python. The parameters will then be fit to the available data using Approximate Bayesian Computation approaches (ABC). The parameterized models from this project will lead to improved quantitative understanding of B cell signaling, which may ultimately impact our understanding of a range of diseases. Students working on this project with Dan Coombs will learn about B cell biology, cell biophysics, image analysis, modeling and scientific computation. There may be an opportunity to assist with handson experimental work as well.
B cell receptors (BCR) are mobile on the surface of B cells. In this primarily experimental project, a student will use twocolour fluorescence microscopy to study the motion of individual receptors either in the presence of a bulklabeled background of receptors or other surface molecules, or in relation to the motion of other nearby receptors. In the first case, the goal will be to study correlations between the mobility parameters of the focal receptor and the density of labelled background. In the second case, correlations will be sought between the mobility parameters of nearby receptors. The experiments will require culture and labeling of cells, imaging, image analysis and then fitting to mathematical models of receptor motion. Students working on this project with Dan Coombs will learn about B cell biology, microscopic imaging and image analysis, modeling and scientific computation.
The development of multicellular organisms is coordinated by both mechanical and biochemical regulation. A recent model of tissue development that coupled mechanical and biochemical regulation demonstrated a surprisingly large variety of dynamical behaviours. Preliminary analysis indicates that the model has a rarely seen codimension two bifurcation that is still in need of elucidation. This project will require a student to carry out numerical bifurcation analysis of the model using both existing bifurcation software (like XPPAUT) and some selfwritten code (in matlab / python). Some background and an interest in learning more about ODEs (MATH 215), bifurcation theory (MATH 345), and numerical methods would be useful.
By 2013, more than 550,000 oil and gas wells had been drilled across the country with the majority located in the Western Canada Sedimentary Basin with 5,00020,000 new wells drilled each year. After a certain operating period, all wells will need to be plugged and abandoned (P&A). As of December 2016, there were 24,802 wells in British Columbia. Of these, 11% were abandoned and 27.4% were inactive.
In Alberta, 151,000 wells have been abandoned which represents 35% of all wells in the province. The consequences of poor P&A are wellbore leakage which can contaminate groundwater, nearsurface ecology and damage to marine ecosystems offshore. Wellbore leakage also contributes to greenhouse gas emissions, may present explosive and/or health risks. P&A is a potential massive societal and environmental problem for Canada.
Wellbore leakage occurs through leakage pathways located within the cement sheath which surrounds the casings. Many operational factors could contribute to the formation of these leakage pathways such as poor primary cementing or repeat pressurization of the casing for the purpose of hydraulic fracturing, just to name a few. Primary cementing can be affected by things such as well orientation, number of cementing stages, well depth, etc.
We are seeking 2 students to work with Ian Frigaard on two streams defined to study this process.
First stream will investigate this process from a fluid mechanics properties. And the second stream aims to get a better understanding of well leakage through the modeling of leakage pathways based on statistical data.
A) Experimental investigation of annular displacement
In the field, after the well is drilled, the casing is lowered into the open well. In this point, the space inside and outside the casing is occupied by the drilling mud that keeps the hydrostatic balance between the hole and the formation. Then, cement slurry is pumped downwards inside the casing, reaches the casing's bottom, and flows up into the annular section displacing the drilling mud upwards. A good seal will not leave residual mud anywhere.
In the lab, we use two flow loops to simulate the field process. We have carefully designed and built the loops to achieve dynamic similarity. We can control the key parameters of the process, such as flow rate, eccentricity, rheology, and fluid's densities. The data acquisition is through imaging with high sensitivity cameras and automated instrumentation. The objective is to capture experimental data relevant to theoretical predictions of the fluidfluid displacement flows under a wide variety of scenarios.
1. What You Will Do:
The student will perform some combination of experimental work. The student will assist in all operations related with the experiment: fluid preparation, running experiment, image processing of the data, rheometry measurements of the fluids and data analysis. The student will learn the physical background to the experiments and may help in design of new components, undertake bits of machining/manufacturing, and implement changes to the current apparatus
2. Supervision Received:
The graduate student mentor will support the student on a daily basis, as will another PhD student involved in the project. Professor Ian Frigaard will facilitate a number of group meetings.
3. Skills for Success:
Active listening, communication, creative thinking, critical thinking, problem solving. Basic programming and machining skills. Interest in fluid mechanics.
B) Data analysis  Oil & Gas Well Data
The research project aims to get a better understanding of well leakage through the modeling of leakage pathways based on statistical data. Provincial regulator such as the British Columbia Oil & Gas Commission (BCOGC) and the Alberta Energy Regulator (AER) keep large databases concerning various well aspects. The student would work on extracting relevant information from these databases and analyzing the data in ways which could contribute to leakage pathways statistical modeling or modeling of well geometry for primary cementing purposes.
1. What You Will Do:
The student will perform some data analysis using either MATLAB or R or other useful software. The student should be able to handle large amount of data. The student should have a general interest in fluid mechanics and be creative and critical thinking, selftaught individual. Programming abilities, i.e.; ability to write code to extract text from PDF is really essential.
2. Supervision Received:
The graduate student mentor will support the student on a daily basis, as will another PhD student involved in the project. Professor Ian Frigaard will facilitate a number of group meetings.
At least one USRA position is offered for talented undergraduate(s) in the group of Leah Keshet to combine mathematical modeling and computational biology. Interested students will be working with other group members on several projects related to cell motility and the collective behaviour of cells in a tissue. Model outcomes will be compared to both controlled single cell experiments (e.g. cells in microfluidic devices), and experiments in full tissues (e.g. developing chick). Image processing is used to extract data from experimental images.
Students will use computational software (such as the opensource package Morpheus), contribute to writing code (guided by a postdoctoral fellow) and to running and testing codes for a variety of insilico experiments.
Students with a computational background and familiarity with Python, MatLab, C++, and similar programming languages, as well as an interest in applied mathematics / mathematical biology are encouraged to apply.
A Sidon set is a set whose pairwise sums a+b are all distinct (other than the trivial a+b = b+a). If we let R(n) denote the size of the largest Sidon subset of {1, 2, ..., n}, then it is known that R(n) is roughly νn for large enough n. Erdös asked whether it is possible that  R(n)  νn  is bounded; such an unlikely result would indicate some extreme regularity to Sidon sets. The student will work with Greg Martin, where we propose to settle this question in the negative. Most constructions of large Sidon sets proceed by constucting a Sidon set modulo some integer q (for example, the construction might take place in a finite field) and then noting that any Sidon set (mod q) remains a Sidon set when considered as a set of integers. Moreover, the (mod q) construction is usually preserved under affine transformations x > ax+b (mod q), a property that can be detected using exponential sums. If one of these affine transformations ends up moving the whole Sidon set so that it does not intersect the interval [m, q], then the resulting Sidon set is still large but is now a subset of {1, 2, ... , m}, resulting in a large enough value for R(m) to answer Erdös's.
With the emergence of big data, information more and more often comes in the shape of a multidimensional array (or tensor). The importance of finding a decomposition of such an object is at least twofold. First, it often uncovers additional information about the incoming data. For example, in the famous Netfix prize problem, finding the decomposition allows us to find different user types. Second, and sometimes more important, the decomposition allows for efficient storage of the tensor.
Despite its usefulness, tensor decomposition remains a hard problem, both computationally and statistically. We will focus on its computational aspects. It has been shown that decomposing a general tensor or finding any of its eigenvectors is NPhard. Although these are hard problems for general tensors, solving them efficiently for specific families is possible.
Elina Robeva aim to hire two students focusing on two different families of tensors: tensors decomposing into an orthogonal tensor network and incoherently decomposable tensors. For each family we will aim to design algorithms for exact decomposition, give an efficiently checkable criterion for when a tensor lies in the family, and describe the eigenvalues and eigenvectors of a tensor in the family.
New measurement technologies like singlecell RNA sequencing are bringing 'big data' to biology. My group develops mathematical tools for analyzing timecourses of highdimensional gene expression data, leveraging tools from probability and optimal transport. We aim to develop a mathematical theory to answer questions like How does a stem cell transform into a muscle cell, a skin cell, or a neuron? How can we reprogram a skin cell into a neuron?
We model a developing population of cells with a curve in the space of probability distributions on a highdimensional gene expression space. We design algorithms to recover these curves from samples at various timepoints and we collaborate closely with experimentalists to test these ideas on real data. We have recently applied these ideas to shed light on cellular reprogramming (to learn more, see here)
We are looking for students with some background in optimization, probability, and mathematical programming. We are looking to hire several students to work on separate aspects of this largescale project.
The time dependent solution of a large class of FokkerPlanck equations for the distribution functions of electrons and/or reactive species can be obtained numerically with an efficient pseudospectral method defined with nonclassical basis polynomials. The solutions are expressed in terms of the eigenfunctions and eigenvalues of the linear FokkerPlanck equation. The FokkerPlanck eigenvalue problem is isospectral with the Schroedinger equation so that the pseudospactral methods developed can be applied to both eigenvalue problems. There are large number of quantum problems of current interest that involve the Yukawa, WoodSaxon and Hulthen potentials.
There are well defined projects for 24 undergraduate summer students working with Bernie Shizgal. A publication on the FokkerPlanck equation with a previous undergraduate student was recently published and can be found at the link;
A publication on the Schroedinger equation with a previous undergraduate student was recently published and can be found at the link;
The pseudospectral methods have been described in the book "Spectral Methods in Chemistry and Physics", Springer 2015
Description: A knot is a nonintersecting closed curve in a threedimensional space, considered up to deforming the curve. Using computer calculations we will investigate properties of modular knots, a family of knots (more precisely) highlighted by É. Ghys.
Interested students should have programming experience. Background in combinatorics or topology is a plus.
In biology, quorum sensing is an intercellular form of communication that bacteria use to coordinate group behavior such as biofilm production and the production of antibiotics and virulence factors. It is also responsible for the mechanism underlying the onset of luminescence production in cultures of the marine bacterium Vibrio fischeri. Luminescence and, more generally, quorum sensing are important for V. fischeri to form a mutualistic symbiosis with a small Hawaiian squid, Euprymna scolopes. During the day the bacterial cells grow to a high density in the light organ of the squid, which then triggers by sunset a luminescence behavior that camouflages the squid at night from its predators by eliminating its shadow in thin pools of ocean water.
In this USRA project, the candidate will work together with my two members of my research group to analyze and perform computer simulations of various nonlinear ODE systems characterizing quorum sensing behavior in 3D domains. These model ODE systems have been recently derived from an asymptotic reduction of more elaborate PDE/ODE systems of quorum sensing. The ideal candidate will have strong calculus skills, a good background in ODEs, some exposure to PDEs, strong computational skills in a scientific programming language such as MATLAB, as well as a willingness to learn some of the biological issues from reading some journal articles with members of my research group. Mathematically, the student will learn techniques in nonlinear ODE dynamics, some bifurcation theory, and be exposed to an interesting interdisciplinary application.
Arrays of boxes filled with positive integers subject to certain rules are called Young tableaux. They generate functions known as Schur functions, which date from Cauchy in 1815, and arise today in many areas including combinatorics, algebra, quantum physics and algebraic geometry. In this project we will investigate a natural generalization of Schur functions and their associated Young tableaux, with the aim of discovering combinatorial rules for them, such as product rules. We may also investigate connections to other areas.
A strong background in abstract algebra such as Math 322 or Math 323 is required, and an aptitude for combinatorics and programming skills will be an asset.
2019
At least one USRA position is offered for talented undergraduate(s) in the group of Leah Keshet to combine mathematical modeling and computational biology. Interested students will be working with other group members on several projects related to cell motility and the collective behaviour of cells in a tissue. Students will use computational software (for example: Cellular Potts Model), contribute to writing code (guided by a postdoctoral fellow) and to running and testing codes for a variety of conditions. Students with a computational background and familiarity with Python, MATLAB, C++, and similar programming languages, as well as an interest in applied mathematics will be seriously considered.
Required skills: Computation using OpenFOAM
By 2013, more than 550,000 oil & gas well had been drilled in Canada. Before hydrocarbons can be produced every well undergoes primary cementing. This operation consists on sealing the annular section between the steel pipe that stabilizes the well (named casing), and the rock formation. The seal should increase production and prevent subsurface fluids from percolating to surface. Nevertheless, gas leakage to surface is common. A public perception is that 1020% of wellbores leak, which has both health & safety consequences and environmental/ecological impact. This project will study this process from a fluid mechanics perspective.
In the field, after the well is drilled, the casing is lowered into the open well. In this point, the space inside and outside the casing is occupied by the drilling mud that keeps the hydrostatic balance between the hole and the formation. Then, cement slurry is pumped downwards inside the casing, reaches the casing’s bottom, and flows up into the annular section displacing the drilling mud upwards. A good seal will not leave residual mud anywhere.
In the lab, we use two flow loops to simulate the field process. We have carefully designed and built the loops to achieve dynamic similarity. We can control the key parameters of the process, such as flow rate, eccentricity, rheology, and fluid’s densities. The data acquisition is through imaging with high sensitivity cameras and automated instrumentation. The objective is to capture experimental data relevant to theoretical predictions of the fluidfluid displacement flows under a wide variety of scenarios.
We are seeking 2 students, who will perform some combination of experimental work and associated computations with the group of Ian Frigaard. Experimentally, the students will assist in all operations associated with the experiment: fluid preparation, running experiment, image processing of the data, rheometry measurements of the fluids and data analysis. The students will need to understand the physical background to the experiments and may need to help in design of new components, undertake bits of machining/manufacturing, and implement changes to the current apparatus. Computationally, the students will run and analyze selected simulations using OpenFOAM.
Interested students should contact Alondra Renteria alondra.renteria.ruiz@gmail.com and Ali Etrati etrati@alumni.ubc.ca with resume and transcript. The intention is that the successful students be funded under the NSERC USRA program, so grades should be good!
Khovanov homology, in its original form devised by Khovanov about 20 years ago, is a combinatorially defined homology theory for knots and links. BarNatan extended this theory to tangles using a certain cobordism category, more precisely a category whose objects are crossingless tangles and whose morphisms are embedded surfaces that interpolate between such tangles. This made calculations of Khovanov homology much more efficient. Very recently, KotelskiyWatsonZibrowius observed that if one specializes to tangles with four ends, a certain version of BarNatan's theory can be interpreted in terms of immersed curves on the 3punctured disc. This is particularly interesting in the light of similar interpretations of Heegaard Floer and instanton homology.
This project has one main goal: compute the immersed curves associated with as many 4ended tangles as possible. A computer program to find the immersed curves from BarNatan's invariant has already been written by Zibrowius. What is missing is a program that computes BarNatan's invariant in this setting. One option would be to adapt one of the already existing implementations of Khovanov homology. Alternatively, it might be easier to just write an implementation of the algorithm from scratch.
Prerequisites for this project with Liam Watson are an excellent grade in a first course in algebraic topology (covering classification of surfaces, simplicial or singular homology and the Euler characteristic) as well as some experience in programming, ideally python and/or C++. Java would be required if we decide to adapt existing implementations of BarNatan's theory.
The matrix completion problem is as follows: Given a subset of entries of a lowrank matrix, the goal is to fill in the missing entries by leveraging the lowrank structure. There are competing convex optimization programs for this, but while one has the strongest theoretical backing (nuclearnorm minimization), another seems to work better in practice (maxnorm minimization). This project with Yaniv Plan studies the difference both theoretically and through numerical simulations. A key will be to understand the underlying geometry of the corresponding convex sets.
The transpose operation makes the ring of n x n complex matrices into an algebrawithinvolution. That is, the ring is an algebra over the complex numbers, and is equipped with an additive self map A > A* that reverses the order of multiplication, and so that A**=A. We will say that an rtuple (A1, A2, ... , Ar) of n x n matrices _generates_ this algebra if no strict subalgebrawithinvolution contains every element of the rtuple. Let X(n,r) denote the space of all such generating rtuples. This project will study the topology of the space X(n,r) and of a related space Y(n,r) where the rtuples are considered only up to an involutionpreserving change of basis. It will study both particularly in the case of small values of n (and r) where it is possible to understand the spaces completely.
The project with Ben Williams will require a strong understanding of linear algebra, knowledge of homology will be highly desirable. Some commutative algebra or algebraic geometry will also be helpful.
In some imaging applications (Xray crystallography, for example) the resulting image essentially captures the magnitude (amplitude) of the Fourier Transform of a function of interest. However, the Fourier transform is complexvalued, so information (the phase) is lost.
The problem can be rephrased as an optimization problem  of selecting a candidate function whose Fourier transform best matches the observed signal. This project with Lior Silberman will investigate whether techniques of regularized optimal transport can be brought to bear on this problem by quantifying distances and gradients in the space of candidate functions.
Background in real analysis and some programming experience are required.
A problem in extremal combinatorics asks how many (discrete) objects are possible subject to some condition. We consider simple matrices which are (0,1)matrices with no repeated columns. We say a simple matrix A has F as a configuration if a submatrix B of A is equal to a row and column permutation of F. Much has been determined about the maximum number of columns an mrowed simple (0,1)matrix can have subject to condition it has no configuration F for some given F. There is an attractive conjecture (A. and Sali) to pursue with some challenging open problems. We will pursue some 4rowed choices for F that when forbidden in A should yield a quadratic bound on the number of columns of A. There are many variations of this problem including considering specially structured families of forbidden configurations. We will have plenty of problems to work on.
Interested students could contact my previous USRA students Santiago Salazar and Jeffrey Dawson and Cindy Tan.
A student might work on a problem from the following small book which is available on line. The background needed is some knowledge of simple systems of ode's and some probability. There are problems at many levels.
[BOOK] Stochastic neuron models PE Greenwood, LM Ward  2016  Springer In this book we describe a large number of open problems in the theory of stochastic neural systems, with the aim of enticing probabilists to work on them. These include problems arising from stochastic models of individual neurons as well as those arising from stochastic models of the activities of small and large networks of interconnected neurons. We sketch the necessary neuroscience background to these problems so that probabilists can grasp the context in which they arise. This project with Priscilla Greenwood focuses on the mathematics of the models and theories.
Elliptic curves are a fundamental object in number theory. The problem of cataloguing elliptic curves (up to various symmetries) is a difficult computational problem and continues to attract significant attention. Recently Bennett developed a connection between elliptic curves over Q and solutions of a family of Diophantine equations called Thue equations. These equations are very difficult to solve rigorously, however by exploiting properties of continued fractions, one can solve them heuristically. In this project we will investigate the link between solutions of Thue equations and continued fractions over other fields  such as the Gaussian integers.
2018
The group of Leah Keshet offers placement for USRA students with computational experience to work on problems in mathematical biology. Our group is interested in modeling cell shape, cell motility, tissue migration and other phenomena associated with developmental and cell biology. We currently work on understanding the behavior of normal and malignant cells, as well as intracellular chemicals that influence cell shape, adhesion, contractility, and motility. .
The student will be in charge of refining and running publicly availble software (Compucell3D, Morpheus) or helping with development and testing of new software, to simulate collective cell behaviour corresponding to mathematical models.
Students with engineeringmath or CSmath double major are especially encouraged to apply. Particulaly promising students who succeed at the USRA research will have an opportunity of being strongly considered for future MSc graduate studies with the Keshet group.
A problem in extremal combinatorics asks how many (discrete) objects are possible subject to some condition. We consider simple matrices which are (0,1)matrices with no repeated columns. We say a simple matrix A has F as a configuration if a submatrix B of A is equal to a row and column permutation of F. Much has been determined about the maximum number of columns an mrowed simple (0,1)matrix can have subject to condition it has no configuration F for some given F. There is an attractive conjecture (A. and Sali) to pursue with some challenging open problems. A variation is to consider a specially structured family of forbidden configurations. We will have plenty of problems to work on.
Interested students could contact my previous USRA students Santiago Salazar and Jeffrey Dawson.
The time dependent solution of a large class of FokkerPlanck equations for the translational distribution functions of electrons and/or reactive species can be obtained numerically with an efficient pseudospectral method defined with nonclassical basis polynomials. The solutions are expressed in terms of the eigenfunctions and eigenvalues of the linear FokkerPlanck equation. The FokkerPlanck eigenvalue problem is isospectral with the Schroedinger equation so that the pseudospactral methods developed can be applied to both eigenvalue problems. There are large number of quantum problems of current interest that involve the Yukawa, Krazner, Pseudoharmonic, WoodSaxon and Hulthen potentials.
There are well defined projects for up to four undergraduate summer students working with Bernie Shizgal. A publication with two undergraduate students was recently published; Shizgal, Ho and Zang, “The computation of radial integrals with nonclassical quadratures for quantum chemistry and other applications”, J. Math. Chem. 55, 413 (2017). The pseudospectral methods have been described in “Spectral Methods in Chemistry and Physics”, Springer 2015
Biologists have discovered remarkable patterns of collective cell migration during early development of animal embryos. For example, the socalled neural crest cells (NCC) migrate in streams along the spine of the embryos of chicks, frogs and zebrafish. Recently, our group built a model to explain NCC collective migration in terms of the chemical signaling inside the cells and between the neighbors.
The project for summer 2018 extends the existing model to explore an intriguing phenomenon: how NCC cells negotiate obstacle courses, as they are shown to be able to do in experiments. The student will generalize our current ODEbased model to account for chemoattraction and filopodial sensing. The project will require MATLAB and Python programming.
We seek a summer student interested in theoretical molecular biophysics. The goal of the project is to understand how baculovirus can break the protein gel filling the pores of the nuclear pore complex and hence enter the cell nucleus. This will be done using molecular simulations.
A background in biophysics and experience in programming/scientific computing (python, C, Linux OS etc) are required. This is a joint project between Prof. Joerg Rottler (Physics) and Prof. James Feng (Mathematics)
The proposed research centres on the problem of tabulating elliptic curves over quadratic fields, to test various conjectures (Birch, SwinnertonDyer, Modularity, etc.) and to complement work done in the LMFDB (the London Database of Lfunction calculations). The work would have a definite computational flavour, but could otherwise be tailored to fit the skillset of the student, involving classical invariant theory (as used to great effect recently by Bhargava), algebraic number theory and complex analysis. It is likely that code would be produced, in one or more of the standard computational algebra packages (Sage, Magma, Pari, etc). Mike Bennett
Oil & gas wells are sealed by a process called primary cementing, in which sequences of fluids are pumped and displace one another along an annular flow path. The fluids are nonNewtonian, have different densities and other rheologies. At UBC we are studying these flows by both experimental means and by modelling/simulation. For this project we seek a motivated individual to help in running fluid flow experiments within our custombuilt flow loops, participating in some construction, design and calibration tasks, possibly running some numerical simulations etc..
The applicant needs to have practical skills, a good physical sense of fluid flows and proficiency in a lab setting. This could be suitable for an Engineering or Science student with some experience and interest in fluid mechanics, plus physical intuition. Ian Frigaard
We will investigate numerically the spectral behaviour of groups defined through topology. In a recent breakthrough KalubaNowakOzawa recently showed that computational techniques can show an eigenvalue gap in the actions of a discrete group, the outer automorphism group of the free group F_5 (more precisely, the group Out(F_5) has "Kazhdan Property (T)").
A related family of groups, "mapping class groups", are connected to the geometry and topology of surfaces, and it is not known whether they have Property (T) or not. We will investigate this question numerically.
Background in group theory and linear algebra (at the level of Math 223,322) is required. Lior Silberman
In some imaging applications (Xray crystallography, for example) the resulting image essentially captures the magnitude (amplitude) of the Fourier Transform of a function of interest. However, the Fourier transform is complexvalued, so information (the phase) is lost.
The problem can be rephrased as an optimization problem  of selecting a candidate function whose Fourier transform best matches the observed signal. We will investigate whether techniques of regularized optimal transport can be brought to bear on this problem by quantifying distances and gradients in the space of candidate functions.
Background in real analysis and some programming expeirence are required. Lior Silberman
2017
A graph, G, is a set of dots, known as vertices, connected to each other by a set of lines, known as edges. A proper colouring of G is a colouring of the vertices such that no two vertices joined by an edge are the same colour. If we are given k colours then the number of ways of colouring G, as a polynomial in k, is known as the chromatic polynomial. If we have infinitely many colours then the chromatic polynomial generalizes to the chromatic symmetric function. This project will aim to generalize the chromatic symmetric function yet further, with the goal of establishing its basic properties, and trying to resolve wellknown conjectures regarding the chromatic symmetric function. One such example is that it is known that if G is a tree, that is a graph with no set of edges forming a cycle, then the chromatic polynomial is the same for every tree with the same number of vertices. However, it is conjectured that two trees are nonisomorphic if and only if they have distinct chromatic symmetric functions. This project will be under the supervision of Stephanie van Willigenburg and Samantha Dahlberg.
A strong background in abstract algebra such as Math 322 or Math 323, an aptitude for combinatorics, and programming skills will be an asset.
This project, which will be jointly supervised by one or more professors in a biology department, will involve computational analysis of multicolour microscopic images of cell surface receptors, likely on B cells. This project will require good programming skills (using MATLAB or Python) and willingness to work closely with the experimental team.
Students could be from Math, CompSci, (Bio)Physics or Engineering.
Minimization problems arise naturally in many branches of mathematics and science. For example, problems in navigation involve finding paths of least length (`geodesics') on the earth's surface. Minimal surfaces, which are twodimensional analogs of geodesics, are minimizers (or simply critical points) of the area function, and arise naturally in material science; for example in fluid interface problems and elasticity problems. A simple physical example of a minimal surface is the soap film that forms after dipping a wire frame into a soap solution. By the laws of surface tension this soap film has the property that it is stable, that is it becomes larger under slight deformations. The theory of minimal surfaces (and submanifolds) has had striking applications, for example to general relativity and low dimensional topology.
This project will involve studying existence of minimal surfaces, properties of minimal surfaces, and applications. Ailana Fraser
A problem in extremal combinatorics asks how many (discrete) objects are possible subject to some condition. We consider simple matrices which are matrices of integer entries with no repeated columns. We say a simple matrix A has F as a configuration if a submatrix B of A is equal to a row and column permutation of F. We typically restrict our attention to (0,1)matrices and much has been determined about the maximum number of columns an mrowed simple (0,1)matrix can have subject to condition it has no configuration F for some given F. We will consider variations where the simple matrices are restricted to have entries in {0,1,2} and have some conjecture to explore arising from joint work with Attila Sali.
Interested students could contact my previous USRA students Foster Tom, Maxwell Allman, Farzad Fallahi, and Santiago Salazar.
Biologists have discovered remarkable patterns of collective cell migration during early development of animal embryos. For example, the socalled neural crest cells (NCC) migrate in streams along the spine of the embryos of chicks, frogs and zebrafish. Moreover, NCCs from different sources manage to stay unmixed while migrating side by side. Later, they seem to be directed to different destinations along the spine, and then toward the front of the body, where they form various tissues and organs.
There are several mysteries about the collective migration. How do cells interact with each other to maintain cohesion among those from the same source, while keeping a boundary between cell clusters from difference sources? How do the cells decide where to stop or turn into a different route? The intensive efforts by biologists have produced some hypotheses. But as these questions involve the intimate coupling between biochemical signaling and cell mechanics, answering them requires the help of quantitative analysis.
We have developed a mathematical model and numerical techniques for modeling the interaction and collective migration of neural crest cells. For this summer project, the student will generalize our existing tool to study two scenarios of interest: (i) Given an externally imposed gradient of chemoattractant, how does a cluster of NCCs behave? (ii) What chemical and geometric cues may guide different streams of NCCs into different migration paths? Our model involves ODEs describing the dynamics of the signaling molecules and molecular motors, as well as ODEs or PDEs governing the mechanical behaviour of the cells and tissues. The computer program includes MATLAB and Python programming.
Displacement of one fluid by another of different properties is a common process in many industrial applications. Our focus is in displacements of miscible fluids in inclined ducts, where density and viscosity difference exists. Depending on the fluid properties, flow rates and inclinations, different flow regimes and behaviours are observed. For this project we seek a motivated individual to help in running a large number of numerical simulations, covering a wide range of parameters in displacement flows. The simulations are in a 2D channel with high aspect ratio and are written using PELICANS, an opensource C++ package. The simulations will be sent to WestGrid and the results will be postprocessed using existing MATLAB codes.
The applicant will need to set the correct parameters for each case, submit the jobs to WestGrid, collect and organize the data, and help with postprocessing the results. Other data management tasks may also be included. Experience with Linux, MATLAB and knowledge of C++ is an advantage.
This could be suitable for an Engineering or Science student with some experience and interest in fluid mechanics, plus physical intuition. Ian Frigaard
Oil and gas wells are cemented to reduce environmental leakage, to strengthen/support them structurally and to zonally isolate production zones in the reservoir. A model of this process has been developed at UBC consisting of 2D hydraulic calculations of viscoplastic fluids in a narrow channel coupled with a vector concentration equation. The model is implemented and solved in MATLAB using finite difference method. For this project we aim to validate this model with experimental/field data that we have available and the main focus will be to aid in this validation. The work will include running different cementing scenarios, postprocessing the outcomes and comparing with the experimental data in an intelligent way.
An ideal candidate: is proficient in MATLAB (reading and compiling MATLAB .m files, troubleshooting MATLAB programs), has strong mathematical background (multivariable calculus, ordinary and partial differential equations), is familiar with numerical algorithms (Numerical derivatives, numerical integration, root finding methods), can demonstrate strong verbal communication, problem solving and critical thinking.
This could be suitable for an Engineering or Science student with some experience and interest in fluid mechanics, plus physical intuition. Ian Frigaard
We are looking for a student to perform computations in linear algebra and finite group actions which will contribute to a project in the representation theory of padic groups. The only required background is linear algebra and group theory (say at the level of Math 223 and Math 322). Familiarity with computer algebra systems such as SAGE is desirable but not required. No knowledge of of more advanced mathematics is needed for the USRA project.
The student will have the opportunity to learn some of the advanced mathematics underlying the project as a whole. Lior Silberman
The behaviour of cells in a tissue depends on their chemical and mechanical interactions, and on stimuli they receive from their environment. Our group studies mathematical models for intracellular signaling and its effect on cell shape, motility, and tissue dynamics. Publicly available software exists for simulating such cellular systems. This project will consist of adapting such software (CompuCell3D, Morpheus, CHASTE, and others) to the specific models for signaling studied by members of our group.
This project is suited for CPSC, MATH or PHYS majors who can demonstrate ability to work on large opensource projects as part of an interdisciplinary team. Specifically, we are looking for motivated students with experience in testdriven development, objectoriented programming (C++, Python or MATLAB) and Linux/Unix server administration. Applicants with scientific computing experience in C++ using template class libraries for numerical methods, linear algebra, mesh manipulation and multicore processing (e.g. Boost, PETSc, MPI, OpenMP, OpenMesh, etc.) are strongly encouraged to apply. Successful accomplishment in this summer work could lead to future research opportunities, including a graduate (MSc.) position in the Mathematical Biology group of Leah Keshet.
The Brownian motion of particles is a basic fundamental physical process. Consider the problem of particle moving randomly on a curved surface, such as a cell wall or material substrate. Starting from a point, what is the average time for such a particle to reach a certain "trapping region"? This is known as the "mean first passage time".
We can approach such problems using partial differential equations (PDEs). In this project, we have several goals: (1) to compute solutions to diffusion PDEs on surfaces using the Closest Point Method; (2) to implement particle simulations based on closest point representations of surfaces; and (3) to investigate the role of curvature in first passage time problems.
Useful skills include proficiency in MATLAB/Octave or Python, a background in numerical algorithms (such as finite differences and interpolation), and some knowledge of differential equations. Knowledge/interest in collaborative software development with Git would be helpful. Colin MacDonald
An etale algebra E over a field F is a finitedimensional algebra of the form E = E1 × · · · × Er, where each Ei is a finite separable field extension of F . If r = 1, i.e., E is a field, the primitive element theorem asserts that E can be generated over F by a single element. If F is an infinite field, the primitive element theorem remains valid for etale algebras. However, it breaks down if F is a finite field. The project is to find the minimal number of generators in this case, under various assumptions on E (e.g., assuming that F  = q and dimF (E) = n is fixed or assuming that q, n and r are fixed).
This problem came up in my joint work with UBC postdoc Uriya First. Uriya plans to be involved in this USRA project in person while he is still at UBC (his appointment ends on June 10). He may also be available remotely after he leaves. We plan to begin with background reading in Galois theory, the classification and basic properties of finite fields, the primitive element theorem for fields and etale algebras, and some related linear algebra (in particular, the Vandermonde determinant and its variants).
Prerequisites are Math 223, 322 and 323 or equivalent. This general topic and the specific question should be accessible to an undergraduate; we are hoping to get a definitive answer to our main question (or at least some variants of it) by the end of the summer. We also hope that working on this topic will prepare a student for a more indepth study of abstract algebra and related areas of pure mathematics. Zinovy Reichstein