# NSERC USRA Projects 2020

**Stan Alama**

My research is in Analysis and Partial Differential Equations (PDE). We encounter PDEs when we try to model natural phenomena in space and time (in physics, biology, or even economics or differential geometry.) The tools used to study PDEs mostly come from Analysis (Real, Complex, Fourier, or Functional Analysis,) and the ideas are extensions of what you first learned in Vector Calculus about functions of several variables. Students looking for a USRA project could pursue one of several directions:

1) The Calculus of Variations. Often solutions of PDEs optimize some energy among all possible functions. Variational problems cover many possible contexts and applications, including geometry (geodesics, minimal surfaces) and physics (waves, classical and quantum mechanics, superconductors).

2) Special solutions to PDE (and ODE): symmetric solutions, traveling or standing waves, periodic solutions.

3) Fourier analysis and Distributions. These are at the core of understanding wave motion, and sound and image processing. They are also powerful tools for understanding the nature of PDEs and their solutions.

You will see many ideas and techniques coming from Math 2XX3, 3A03, 3F03, 3FF3, 3X03.

**Hans Boden **

My research is in the areas of geometric topology and low-dimensional topology, and I would be willing to supervise an undergraduate summer research projects on a range of topics, including knots, links, braids, 3-manifolds, invariants in low-dimensional topology, Morse theory, combinatorial group theory, computational algebra, geometry and topology of orbifolds, gauge theory, and moduli spaces of vector bundles over algebraic curves. The actual topic would be tailored to student's interest and background, and the goal would be to explore some open problems in any of these exciting areas. Below are some titles of previous USRA applications that I agreed to supervise.

1. Invariants of virtual and welded knots

2. Computational algebra and the braid group

3. Intersection theory for algebraic curves

4. Automatic structures in low-dimensional topology

The successful applicant would work directly with me for a period of 16 weeks. The first 4-6 weeks would consist of a crash course in the area of research, and this would involve reading several key books or articles in the area. The second part of the project would be an open-ended investigation into some open problems.

Minimum course requirements: Math 2R03.

**Ben Bolker**

I am interested in a very broad range of topics in mathematical and statistical biology. Particular recent interests include: (1) approximate models for the dynamics of interspecific interactions (especially competition, but also predator-prey and epidemiological systems) in continuous space, the integration of spatial environmental variability into such models, and the estimation of parameters; (2) the evolution of virulence, resistance and tolerance of pathogens and their hosts; (3) the development of practical and polished tools for fitting biological models to data, especially in the area of generalized linear mixed models and their extensions. I am looking for students with intellectual curiosity, courage, and independence; we will start with directed readings to get you up to speed on the biological, mathematical, and statistical concepts and tools relevant to the project.

There are no set minimum course requirements, but some combination of the following courses would be useful: Math 1B03, 2R03, 2C03, 3F03; Stats 2D03, 2MB3, 3A3, 3D3. Previous programming experience is not absolutely necessary, but you must be comfortable with computing and previous experience with the R language would be a bonus.

**Lia Bronsard **

In the field of Partial Differential Equations and the Calculus of Variations, we are interested in studying qualitative properties of solution of equations arising in various contexts: physics, geometry, biology, or economics, for instance. In the Calculus of Variations, the problem is to minimize a quantity (called a functional,) such as arclength of curves on surfaces, and use analysis and ODE/PDE techniques to find the qualitative property of the minima. We can also study the dynamics of systems which decrease the values of the functional in time, and approach the optimal solutions asymptotically. For the example of arclength, this dynamical process is called motion by curvature. In this way, analysis, physics, and geometry enter into the study of these equations. Another examples comes from the study of vortices, which are point singularities with quantized winding numbers, occuring in the Ginzburg-Landau model of superconductors. Again, the optimal shape of vortex solutions is a minimization problem in the Calculus of Variations, and there are associated equations governing the motion of vortices. The methods used will involved vector calculus, real analysis, ODE and PDE, and maybe some differential geometry. The normal prerequisites are 2X3-2XX3, although 3A3, 3F3 and 3FF3 would be desirable.

**Jonathan Dushoff**

I am a theoretical biologist. Most of my work is on the evolution and spread of infectious diseases that affect humans, including HIV, influenza, malaria, canine rabies and Ebola. To work with me, you should be willing to engage with data and bringmathematical tools to bear on scientific questions. Programmingexperience is a plus. For more information, please see:http://lalashan.mcmaster.ca/theobio/DushoffLab/.

**David Earn **

I apply mathematics to biological science. Most students who I supervise work on the epidemiology of infectious diseases, but some work on conservation of endangered species, evolution of animal behaviour, music perception, or music theory. If you are interested in another area of biology, and have an idea that you would like to investigate using mathematical models, then I would be happy to discuss it. You might think that to work with me you need a formal background in biology, but that's not true. You just need to be interested in biology. You do need to have taken some mathematics courses and, most importantly, you need to be keen to use mathematical models to contribute to our understanding of biology. Some knowledge of computer programming is usually very helpful.

One of my long-term goals is to digitize a wide variety of extremely valuable historical disease data sets and make them available online at the International Infectious Disease Data Archive (http://iidda.mcmaster.ca). Undergraduates at any level can get involved in the collection and digitization of these data. Senior undergraduates can potentially get involved in mathematical modelling of disease dynamics if they have done well in an advanced course in ordinary differential equations, such as Math 3F03. It is also extremely advantageous to have taken courses in mathematical modelling or mathematical biology (e.g., Math 3MB3 and/or Math 4MB3). Projects involving statistical analysis of disease data are also possible for individuals who have a solid background in statistics (e.g., an A grade in one or more Level III Stats courses).

Further information on undergraduate research opportunities with David Earn can be found on his web site at https://davidearn.mcmaster.ca/opportunities/for-undergraduate-students

**Megumi Harada **

My research is in equivariant symplectic geometry in a broad sense. I am particularly interested in connections between symplectic geometry and closely related fields such as algebraic geometry, geometric representation theory, combinatorics, and equivariant algebraic topology. Particular examples of topics of recent interest to me are Schubert calculus, Hessenberg varieties, Newton-Okounkov bodies, and toric degenerations. Many of these research areas have introductory treatments (lecture notes, textbooks, expository articles) which would be accessible to an advanced and motivated undergraduate; they also have a wealth of open problems which would be amenable to explicit computational exploration, suitable for a summer project. However, the actual topic would be tailored to student's interests and background.

Minimum course requirements: Math 3E03.

Possible number of students: 1

**Paul McNicholas**

My research is in computational statistics and data science, with a focus on classification and clustering using (possibly matrix variate) mixture models. Several projects in these directions would be suitable for an undergraduate student. Most of these projects involve a combination of methodological and computational work; however, projects focusing primarily on one or the other are also available. Many of the undergraduate researchers I have previously advised have carried out at least some of their work in conjunction with a postdoctoral fellow or a Ph.D. student within my research group, in addition to meeting with me regularly, and such an arrangement is likely for future students. Further information about me, my research, and my research group is available at www.paulmcnicholas.info

Minimum course requirements: two or more STATS courses at the 300 or 400 level.

Possible number of students: 2

My research focus is on data science and the computational aspects of statistics, with a particular focus on the analysis of modern data sets, i.e., so-called big data, and evolutionary computation, e.g., cellular automata. A number of projects in these directions would be suitable for an undergraduate student. Such projects are computationally intensive, yet usually involve a combination of methodological and computational work. Further information about me and my research is available at https://ms.mcmaster.ca/~sharonmc/.

Minimum course requirements: None, but some knowledge of programming is required. The student should be comfortable programming in at least one data science language such as R, Julia, or Python.

Possible number of students: 1

My research specialties are algebraic and geometric topology. Student research projects could cover a variety of topics including knot and link theory, invariants of low dimensional manifolds, equivariant topology (the study of symmetries of a space) and real life applications of algebraic topology (for example, to shape recognition or to sensor networks).

The successful applicant would work directly with me for a period of 16 weeks. The initial phase of the project would involve acquiring the necessary background by means of reading suitable introductory textbooks, survey articles and relevant journal articles. After that, the focus would be on the exploration of some open problems.

Minimum course requirements: Math 3E03.

Possible number of students: 1

**Dmitry Pelinovsky**

“Existence and stability of nonlinear waves”

The summer project is available in the area of nonlinear partial differential equations and discrete dynamical systems. We are interested in the basic questions on the existence and stability of particular periodic or solitary wave solutions in these dynamical systems. These questions clarify the role of these particular solutions in the global dynamics of a given dynamical system.

Particular problems depend on the background and interests of the successful NSERC USRA student. The most recent topics of our interests include periodic waves in the integrable (NLS and KdV) equations, standing waves on quantum graphs, and justification of amplitude equations in the nonlinear lattice equations.

The successful summer student will be involved in reading the relevant

literature, performing numerical computations, and discussing the analytical issues. A background in differential equations (MATH 3DC3, 3F03, 3FF3), numerical methods (MATH 2T03, 3Q03), and analysis (Math 3A03, 3X03) is expected, but is not required.

**Bartosz Protas **

Topics in Vortex Dynamics

One of the main research themes in my group concerns problems of vortex dynamics. Vortices are a class of solutions of equations describing the motion of an inviscid fluid. As such they represent idealized models of many natural phenomena ranging from swirling motion in the bathtub to cyclones occurring on the continental scale. There also arise in emerging areas such as superfluids. The principal open questions in this field are related to the existence of equilibrium solutions in domains with different geometries, stability of such equilibria and also methods for controlling vortex motion. These problems are typically addressed using a combination of mathematical analysis and methods of scientific computing.

The goal of the summer project would be to explore some classical and some open problems in vortex dynamics. As a first step, the student would need to do some reading to acquire necessary background in theoretical fluid mechanics and computational mathematics. The main part of the project will involve development of computational techniques using MATLAB and/or Maple. These tools will then be applied to investigate the properties of some selected vortex systems combining analytical and computational insights. The anticipated duration of the project will be 16 weeks.

Course requirements: Math 1MP3, Math 3F03, Math 3NA3 (Math 4NA3 would also be very useful) Possible number of students: 1

Matt Valeriote

"Algebra, Logic and the Constraint Satisfaction Problem "

Many interesting and important questions from computer science, combinatorics, logic, and database theory can be expressed in the form of a constraint satisfaction problem. Recently an algebraic approach to settling some central conjectures in this area have been developed and have led to the investigation of some novel properties of finite algebraic systems.

The proposed project will involve experimenting with small algebraic systems to test several conjectures and open problems that are concerned with the existence of certain derived operations of an algebra that help to govern the types of solutions that may exist for associated instances of the constraint satisfaction problem. There are a couple of computational tools that may be employed in this project, but during the initial phase of the project a fair amount of background reading will need to be done.

**Adam Van Tuyl**

This project explores the interplay between graph theory, abstract simplicial complexes, and abstract algebra, and will introduce the student to Stanley-Reisner theory. Very roughly speaking, Stanley-Reisner theory allows one to associate an abstract simplicial complex (an object with a lot of combinatorial information) with an algebraic object, specifically, a monomial ideal. This "dictionary" between two areas allows one to study the same object in two different ways.

The specific goal of this project is to examine the algebraic properties of a family of simplicial complexes constructed from a graph. The elements of this simplicial complex contain all the subsets of the vertices of the graph such that the induced graph on those vertices has independence number at most n. Although the topology of these complexes has been studied, little is known about the associated algebraic structure. This research project hopes to illuminate these features.

Interested students should have completed at least one course in abstract algebra (e.g. Math 3GR3). Graph Theory (Math 3V03) is recommended, but not required. In addition, a portion of this project will be devoted to generating examples using specialized computer algebra programs. This project will be jointly supervised by Adam Van Tuyl and Kevin Vander Meulen (Redeemer University College). For examples of similar projects given in the past, see Dr. Van Tuyl's web page.

**McKenzie Wang **

My research projects deal with understanding equations or inequalities satisfied by the curvature

tensor of Riemannian manifolds. While it takes some preparations to explain how the curvature tensor is defined, the interesting equations or inequalities can be written down when the spaces have a lot of continuous symmetries. So it is possible to participate in research by analysing these equations or inequalities at the same time as one reads up on where they come from.

For example, for a surface of revolution in R^3 to have constant Gaussian curvature, the relevant equation is just the ordinary differential equation F’’ + c F = 0 that one studies in Math 2C3, where the constant c is the value of the curvature and F is a function of one variable.

A more fancy example is the Einstein condition for SU(3) invariant metrics on the twistor space of CP^2. The equations are simply the system consisting of

6/a + (a/bc) - (c/ab) – (b/ac) = E

and the two other equations obtained from the above by cyclically permuting the variables a, b, c. (E is a fixed but arbitrary positive constant.) You have to find solutions with a, b, c all positive. Clearly, you can study the system without knowing what the twistor space and what

the space CP^2 are.

For a summer research project, I will suggest appropriate curvature equations or inequalities for you to study based on your expertise and interests. If you are strong in differential equations and analysis, the curvature equations will be systems of nonlinear ordinary differential equations.

If you are strong in algebra and topology, the curvature equations will be systems of algebraic equations. If you are strong in numerical computations, one can also study numerical solutions of

curvature equations in cases where an existence proof would be difficult to find. While you analyse the equations or inequalities, you can read up on the theoretical background necessary for deriving the equations and/or inequalities.

The names and projects of previous USRA students can be found on my webpage whose link is in the Math Dept Faculty listing.

**Gail Wolkowicz **

My students and I have been formulating and analyzing models motivated by questions in ecology and epidemiology. Applications include pest control, prevention of species' extinction, and control or eradication of certain diseases. Recently I am involved in different aspects of modelling Dengue fever,a mosquito borne disease responsible for infecting 50-100 million people annually resulting in 10,000 infant deaths. Another project that I am about to become involved in, with 16 other researchers from mathematicians to statisticians to biologists, involves the study of the effect of individual variation in population models.

Formal training in biology is not necessary. However, some experience in modelling is desired. Courses such as M2E03, M3F03, and M3N03 and even M4G03, though not necessary, would be helpful to contribute to the more theoretical aspects of my research program. On the other hand, someone with excellent computer skills would be able to help with the more computational aspects of my program.