Proposed research projects for Summer 2016

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 (2015, 2014, 2013, 2012) and contact the professors if they are not listed here.
Supervisor Name
Research Project
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 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


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 and
Andrew 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
TITLE: 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.

Projects from last year: 2015