Updated USra

### Index

Note to students: This list will be continuously updated with new projects until the application deadline in February. You can also look at projects from previous years and contact the professors if they are not listed here.

Supervisor Research project

Summer 2017

Steph Van Willigenburg & Samantha Dahlberg

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.

Dan Coombs

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.

Ailana Fraser

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.

Richard Anstee

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.

James Feng

COMPUTER MODELING AND SIMULATION OF COLLECTIVE MIGRATION OF NEURAL CREST CELLSEXTREMAL COMBINATORICS DESCRIPTION

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.

Ian Frigaard & Ali Etrati

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.

Ian Frigaard & Amir Maleki

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.

Lior Silberman & Avner Segal

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.

Leah Keshet

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.

Colin MacDonald

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.

Zinovy Reichstein

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.

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Summer 2016

Richard Anstee

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.

Daniel Coombs

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.

Eric Cytrynbaum

James Feng

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".

Kalle Karu

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.

Leah Keshet

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.

Brian Marcus
A. Rechnitzer

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.

Greg Martin

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.

Dale Rolfsen

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.

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Summer 2015

Richard Anstee

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.

Daniel Coombs

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).

James J. Feng

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".

Leah Keshet

Leah Keshet
Eric Cytrynbaum

Lior Silberman

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Summer 2014

Richard Anstee

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.

George Bluman
Zinovy Reichstein

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.

Daniel Coombs

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.

Ian Frigaard
Kamran Alba

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.

Stephen Gustafson

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.

Kalle Karu

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).

Leah Keshet

Fok-Shuen Leung

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.

Greg Martin

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.

anthony Peirce

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.

http://www.math.ubc.ca/~peirce

Lior Silberman

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.

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Summer 2013

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.

Richard Anstee

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.

James J. Feng

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".

Ian Frigaard

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.

Julia Gordon
sujatha Ramdorai

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.

Christoph Hauert
Wes Maciejewski

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.

Ed Perkins

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.

Lior Silberman

Michael Ward

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.

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Summer 2012

Omer Angel
Lior Silberman

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.

Richard Anstee

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.

Bud Homsy

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.

Kalle Karu

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.

Leah Keshet

Math Biology

Fok-Shuen Leung

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.

Akos Magyar
M. Pramanik

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.

Brian Marcus
A. Rechnitzer

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.

Anthony Peirce

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).

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Summer 2011

Richard Anstee

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.

Neil Balmforth

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)

Martin Barlow

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.

Michael Bennett
Greg Martin

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.

George Bluman

Jim Bryan

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

James J. Feng

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.

S. Gustafson
Tai-Peng Tsai

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.

Bud Homsy
Harish Dixit

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.

Kalle Karu

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.

Anthony Peirce

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.

http://www.math.ubc.ca/~peirce

M. Pramanik

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.

D. Schoetzau

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.

Zinovy Reichstein