Generalized skew Schur functions

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.

Extremal Combinatorics

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 m-rowed 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 4-rowed choices for F that when forbidden in an m-rowed 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.

Osborn’s Model

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.

Numerical implementation of correction method for the Ewald Sphere curvature

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 cryogenic-electron microscopy (cryo-EM) data. For the following project, Dr. David Dynerman (Chan Zuckerberg Biohub, San Francisco, USA) will be co-supervisor. 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.

Our group commits to the “Inclusive Excellence”, defined by UBC’s Equity Diversity and Inclusion endorsement, and strongly encourages students from underrepresented groups to apply.

Project Abstract:

As the level of detail in 3D models produced by cryogenic-electron microscope (cryo-EM) 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 cryo-EM images without relying on the standard linear assumption. Humorously, our approach to correct this non-linear 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 cryo-EM field.

Computer simulation of cell polarization and migration in 3D

Biologists have discovered remarkable patterns of cell migration in many essential biological processes, including development of embryos and wound healing. For example, the so-called 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 cell-cell 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.

Minimal Surfaces and Eigenvalue Problems

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 two-dimensional 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.

Complex Fluids Lab

Canada has the third-largest 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 time-dependent 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.

Interested students should contact Masoud Daneshi (masoud.daneshi@ubc.ca) with resume and transcript. The intention is that the successful students be funded under the NSERC USRA program

Computational modelling of single and collective cell behaviour

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 open-source 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 case-studies.

Dynamics of Deterministic and Stochastic Neural Models

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 Morris-Lecar 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 on-line)

Symmetry-adapted inference models for 3D point cloud data

Inference problems involving 3-dimensional point data (i.e., data where each observation is comprised of a collection of points in 3d-euclidian 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 3-dimensional 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 inter-atomic 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.
For informal enquiries, please contact Matthias Sachs (msachs@math.ubc.ca) and Christoph Ortner (ortner@math.ubc.ca)

Accelerated Geometry Optimisation for Molecular Modelling

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 multi-grid 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 real-world 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/
For informal enquiries please contact Christoph Ortner (ortner@math.ubc.ca)

Optimality of heavy tailed random matrix inequalities

In compressed sensing and it's generalizations, one captures linear measurements of a signal (i.e., an n-tuple of real numbers) which has some known structure, i.e., the signal is known to belong to a particular subset of n-tuples 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 in-depth understanding of sub-Gaussian random variables and high-dimensional geometry; ideally it results in a publication. We believe that the load is most appropriate for two USRA students.

Variations on the theme of the Nullstellensatz

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 p-closed 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 p-closed fields and central simple algebras,
3. trying to generalize the above preprint.

Prerequisites: Strong background and interest in abstract algebra, preferably Math 423.

Constructing meaningful tensor networks using hypergraphs

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.

Nonnegative tensor rank and total positivity

With the emergence of big data, information more and more often comes in the shape of a multi-dimensional array (or tensor). The importance of finding a decomposition of such an object is at least two-fold. 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 NP-hard 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.

Decomposing structured tensors

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.

Analyzing developmental stochastic processes with optimal transport

New measurement technologies like single-cell RNA sequencing are bringing ‘big data’ to biology. My group develops mathematical tools for analyzing time-courses of high-dimensional 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 high-dimensional 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 large-scale project.

The Study of ODE Quorum Sensing Models of Collective Behavior of Bacteria

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 3-D 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

Elliptic curves and congruence class bias in the Lang-Trotter conjecture

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 prime-indexed 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.

Fractional Reaction-Diffusion Systems

We want to investigate reaction-diffusion systems with nonlocal diffusions. Nonlocal diffusions are ubiquitous in nature. In this project we will study several prototype fractional Gierer-Meinhardt, Schnakenberg, Gray-Scott, 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.

Extremal Combinatorics

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 m-rowed 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. We will have plenty of problems to work on.

Localization of eigenfunctions via an effective potential

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.

Building and fitting a cell-scale model of B cell receptor mobility and clustering

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 will learn about B cell biology, cell biophysics, image analysis, modelling and scientific computation. There may be an opportunity to assist with hands-on experimental work as well.

Two-colour particle tracking microscopy to elucidate B cell receptor mobility

B cell receptors (BCR) are mobile on the surface of B cells. In this primarily experimental project, a student will use two-colour fluorescence microscopy to study the motion of individual receptors either in the presence of a bulk-labelled 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 labelling of cells, imaging, image analysis and then fitting to mathematical models of receptor motion. Students working on this project will learn about B cell biology, microscopic imaging and image analysis, modelling and scientific computation.

Numerical Bifurcation Analysis of a Cell Mechanics Model

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 co-dimension 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 self-written 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.

Investigating the properties of the ribosomes and their impact on translation dynamics across scales and systems

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 hight hroughput 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.

Experimental & Computational Fluid Dynamics

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 5,000-20,000 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, near-surface 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 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 fluid-fluid 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.

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, self-taught 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.

Computational and Mathematical Models in Cell Biology

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 open-source package Morpheus), contribute to writing code (guided by a postdoctoral fellow) and to running and testing codes for a variety of in-silico 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 question of Erdös on Sidon sets

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. 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

Generalized gambler's ruin problem

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).

Decomposing structured tensors

Analyzing developmental stochastic processes with optimal transport

New measurement technologies like single-cell RNA sequencing are bringing 'big data' to biology. My group develops mathematical tools for analyzing time-courses of high-dimensional 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 high-dimensional 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 large-scale project.

Pseudospectral methods of solution of the Schroedinger and Fokker-Planck equations.

The time dependent solution of a large class of Fokker-Planck equations for the distribution functions of electrons and/or reactive species can be obtained numerically with an efficient pseudospectral method defined with non-classical basis polynomials. The solutions are expressed in terms of the eigenfunctions and eigenvalues of the linear Fokker-Planck equation. The Fokker-Planck 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, Wood-Saxon and Hulthen potentials.

There are well defined projects for 2-4 undergraduate summer students. A publication on the Fokker-Planck equation with a previous undergraduate student was recently published and can be found at the following link;
https://doi.org/10.1016/j.physa.2019.01.146

A publication on the Schroedinger equation with a previous undergraduate student was recently published and can be found at the following link;
https://doi.org/10.1016/j.comptc.2019.01.001

The pseudospectral methods have been described in the book "Spectral Methods in Chemistry and Physics", Springer 2015 - http://www.springer.com/gp/book/9789401794534

Statistics of Modular Knots

Description: A knot is a non-intersecting closed curve in a three-dimensional space, considered up to deforming the curve. Using computer calculations we will investigate properties of modular knots, a family of knots (more precisely, links) highlighted by É. Ghys.

Interested students should have programming experience. Background in combinatorics or topology is a plus.

The Study of ODE Quorum Sensing Models of Collective Behavior of Bacteria

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 3-D 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.

Tabulation of Conway Tangles

The study of homological invariants of knots has benefited from the existence of knot tables (see, for instance, katlas.org). 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.

Combinatorial rules for generalized Schur functions

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.

MATHEMATICAL MODELING AND COMPUTATIONAL BIOLOGY

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.

FLUIDS PROJECT

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 10-20% 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 fluid-fluid displacement flows under a wide variety of scenarios.

We are seeking 2 students, who will perform some combination of experimental work and associated computations. 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 analyse 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!

CURVES ASSOCIATED WITH TANGLES

Khovanov homology, in its original form devised by Khovanov about 20 years ago, is a combinatorially defined homology theory for knots and links. Bar-Natan 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, Kotelskiy-Watson-Zibrowius observed that if one specialises to tangles with four ends, a certain version of Bar-Natan's theory can be interpreted in terms of immersed curves on the 3-punctured 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 4-ended tangles as possible. A computer program to find the immersed curves from Bar-Natan's invariant has already been written by Zibrowius. What is missing is a program that computes Bar-Natan'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 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 Bar-Natan's theory.

STABILITY OF MATRIX COMPLETION WITH CONVEX OPTIMIZATION

The matrix completion problem is as follows: Given a subset of entries of a low-rank matrix, the goal is to fill in the missing entries by leveraging the low-rank structure. There are competing convex optimization programs for this, but while one has the strongest theoretical backing (nuclear-norm minimization), another seems to work better in practice (max-norm minimization). We study the difference both theoretically and through numerical simulations. A key will be to understand the underlying geometry of the corresponding convex sets.

SPACES OF GENERATORS FOR MATRIX ALGEBRAS WITH INVOLUTION

The transpose operation makes the ring of n x n complex matrices into an algebra-with-involution. 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 r-tuple (A1, A2, ... , Ar) of n x n matrices _generates_ this algebra if no strict subalgebra-with-involution contains every element of the r-tuple. Let X(n,r) denote the space of all such generating r-tuples. This project will study the topology of the space X(n,r) and of a related space Y(n,r) where the r-tuples are considered only up to an involution-preserving 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 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.

PHASE RETRIEVAL THROUGH OPTIMAL TRANSPORT

In some imaging applications (X-ray 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 complex-valued, 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 experience are required.

EXTREMAL COMBINATORICS

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 m-rowed 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 4-rowed 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.

The Mathematics of Stochastic Neuron Models

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. We focus on the mathematics of the models and theories.

Elliptic curves and continued fractions

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.

COMPUTATIONAL AND MATHEMATICAL BIOLOGY OF CELLS

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 engineering-math or CS-math 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.

EXTREMAL COMBINATORICS

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 m-rowed 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 previous USRA students Santiago Salazar and Jeffrey Dawson.

PSEUDOSPECTRAL METHODS OF SOLUTION OF THE SCHROEDINGER AND FOKKER-PLANCK EQUATIONS

The time dependent solution of a large class of Fokker-Planck equations for the translational distribution functions of electrons and/or reactive species can be obtained numerically with an efficient pseudospectral method defined with non-classical basis polynomials. The solutions are expressed in terms of the eigenfunctions and eigenvalues of the linear Fokker-Planck equation. The Fokker-Planck 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, Pseudo-harmonic, Wood-Saxon and Hulthen potentials.

There are well defined projects for up to four undergraduate summer students. 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 - http://www.springer.com/gp/book/9789401794534

COMPUTER SIMULATION OF COLLECTIVE MIGRATION OF NEURAL CREST CELLS IN AN OBSTACLE COURSE

Biologists have discovered remarkable patterns of collective cell migration during early development of animal embryos. For example, the so-called 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 ODE-based model to account for chemoattraction and filopodial sensing. The project will require MATLAB and Python programming.

HOW THE BACULOVIRUS GETS INTO CELL NUCLEUS: MOLECULAR DYNAMICS SIMULATIONS

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)

ELLIPTIC CURVES OVER NUMBER FIELDS

The proposed research centres on the problem of tabulating elliptic curves over quadratic fields, to test various conjectures (Birch, Swinnerton-Dyer, Modularity, etc.) and to complement work done in the LMFDB (the London Database of L-function calculations). The work would have a definite computational flavour, but could otherwise be tailored to fit the skill-set 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).

CEMENTING OF OIL & GAS WELLS

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 non-Newtonian, 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 custom-built 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.

PROPERTY (T) FOR MAPPING CLASS GROUPS

We will investigate numerically the spectral behaviour of groups defined through topology. In a recent breakthrough Kaluba--Nowak--Ozawa 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.

PHASE RETRIEVAL THROUGH OPTIMAL TRANSPORT

In some imaging applications (X-ray 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 complex-valued, 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.

NEW GENERALIZATIONS OF THE CHROMATIC POLYNOMIAL

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 well-known 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 non-isomorphic 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.

ANALYSIS AND MODELING OF SUPERRESOLUTION MICROSCOPY DATA FOR CELL SURFACE RECEPTORS.

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.

MINIMAL SURFACES

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 two-dimensional 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.

EXTREMAL COMBINATORICS

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 m-rowed 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.

COMPUTER MODELING AND SIMULATION OF COLLECTIVE MIGRATION OF NEURAL CREST CELLS

Biologists have discovered remarkable patterns of collective cell migration during early development of animal embryos. For example, the so-called 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.

NUMERICAL SIMULATION OF FLUID-FLUID MIXING AND DISPLACEMENT IN CHANNELS

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 open-source C++ package. The simulations will be sent to WestGrid and the results will be post-processed 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 post-processing 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.

VALIDATION OF A MODEL FOR CEMENTING OF OIL/GAS WELLS

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, post-processing 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 (multi-variable 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.

DECOMPOSITION OF PRINCIPAL SERIES REPRESENTATIONS VIA ELEMENTARY METHODS

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 p-adic 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.

SIMULATING THE INTERACTIONS AND MIGRATION OF MULTICELLULAR TISSUES

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 open-source projects as part of an interdisciplinary team. Specifically, we are looking for motivated students with experience in test-driven development, object-oriented 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 multi-core 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.

COMPUTING FIRST PASSAGE TIMES AND RANDOM WALKS ON SURFACES

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.

MINIMAL NUMBER OF GENERATORS FOR AN ETALE ALGEBRA

An elate algebra E over a field F is a finite-dimensional 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 re-mains 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 in-depth study of abstract algebra and related areas of pure mathematics.

Title: Extremal Combinatorics

Description: The problem of forbidden Berge subgraphs will be considered. The relationship with the problem of Forbidden Configurations will be explored. We say a (0,1)-matrix A has F as a configuration if a submatrix B of A is equal to a row and column permutation of F. We say a (0,1)-matrix A has F as a Berge subgraph if a submatrix B of A has 1's in the positions of the 1's of a row and column permutation of F (B may also have 1's in other positions).

Let F,m be given. The extremal problems consider for an m-rowed matrix A how many different columns can you have without having F as a configuration or perhaps without having F as a Berge subgraph. The student will consider some recent papers in this area to seek connections between the two problems.

Interested students could contact my previous USRA students Foster Tom, Maxwell Allman and Farzad Fallahi.

Analysis and modeling of superresolution microscopy data for cell surface receptors.

This project, which will be jointly supervised by Michael Gold in the Cell Biology and Immunology department, will involve computational analysis of multicolour microscopic images of cell surface receptors on B cells. Further details available on request. This project will require good programming skills (using Matlab or Python) and willingness to work closely with the experimental team.

An active fluid model for the fruit fly germband during embryonic development.

A key event in the development of the fruit fly embryo is the formation of the head-tail axis along the body, partly thanks to the elongation of an epithelial tissue known as the germband. By interdigitating cells along the transverse direction, the germband extends by a factor of two along the head-tail axis. The objective of this project is to explore this extension process by mathematical and computation tools.

My group has worked on vertex models that represent the tissue as an elastic network of polygonal cells, on which myosin motors exert contractile force to drive the rearrangement of the cells. In this USRA project, we will explore a complementary approach by viewing the tissue as an effective fluid driven by distributed contractile forces. More specifically, we model the tissue as an active viscoelastic fluid whose motion is driven by distributed body forces and moments that induces locally elongation along the head-tail axis and contraction in the transverse direction. The interesting questions include the following: What magnitude of active forces will generate the observed amount of axial extension? What are the roles of the extrinsic forces pulling on the tissue? Can this model quantitatively predict the outcome of the germband extension?

The project will involve numerical solution of PDEs describing the fluid dynamics of the tissue. Ideally the student has had exposure to fluid mechanics and numerical methods. Knowledge of cell biology is not essential but will be a plus. The student will help build the model and carry out computations to explore its predictions. See more background information on my webpage http://www.math.ubc.ca/~jfeng/ under "Research" and "Publications".

Combinatorics of the Bruhat graph

Let S_n be the symmetric group, the group of permutations of the set {1,2,...,n}. The Bruhat graph of S_n is a graph with vertices the elements of S_n, with two vertices connected by an edge if one permutation is obtained from the other by switching two numbers. See the picture for part of the Bruhat graph of S_4 (the picture only shows 18 out of the 24 elements of S_4).

Bruhat graphs are related to the geometry of flag varieties in algebraic geometry, to representation theory of Lie groups, and to other parts of mathematics. A problem in combinatorics asks to enumerate all paths in the Bruhat graph. It is known that this enumeration can be encoded in a polynomial, called the complete cd-index. The main open conjecture states that this polynomial has non-negative coefficients. The goal of this project is to work towards proving this conjecture.

Some elementary group theory will be needed (such as taught in math 322 or 422). The main problem is combinatorial and does not need any higher algebra.

Title: Understanding cell topology and geometry in tissue simulations

Description: This summer research position is suitable for a CPSC/MATH USRA student with experience in Python, Object Oriented Programming (C++ or Java) and working in a UNIX environment. The student will be responsible for implementing algorithms from computational geometry to automatically find and classify cells according to their phenotype based on measurements of shape, size, polarity, movement speed etc. Our group is interested in modeling cell division, cell motility, tissue migration and other phenomena associated with morphogenesis in diverse biological and physiological systems. The USRA student will learn to use open-source tools for extracting rich information from simulations of biological systems under the guidance of current group members. Opportunity to interact with both mathematical modelers and biological researchers will be provided.

Prior experience with open-source development, computational geometry, statistics or machine learning is an asset.

Capacity of constrained systems.

A constrained system X is a set of d-dimensional arrays defined by a small set of forbidden patterns. A simple example is the set of arrays of 0's and 1's on the vertices of the d-dimensional lattice such that two adjacent vertices cannot both be 1. An array is allowed if it does not contain a forbidden pattern. The capacity of X is defined as the asymptotic growth rate of the number of allowed arrays on large d-dimensional cubes. The capacity is closely related to notions of entropy in information theory and ergodic theory and free energy and pressure in statistical physics.

When d = 1, the arrays are simply sequences and there is a simple formula for capacity given as the log of the largest eigenvalue of a matrix associated with the set of forbidden patterns. For d > 1, there is no general formula for the capacity of a constrained system and the exact value of capacity is known only for a handful of systems. Nevertheless, there are very good methods of approximating capacity. The methods are based on ideas in linear algebra, combinatorics and probability.

The goal of this project is to compare approximations given by different methods and to explore some related problems of theoretical interest. The student(s) will first learn the basics of capacity and entropy, applications inside and outside of mathematics, and methods for computing approximations.

Background in programming and linear algebra is essential. Background in probability and combinatorics would be helpful.

Statistics of the multiplicative group

For every positive integer n, the quotient ring Z/nZ is the natural ring whose additive group is cyclic. The "multiplicative group modulo n" is the group of invertible elements of this ring, with the multiplication operation. As it turns out, many quantities of interest to number theorists can be interpreted as "statistics" of these multiplicative groups. For example, the cardinality of the multiplicative group modulo n is simply the Euler phi function of n; also, the number of terms in the invariant factor composition of this group is closely related to the number of primes dividing n. Many of these statistics have known distributions when the integer n is "chosen at random" (the Euler phi function has a singular cumulative distribution, while the Erdos-Kac theorem tells us that the number of prime divisors follows an asymptotically normal distribution). Therefore this family of groups provides a convenient excuse for examining several famous number theory results and open problems.

The goal of this USRA project is to write an expository article on what is known about these statistics of the multiplicative group. Of course this goal provides a concrete body of knowledge to assimilate and master; this knowledge is quite accessible for strong undergraduate students. Ultimately we would hope to publish a joint paper in the American Mathematical Monthly; this would require a very well-written article that is both mathematically rigorous and also inviting to the non-specialist reader. The successful candidate for this project should have a solid understanding of number theory (say MATH 312) and also solid skills in algebra (say MATH 322, although that indicates more the level of mathematical maturity than a catalog of necessary content); some of the required techniques can be learned during the project. At least as importantly, the candidate should have excellent mathematical writing skills: composing proofs should be seen not just as a challenge to overcome but also as an opportunity to enlighten and even entertain potential readers.

Free group orderings and automorphisms

Free groups F_n are basic objects in the theory of groups and also arise in topology, analysis and many other branches of mathematics. Free groups are orderable, in the sense that the elements can be linearly ordered in a way that is invariant under multiplication. In fact there are uncountably many such orderings of F_n if n>1. A fundamental question is whether a given automorphism will respect some ordering of F_n. This has application to knot theory and other aspects of low dimensional topology.

The project will involve study of the literature on orderable groups, the automorphism group Aut(F_n) and spaces of orderings of F_n and other groups. It will also involve writing computer programs which can test whether an automorphism preserves some order. One goal will be to gain understanding of (and perhaps solve) the question of the structure of the space of orderings of F_n, which is conjectured to be a Cantor set.

Forbidden Families of Configurations.

The general area of investigation is extremal combinatorics. The goal is to expand on the M.Sc. work of Christina Koch and also continue the work of Maxwell Allman and myself on bounds for a special family of configurations. The following is a typical problem given in matrix language. Let m be given and let F be a given kxt (0,1)-matrix. Let A be an mxn (0,1)-matrix with no repeated columns and no submatrix that is a row and column permutation of F. We seek bounds on n in terms of m,F. (so called problem of Forbidden Configurations). It is perhaps surprising that n < cm^k, for some c, but we can do even better for many F.

Interested students could contact my previous USRA students: Foster Tom and Maxwell Allman.

Cluster Analysis of Super Resolution Fluorescence Images.

This project is to work on novel methods for interpreting data from a very modern microscopic imaging technique, dSTORM. The images, which are taken at UBC, are large - each contains ~1-4 x 106 data points. The points (each representing light emitted by a single fluorescent molecule attached to a defined protein in a cell) are, as expected, distributed in discrete clusters, but the shapes of the clusters are highly irregular. The goal of this project will be to learn about existing methods, understand why they fail for automatic detection and classification of the observed clusters, and then develop new approaches to this problem. The lab produces novel code for displaying and analysing data acquired from both light and electron microscopes and we would prefer an individual who is familiar with the programs we are using; C++, Matlab and OpenGL. This project will be joint supervised by Edwin Moore (Cell and Physiological Sciences) and Dan Coombs (Mathematics).

Molecular-scale simulation of calcium ions within cardiac tissue.

We want to model calcium ion movement and interactions with the resident proteins within the dyadic cleft (which is within cardiac cells). Input data would be the geometry of the cleft, the position of the relevant proteins and their assumed behaviour in response to calcium ions as well as to other intracellular signaling molecules. The output of the model would be a ‘calcium spark’, which is the calcium transient produced by a single dyadic cleft. Since the volume of the dyadic cleft is measured in femtolitres, the model will be constructed using stochastic approaches. Experimental results show that the positions of the relevant molecules within the cleft are subject to changes in response to both physiological and pathological factors. Changes in the molecules’ positions are also correlated with changes in both the magnitude and kinetics of the calcium spark. The goal of this project is to duplicate the experimental results and to make testable predictions. This project will be joint supervised by Edwin Moore (Cell and Physiological Sciences) and Dan Coombs (Mathematics).

Modeling collective migration of cells during embryonic development.

Biologists have discovered remarkable patterns of collective cell migration during early development of animal embryos. For example, the so-called neural crest cells (NCC) migrate in streams along the spine of the embryos of chicks, frogs and zebrafish. The migration is very rapid, and resembles metastasis of cancer cells so much that NCC migration has been used as a model for the latter. 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.

In collaboration with developmental biologists, we have been developing mathematical models on various morphogenetic processes that test the existing hypotheses and strive for a clear in-depth quantitative understanding. These models typically involve 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. This USRA project will study the signaling pathways controlling the cell-cell communication during collective cell migration, and explore how the chemo-mechanical coupling leads to different patterns. The student will help build the models and carry out computations to explore their predictions. See more background information on my webpage http://www.math.ubc.ca/~jfeng/ under "Research".

The general area of investigation is extremal combinatorics. The work will involve exploring problems of forbidden families of configurations. The goal is to expand on the M.Sc. work of Christina Koch and also continue the work of Lu and myself on bounds for a special family of configurations which made use of Ramsey Theory. The following is a typical problem given in matrix language. Let m be given and let F be a given kxt (0,1)-matrix. Let A be an mxn (0,1)-matrix with no repeated columns and no submatrix that is a row and column permutation of F. We seek bounds on n in terms of m,F (so called problem of Forbidden Configurations). It is perhaps surprising that n < cm^k, for some c, but we can do even better for many F.

Interested students could contact my previous USRA students: Ron Estrin and Foster Tom.

Title: The mathematics of parallel parking

The student will study a mathematical model of parallel parking. This project will combine the theory of Lie algebras (in particular, computations in enveloping algebras) with methods of applied mathematics aimed at practical applications. We are looking for a mathematics undergraduate with a strong physics background. During the project the student will learn about Lie algebras and the computation of global groups resulting from their commutators, including applications to a model for parallel parking. This model has free modelling parameters which require investigation by the student. The student will also do a literature survey and consequent study of existing mathematical models for parallel parking.

Title: An age-structured model of mosquito biopesticides and malaria control

Description: Bio-pesticides are natural pathogens that are called into service to control pest populations. An important class of biopesticides are fungal pathogens of mosquitos. These fungi can replace or be used in conjunction with chemical pesticides in controlling the incidence of malaria. Additionally, genetic engineering of the fungi has been performed and variants have been produced that are shown to control malaria parasites within the mosquito. This raises the possibility of reducing the burden of malaria by curing the mosquitos. However, there are some potential issues regarding the effectiveness of these biopesticides that are related to the precise timing of the malaria infection process, the fungal pathogen infection process, and the life-stages of the mosquito. In this project, a new age-structured model of these effects will be developed and analyzed with the goal of understanding (a) the possible efficacy of existing biopesticides of this type, and (b) how future biopesticides might be optimally designed to control the impact of malaria.

Project 1: Mixing and displacement in pipe flows

We seek a motivated individual to help in modifying an existing experimental apparatus in order to be able to conduct experiments involving two-fluid displacement flows in an inclined pipe. The applicant will need to understand the flow experiments to be run, help in design of new components and other modifications, undertake bits of machining and/or manufacturing, and implement the changes to the apparatus, all under supervision. Once modified, the person will assist in various operations associated with the flow loop: mixing and fluid preparation, operating the flow loop, running careful experiment, image processing of the data. Machining and instrumentation experience, data acquisition, etc. are considered as advantages. In some stages of the project the individual might be asked to run Computational Fluid Dynamics (CFD) codes which require programming skills and data analysis.

Displacement of one fluid by another is a common process in industrial applications, where the fluids are not always Newtonian and where a range of fluid properties and densities are used. Here we focus on pipe flow displacements in inclined pipes, where there is also a significant density difference. Depending on the fluid properties and flow rates the fluids either mix, or displace with a clean interface, or stratify during the displacement. We seek to understand these flows mostly experimentally also partly computationally though numerical simulations.
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Project 2: Gas migration in viscoplastic fluids

We seek a motivated individual to help run experiments in a small-scale apparatus. The apparatus is made of an acrylic container and involves viscoplastic fluids, ultra-sensitive pneumatic components and high speed flow imaging. The applicant will need to understand the flow experiments to be run and may need to help in design of new components, undertake bits of machining and/or manufacturing, and implement the changes to the apparatus, all under supervision. The person will assist in various operations associated with the experiment: mixing and fluid preparation, running careful experiment, image processing of the data and rheometry measurements of the non-Newtonian fluids. Experience with pneumatic, machining and instrumentation, data acquisition etc. is an advantage. In some stages of the project the individual might be asked to run Computational Fluid Dynamics (CFD) codes which require programming skills and data analysis.

Through fundamentally studying the topic of gas bubble movement in a viscoplastic bed with intend to resolve the gas-migration problem in cemented oil & gas wells. After drilling oil & gas wells, the annulus section between the production casing and rock formation is cemented using cement slurry. The slurry is then left to set and solidify. In this stage of the process gas may enter the cemented annulus creating channels that provide an undesirable flow path of the reservoir fluids including hydrocarbons into the wellbore and near-surface environment. Our aim is to study this problem mostly experimentally (also partly computationally and analytically) in order to finally design the cement slurry fluid such that it minimizes the gas intrusion from formation into the wellbore. This will in return, decreases the environmental impacts and increases the well productivity.
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The positions are likely to appeal to applied mathematics or engineering physics students. The mathematical content of the positions lies in understanding physical fundamentals, in data analysis and some computation.

Title: Soliton Stability: Analysis and Numerics

Background: Many partial differential equations of physical importance -- examples include nonlinear Schroedinger and Korteweg-de Vries equations -- describe dispersive waves, but nevertheless possess "soliton" solutions whose spatial profiles are constant in time. Some of these solitons are well-known to be stable (hence physically observable) against certain mild perturbations, but with the exception of one or two very special "completely integrable" cases, the response of a soliton to rougher (high frequency) noise is not understood.

Project: This project has both computational and theoretical elements (and could in principle be suitable as a "team" project). Building on existing numerical schemes and codes, we aim to compute the response of one-dimensional periodic solitons to various classes of noise, and propose some conjectures based on these computations. On the analysis side, we will review the few proofs available in the literature of stability against rough perturbations, and propose analytical approaches to our conjectures.

The general area of the research is lattice polytopes, cones over polytopes, and fans obtained by gluing such cones. There are several problems that can be chosen according to student's background knowledge and interests. The problems are about subdivisions of polytopes (triangulations), counting lattice points inside a polytope, and polytopes with special properties, such as nonsingular polytopes, for example. Techniques used in the study involve combinatorics, linear algebra, and some abstract algebra (group and ring theory).

Project 1: Simulation of interacting cells in epithelium.
Project 2: Analysis and simulation of molecular motors on microtubule tracks.

Title: Visualizing calculus from the completeness of R to Taylor's Remainder Theorem.

Description: Consider the content of a "standard" pair of first-year courses in differential and integral calculus. What is the logical structure of this content, and how can it be visualized meaningfully? The goal of this project is to create an online setting that will provide this visualization. This will likely take the form of a navigable directed graph in which clicking on an edge will yield an explicit proof of implication.

Programming experience is strongly preferable but not necessary. The candidate should have good mathematical writing skills and enjoy reworking proofs.

ABC Triples.

The ABC conjecture is an extremely ambitious assertion in number theory, roughly saying that three numbers that are additively related cannot all have unusual factorizations. More precisely, consider triples of positive integers a, b, c with a+b=c, and let R be the product of all the distinct primes that divide a, b, or c. For any positive number epsilon, the ABC conjecture asserts that there are only finitely many such triples (a,b,c) such that R < c^{1-epsilon}. For example, taking a=1 and b=2^m, the ABC conjecture says that numbers of the form 2^m+1 are "almost squarefree".

It might be considered annoying that we need the epsilon in that statement; however, the "naive ABC conjecture" - namely the assertion that R must be at least c - is certainly false. We know several constructions that give infinitely many counterexamples to this naive statement. Some of the simplest such constructions, however, are "folklore examples" and quite hard (or impossible) to find in the literature. People have also found it interesting to search for numerical examples of triples (a,b,c) where R is smaller than c (even though no finite set of such examples can prove or disprove the full ABC conjecture). Whether they come from theoretical constructions or numerical searches, triples (a,b,c) with a+b=c where R < c are called "ABC triples".

The goal of this USRA project is to write an expository article on what is known about ABC triples. Of course this goal provides a concrete body of knowledge to assimilate and master; this knowledge is quite accessible for strong undergraduate students. Ultimately we would hope to publish a joint paper in the American Mathematical Monthly; this would require a very well-written article that is both mathematically rigorous and also inviting to the non-specialist reader. The successful candidate for this project should have a solid understanding of number theory (say MATH 312) and also solid skills in analysis (possibly MATH 320, although that indicates more the level of mathematical maturity than any specific content); some of the required techniques can be learned during the project. At least as importantly, the candidate should have excellent mathematical writing skills: composing proofs should be seen not just as a challenge to overcome but also as an opportunity to enlighten and even entertain potential readers.

Novel Approximation Schemes to model Hydraulic Fracture

Propagation

Hydraulic fractures (HF) are a class of tensile fractures that propagate in brittle materials by the injection of a pressurized viscous fluid. Examples of HF occur in nature as well as in industrial applications. Natural examples of HF include the formation of dykes by the intrusion of pressurized magma from deep chambers. They are also used in a multiplicity of engineering applications, including: the deliberate formation of fracture surfaces in granite quarries; waste disposal; remediation of contaminated soils; cave inducement in mining; and fracturing of hydrocarbon bearing rocks in order to enhance production of oil and gas wells. Novel and emerging applications of this technology include CO2 sequestration and the enhancement of fracture networks to capture geothermal energy. They have recently received considerable attention in the media due to the intense hydraulic fracturing of horizontal wells in order to release the natural gas embedded in shale-like rocks – a procedure referred to as “fracking.”



We plan to investigate the numerical solution of an integral equation that governs the propagation of a fracture in an elastic medium under conditions of plane strain. We will consider a collocation scheme to solve this integral equation. Of particular interest is the robustness of the solution to mesh refinement. The objective of this project is to devise autonomous mesh refinement strategies that will be able to achieve uniformly convergent schemes that are much more efficient than can be achieved using a uniform mesh. The new adaptive scheme scheme will then be used to solve the dynamic model for a hydraulic fracture propagating in an elastic medium. It is proposed that a computer code will be developed in MATLAB and the numerical solutions will be checked against existing asymptotic solutions.



For more information please check my web site:
http://www.math.ubc.ca/~peirce

Project 1. Computation of eigenfunctions on polygonal domains

Abstract: Motivated by the Polymath7 project and the collocation method we will investigate a finite-element method for computing approximate eigenfunctions on plane domains.  Some programming background required.

Project 2. Topology of modular links

Abstract: We will use computer calculations to investigate topological invariants of the spaces obtained by removing closed geodesics from the unit cotangent bundle of the modular surface.  Programming experience will be required (familiarity with python an advantage); background in group theory, algebra and topology will be useful.  For pictures of what the spaces look like see here.

Symmetries of spaces: groups and topology

A mixture of topology, geometry and group theory will be used to investigate symmetries of Riemann surfaces and other familiar examples.

The general area of investigation is extremal combinatorics. The following is a typical problem given in matrix notation. Let F be a given kxt (0,1)-matrix and let A be an mxn (0,1)-matrix with no repeated columns and no submatrix F. There is a conjecture of Anstee, Frankl, Furedi and Pach that claims that there is a constant c(F) so that n < c(F) x m^k. The immediate goal is asymptotic bounds for 2-rowed forbidden submatrices completing the results of Ronnie Chen and Ron Estrin. The work will focus on specific F in order to gain insight. A related problem is where we forbid any row and column permutation of F (so called problem of Forbidden Configurations).

Interested students could contact my previous USRA students: Ronnie Chen and Ron Estrin.

Title: Modeling morphogenesis of the fruit fly

Biologists have studied the fruit fly Drosophila extensively as a model system, and have accumulated a considerable qualitative understanding. The shape evolution and growth of the fly's embryo has garnered particular attention. These are complex processes that are controlled by biochemical signals upstream, and actuated through physical forces and mechanical deformation downstream. Because of this complexity, it is difficult to draw definite conclusions based on observations and experiments alone. In many cases, conflicting hypotheses have been proposed to rationalize the observations.

In collaboration with developmental biologists, we have been developing mathematical models on various morphogenetic processes that test the existing hypotheses and strive for a clear in-depth quantitative understanding. These models typically involve 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. This USRA project will study the signaling pathways controlling the dorsal closure process as well as the chemomechanical coupling during episodes of cell rearrangement known as intercalation. The student will help build the models and carry out computations to explore their predictions. See more background information on my webpage http://www.math.ubc.ca/~jfeng/ under "Research".

Title: Displacement and mixing flows

The group has been carrying on research in this area for the past 5 years, combining a blend of analytical, computational and experimental methods. The basic setup is that one dense fluid is pushed along a pipe, driving a less dense fluid beneath it. The fluids can have different viscosities as well as densities, plus the pipe can be inclined at any angle. The range of flows observed is very wide and is not yet fully understood. It ranges from structured laminar multi-layer flows, through inertial driven partial mixing to full turbulence, driven by buoyancy or flow rate.  The main motivation for our study is to help understand mixing phenomena that occur during displacement of fluids in oil wells. This project will suit someone with some lab experience who has a practical ability to construct things. The student will work as part of the team in modifying the experimental apparatus and conducting experiments, under supervision. As the project progresses the student will be exposed to the types of techniques used to analyse these flows.

Modern computational methods allow us to test out and compute various aspects learnt in Abstract Algebra. These are very relevant for Algebraic Geometry. We shall explore a few such topics, both the abstract aspects and using mathematical computational software to do a few explicit computations.

Title: Evolution in Heterogeneous Environments

Description: Typical evolutionary models assume that the environment is constant/homogeneous over space and time. In reality, an environment may be heterogeneous in a number of ways: sites may be resource-rich or poor, more-connected or less, and the size of the population may fluctuate. Understanding the effects of heterogeneity is a step towards further understanding the natural world. The USRA student will investigate how various aspects of heterogeneity affect the outcome of an evolutionary process. We will consider two setups: the invasion of an advantageous mutant with constant fecundity and evolutionary game theory. The student will perform literature reviews, write computer simulations, and assist in communicating the results of the research.

Title: Stochastic epidemic models, bond percolation and critical values.

In one class of stochastic models for the spread of an infectious disease, carriers of the disease infect nearby susceptible individuals with probability p. Infected individuals then recover and are immune to further infection.  There is a critical value of p, depending on the range R of infection and underlying lattice, above which the disease can spread without bound and below which the disease will die out with probability one.  Bond percolation is a model in statistical physics in which each bond of range R is open or closed independently with probability p. Again there is a critical value of p above which there may be an infinite connected set of open bonds (water can flow to infinity), and below which there can be no such collection.  The two models are closely related.  First, a prospective student has to learn enough probability to understand these models.  In work with Steve Lalley and Xinghua Zheng, we have some conjectures on the behaviour of the critical p's as the range becomes large.  Then the student will run some simulations to test these conjectures.  The student will also write up a short report on the results. Familiarity with computer simulations is required as is exposure to some probability at an undergraduate level.

Title: Stability of Localized Patterns for a Reaction-Diffusion System  in Biology with Time Delay

 There are numerous examples of morphogen gradients controlling long  range signalling in developmental and cellular systems. We propose  to investigate one such two-component reaction-diffusion model that  includes the effect of a time delay due to the time needed for gene  expression. I am motivated by a recent full numerical study of such  a system with Gierer-Meinhardt kinetics that was undertaken in  S. Lee et al. (Bulletin of Mathematical Biology, 72(8), (2010),  pp. 2139-2160). I have recently realized, that by making specific  convenient choices in the powers of the nonlinearities in the  Gierer-Meinhardt kinetics, it is possible to undertake a complete  stability theory for a localized state in this model. This is done  by first deriving a nonlocal eigenvalue problem that represents a  non-self adjoint rank one perturbation of a Sturm Liouville-type  operator. Secondly, one can derive a transcendental equation for any  discrete eigenvalues of this problem, which includes the effect of  the delay time for gene expression. Complex variable methods on this  transcendental equation can then be used to predict Hopf  bifurcations in the model. Techniques from Math 345, 300, 301, and  316 will be used in the analysis.

Title: Value distribution of eigenfunctions on random graphs

Description: We wish to explore the value distributions of eigenfunction of random matrices, especially those associated to random regular graphs. We hope to obtain both experimental and analytical results.

The experimental part will involve numerical linear algebra; experience with Octave, Matlab or the like would be helpful.

The problem area is Extremal Hypergraph Theory although I prefer to use the language of matrix theory. The following is a typical problem. Let F be a given kxt (0,1)-matrix and let A be an mxn (0,1)-matrix with no repeated columns and no submatrix F. There is a conjecture of Anstee, Frankl, Furedi and Pach that claims that there is a constant c(F) so that n < c(F) x m^k. One can establish that n < m^{2k-1-e} where e is a function of k,t and is small. The goal would be to improve this result. The work will focus on specific F in order to gain insight. A related problem is where we forbid any row and column permutation of F (so called problem of Forbidden Configurations).

Interested students could contact my previous USRA students: Connor Meehan and Ronnie Chen.

Title: Studies of the attraction of floating particles

Background:

Particles at an interface get attracted to each other and form aggregates and clusters. This is commonly referred to as the “Cheerios effect”, owing to the tendency of pieces of cereal floating on milk to stick together.  It is known that the attractive force is due to the combination of surface tension and the curvature of the meniscus separating the particles.  In spite of this basic fact, prevailing theories of particle attraction can handle only a few particles and often make restrictive assumptions. 

Our interest in this subject derives from an interest in understanding how the “Cheerios effect” influences coating flows.  In coating applications, (such as coating of memory devices and other recording media, producing photographic film, etc.), a thin layer of fluid is deposited by a flow onto a solid substrate.  It has been found recently that particles lying at the interface of the liquid being coated can modify the coating properties to a significant degree. This project will aim to understanding the forces present at an interface due to the presence of particles in simple flow topologies.

Project Description:

This project has both experimental and computational components.  The overall objective is to establish the quantitative relationship between the particle configuration, the interparticle spacing, and the force of attraction.  On the experimental side, simple table-top experiments with well characterized particles and fluids in controlled conditions will be carried out.  Fundamental data on the position, velocity and acceleration of particles as they come together will provide important tests of theories.  On the computational side, the partial differential equations governing interface dynamics will be solved numerically using available Open Source software like Surface Evolver for groups of multiple particles.

There are several problems that can be chosen according to student's background knowledge and interests. The general area is the geometry of polytopes and fans.  Topics include formulas for lattice point counting, face enumerations, subdivisions of fans, tropical geometry.

The necessary background is linear algebra, elementary combinatorics, and preferably some algebra.

Math Biology

Title: Mathematics modules in Science One

Description: Science One is a highly selective first-year program taught by eight faculty members from the Mathematics, Biology, Chemistry, and Physics & Astronomy departments. These subjects are taught in parallel with occasional crossover topics. The goal of this project is to rewrite the mathematics curriculum to be completely integrated with other subjects. The student would help design modules with broad themes -- for example, "Decay", "The Normal Distribution", "Evolutionary Processes" -- but specific mathematical content. A background in Science One is preferable but not necessary.

Title: Problems in density Ramsey theory.

Description: Ramsey theory is a beautiful area at the interface of analysis, combinatorics and number theory whose aim is to show that large but otherwise arbitrary sets of integers or integer points necessarily contain highly regular structures. Its basic results are both natural and easy to formulate, however their proofs can be surprisingly difficult and varied. A famous example is Roth's theorem that states that if a set contains a "positive proportion" of the integers then it must contain three equally spaced points. Another is van der Courput's theorem which establishes the same for the set of primes.

There has been remarkable progress over the past decade in understanding the underlying principles behind the different approaches in Ramsey theory, and by now a number of problems can be formulated whose solution does not require excessive formal knowledge. The aim of this project is to introduce the students to the basic analytic techniques and possibly try them on some open questions in this area.

APPROXIMATING ENTROPY OF MARKOV RANDOM FIELDS

The entropy of a stationary random process quantifies the degree of randomness of the process. Entropy is important in information theory, where it governs optimal rates of data compression and data transmission, in statistical physics, where it represents disorder in physical systems, and dynamical systems where it quantifies the complexity of a system.

In the case of a one-dimensional stationary Markov chain, there is a simple, exact formula for entropy in terms of the transition probabilities of the chain. In two dimensions, the Markov property generalizes to the notion of a stationary Markov random field (MRF), which can be viewed as a random process of arrays on the integer lattice.  While MRF`s can be very concrete, there is no known formula for the entropy. However, there have been many methods developed for approximating entropy of MRF`s.  These methods use linear algebra, probability and combinatorics.

In this project, the student will first learn the basics of entropy in theory and practice and the methods for computing approximations.  Then the student will write computer programs to compare the performance of the methods.

Background in computer programming, preferably C++, and linear algebra is required. Background in probability and combinatorics would be helpful.

1) NUMERICAL SOLUTION OF A SEMI-INFINITE HYDRAULIC FRACTURE PROPAGATING IN AN ELASTIC MEDIUM:
Hydraulic Fractures (HF) are created in the oil and gas industry to enhance the production of hydrocarbons. This has been the subject of some controversy recently due to the potential negative impacts of this process also known as "Fracking". Our goal is to better understand the mechanics of these fractures that are generated in brittle rock formations by the injection of a high pressure viscous fluid.
This project aims to investigate various numerical schemes to solve the problem of a semi-infinite Hydraulic Fracture propagating in an elastic medium at a constant velocity V. This problem is important from a theoretical point of view as it forms the fundamental solution for many tip asymptotic solutions for finite fractures, which can be obtained via an appropriate re-scaling of the variables. The project will involve partial differential equations, complex analysis, and numerical analysis using the MATLAB programming language, which the USRA will learn during the course of the project. A strong background in PDEs (Math257/316, Math 300, and Math 405 is a recommendation).

2) DEVELOPMENT OF AN EFFICIENT CUDA IMPLEMENTATION OF AN EXTENDED FINITE ELEMENT (XFEM) ALGORITHM FOR MODELING HYDRAULIC FRACTURES:
This project involves the development of an object-oriented set of CUDA routines for the efficient implementation of an existing Extended Finite Element Algorithm Developed at UBC (in the MATLAB language) for modeling propagating Hydraulic Fractures. We have recently acquired a GPU server comprising 2 GPU boards with 480 GPU processors and 6 GB of RAM each as well as 12 Xeon Multicore CPUs with 96 GB of shared RAM. The project will be ideal for a combined Honors student in Computer Science and Mathematics. Being a development of C++, CUDA is a language developed to exploit the multi-threading capabilities of the CGU architecture processors. The USRA will learn both the numerics behind the XFEM algorithm as well as the CUDA programming language. (A strong background in numerics Math405 - or equivalent courses in CS and C++ programming are recommended).

The problem area is Extremal Set Theory although I prefer to use the
language of matrix theory. Let F be a given kxt (0,1)-matrix and let A be an mxn (0,1)-matrix with no repeated columns and no submatrix F. There is a conjecture of Anstee, Frankl, Furedi and Pach that claims that n is O(m^k). One can establish that n is O(m^{2k-1-e}) where e is a function of k,t and is small. The goal would be to improve this result. The work will focus on specific F in order to gain insight.

1) Modelling Mudflow

Mudflows and debris flows are significant geological hazards (USGS, http://vulcan.wr.usgs.gov/Glossary/Lahars/images.html). Mud itself is a ``viscoplastic fluid'', meaning that the material will remain solid unless the forces acting upon it exceed a threshold, the yield stress; once this threshold is reached, the material flows like a fluid. The goal of this project is to analyse a theoretical model for the sudden release of a pile of mud (the dambreak problem) and find its final shape, extracting predictions for the ``yield surfaces'' that separate the regions that flowed from those that remained rigid. The project will build an experiment to compare with the theory, using a transparent analogue laboratory fluid (Carbopol). Markers placed in the fluid will be tracked to determine the flowing and stagnant regions, and gain insight into the flow history.

Refs: Viscoplastic dambreaks and the Bostwick Consistometer,
(www.math.ubc.ca/~njb/Research/bosto.pdf)


2) Dynamics of granular toys

Piles of granular materials display fluid-like behaviour when they avalanche under gravity, yet behave like solids once they come to rest. This project will explore one or more of the following granular ``toys'', each of which is meant to illustrate and study the mechanics of flowing granular media.
a) the granular pencil on an inclined plane:
the threshold for the onset or cessation of avalanching depends
on how superposed layers of grains flow over one another.
Idealized to the extreme, one can explore how a grain rolls
irregularly over a layer of static grains, or equivalently a polygon
(pencil) rolls down an incline.
b) rocking and rolling granular drums:
cylinders either empty or completely full of grains rolls
like solid objects down an incline. If the cyliner is partially full,
on the other hand, the dynamics is far more complicated, becoming unsteady
and erratic as grain avalanche within the rolling drum.
c) laboratory bulldozers:
granular currents can be established by placing a stationary
wall above a moving conveyor belt; the bulldozing action of the wall
can be steady or unsteady, depending on the belt speed and whether the
granular medium avalanches or flows as a whole.
The goal of the project is to build simple laboratory experiments
and theoretical models for each of these toys.

Refs: Granular dambreaks
(www.math.ubc.ca/~njb/Research/gran.pdf)


3) Shear flow instability and vortex pairing

As illustrated by the meandering of the Gulf Stream, sheared fluid flow is often unstable, with jets and shear layers rolling up into arrays of vortices.
(http://earthobservatory.nasa.gov/IOTD/view.php?id=5432
 http://weathervortex.com/wakes.htm)
It is commonly believed that these vortices suffer a secondary instability once they are formed wherein vortices pair up, interact and merge together. The purpose of this project is to demonstrate that this pairing does NOT always occur, and periodic arrays of vortices can sometimes be stable. The project will derive a reduced model for shear instability and then exploit it to build vortex equilibria and test their stability towards perturbations that seek to pair and merge the vortices. The problem has a counterpart in plasma dynamics, and will involve some numerical work. However, the numerical scheme that will be used is straightforward.

Refs: Dynamics of Vorticity Defects in Shear.
(www.math.ubc.ca/~njb/Research/defect.pdf)
Onset of meandering in a barotropic jet
(www.math.ubc.ca/~njb/Research/jet1.pdf)

Title: Branching random walk.

The project will study random networks arising from branching random
walks. We are interested in the electrical resistance of the network,
and time it takes a random walk in the network to move a distance R
from the origin.

Title: Computational problems related to Egyptian fractions

Background: Egyptian fractions, that is, sums of reciprocals of positive integers, arise in a variety of contexts, ranging from recreational number theory to the so-called optic equation. Recent work in this area has typically used sophisticated sieve methods; these results have usually described "average behaviour" of egyptian-fraction representations of rational numbers, without necessarily explicitly determining extremal cases.

Project Description: The project is primarily computational in nature, based upon understanding and implementing an algorithm designed to find egyptian-fraction representations of a given positive rational number, with specified bounds upon the number and size of terms. From this algorithm, one should hope to explicitly quantify results of Martin, related to old questions of Pal Erdos.

Scattering of Waves.

Title: Quivers:  geometry, algebra, combinatorics, and physics.

Title: Particle-based computer simulation of flow

We wish to explore how blobs of soft material interact with each other when subject to forcing, and how they yield to flow like a fluid. For this, the student will integrate and adapt computer programs to implement different models of internal forcing. The computation will be based on the so-called smoothed particle hydrodynamics, and will, we hope, yield an elegant mesoscopic model for the flow of soft materials.

Title: Numerical study of soliton dynamics

Note: This project has both computational and theoretical elements.
It would be suitable as either an individual or "team" project.

Background:

There are many partial differential equations of physical importance, such as nonlinear Schroedinger and Korteweg-de Vries-type equations, whose solutions have a "dispersive wave" nature, but which possess solitary wave ("soliton") solutions whose spatial profiles are constant in time. There is a large literature on the analytic and numerical study of the stability properties of these solitons. There are comparatively few results on the interaction of solitons, except for those few special equations which are ``completely integrable", and for some recent literature on collisions of 2 solitons whose sizes are almost the same, or else have very different magnitudes.

Project:

Set the domain to be the circle, for simplicity of numerical computation. We propose to consider dispersive equations which posses a branch of solitary waves, one part of which is stable, the other part unstable. We plan to first research and develop numerical methods to study the dynamics of solutions near this branch, particularly near the stability-instability transition, where we expect to observe some rich phenomena. We will then propose some conjectures, gather numerical evidence to support them, and try to prove them.

Title: Studies of the attraction of floating particles

Background:

Particles at an interface get attracted to each other and form aggregates and clusters. This is commonly referred to as the “Cheerios effect”, owing to the tendency of pieces of cereal floating on milk to stick together.  It is known that the attractive force is due to the combination of surface tension and the curvature of the meniscus separating the particles.  In spite of this basic fact, prevailing theories of particle attraction can handle only a few particles and often make restrictive assumptions. 

Our interest in this subject derives from an interest in understanding how the “Cheerios effect” influences coating flows.  In coating applications, (such as coating of memory devices and other recording media, producing photographic film, etc.), a thin layer of fluid is deposited by a flow onto a solid substrate.  It has been found recently that particles lying at the interface of the liquid being coated can modify the coating properties to a significant degree. This project will aim to understanding the forces present at an interface due to the presence of particles in simple flow topologies.

Project Description:

This project has both experimental and computational components.  The overall objective is to establish the quantitative relationship between the particle configuration, the interparticle spacing, and the force of attraction.  On the experimental side, simple table-top experiments with well characterized particles and fluids in controlled conditions will be carried out.  Fundamental data on the position, velocity and acceleration of particles as they come together will provide important tests of theories.  On the computational side, the partial differential equations governing interface dynamics will be solved numerically using available Open Source software like Surface Evolver for groups of multiple particles.

Title: Vector bundles on toric varieties

Toric varieties are algebraic varieties constructed from combinatorial data, such as polytopes and polyhedral cones. Similarly, vector bundles on a toric variety are described by combinatorial and linear algebra data. A main open problem in the field is to determine if a toric variety has any nontrivial vector bundles. The goal of this  project is to study examples of toric varieties and look for nontrivial vector bundles on them. The main tools are combinatorics (of convex polytopes and cones), linear algebra, and possibly computer algebra.

Title: Novel Approximation Schemes for Hydraulic Fractures

Hydraulic fractures (HF) are a class of tensile fractures that propagate in brittle materials by the injection of a pressurized viscous fluid. Examples of HF occur in nature as well as in industrial applications. Natural examples of HF include the formation of dykes by the intrusion of pressurized magma from deep chambers. They are also used in a multiplicity of engineering applications, including: the deliberate formation of fracture surfaces in granite quarries; waste disposal; remediation of contaminated soils; cave inducement in mining; and fracturing of hydrocarbon bearing rocks in order to enhance production of oil and gas wells. Novel and emerging applications of this technology include CO₂ sequestration and the enhancement of fracture networks to capture geothermal energy.

We plan to investigate the numerical solution of an integral equation that governs the propagation of a fracture in an elastic medium under conditions of plane strain. We will consider a piecewise cubic Hermite approximation to the crack-opening-displacement which is used along with a collocation technique on a uniform mesh. We will also consider various approximation strategies for a propagating fracture, which does not coincide with the prescribed uniform mesh. The Hermite scheme will then be used to solve the dynamic model for a hydraulic fracture propagating in an elastic medium. It is proposed that a computer code will be developed in MATLAB and the numerical solutions will be checked against existing asymptotic solutions.

For more information please check my web site:
http://www.math.ubc.ca/~peirce

Title: Patterns in sparse sets

A set is sparse or thin if its "size" (appropriately construed) is small in relation to the ambient space that it belongs to. For example, the standard Cantor middle-third set is sparse on the real line (in what sense?). The primes are sparse on the integers, but not as sparse as say the perfect squares. It seems, superficially at least, that the sparser a set is, there is less of it to actually get a handle on, which is why it is all the more striking when it turns out that some of these sets have lots of geometric patterns and structures in them. A thriving area of research touching upon many branches of mathematics centers on understanding the properties that ensure such structures.

This project is concerned with identifying patterns in some specific examples of sparse sets. The theoretical component of the project will involve analyzing some very concrete sets, such as the Cantor set or its higher dimensional counterpart, the Sierpinski gasket. There will also be a computational aspect of the project, where we will formulate numerical experiments to address some long-standing open questions in geometric measure theory.

Experience with a mathematical computational system (Maple, Mathematica or Matlab) will be required for the experimental part of the project.

Title: Exactly divergence-free discretizations for buoyancy driven flow problems

Description: We plan to develop, implement and numerically test novel
finite element methods for the numerical simulation of buoyancy
driven flow problems, in which the incompressible (Navier-)Stokes
equations are coupled with a diffusion equation. This project
is part of a bigger research effort devoted to the understanding and advancing of exactly divergence-free finite element methods in the context of multi-physics flow problems.

More specifically, starting from an existing (Navier-)Stokes solver
and an existing elliptic solver (with codes available in MATLAB), we
plan to develop suitable discretizations of the coupling terms, and
to devise a solver for the fully coupled problem by employing
suitable linearization and iteration techniques. We wish to perform numerical benchmark and accuracy tests, and to simulate stationary
non-isothermal flows.

This project has a strong computational component and involves computer
programming in MATLAB. Some background in computational PDEs would be a helpful, but not absolutely necessary.

Cubic hypersurfaces and a conjecture of Cassels and Swinnerton-Dyer