Math Grad Committee

Arithmetic structures in random sets - by Mariah Hamel.

In this talk, we will discuss additive properties of dense subsets of random sets. Sarkozy's theorem for squares states that for any fixed positive number delta there is a large integer N_0 such that if N > N_0 and if A{1,...,N} with |A| >= delta N, then A must contain two elements whose difference is a perfect square. We extend this result to a random setting, where instead of requiring that A have positive relative density in {1,...,N}, we require that A has positive relative density in a random set. If time permits we will discuss a similar result which ensures long arithmetic progressions in sumsets of subsets of random sets. This is joint work with Izabella Laba.

The artic circle phenomenon - by Sunil Chhita.

The Arctic Circle phenomenon is the amazing result that polar regions are formed when randomly tiling an Aztec diamond with dominos. We shall first introduce the concept of domino tilings, then provide connections to the total asymmetric exclusion process in discrete time and to Young diagrams.

Prime divisors of certain quartic linear recurrences - by Amy Goldlist.

Recurrence sequences appear in many areas of mathematics and are studied by many. A recurrence sequence of order n is defined by a polynomial of degree n, and n initial values. Given a sequence, there are several important questions which one can ask about the sequence: 1 - Which primes divide at least one number in the sequence? What is their (relative) density in the set of all primes?; and 2 - Which primes divide several consecutive numbers in the sequence? What is their density in the set of all primes?.

This last question is the one I will address. Though density is at heart an analytic problem, we will explore ways of rephrasing density questions in an algebraic way, using everyone's favourite go-between theorem, the Chebotarev Density Theorem.

Finally, here is a justification of why this topic is important, copied directly from my NSERC application: "The theory of recurrence sequences is a subject deeply intertwined with such diverse mathematical topics as graph theory, Diophantine equations and dynamical sequences. Linear recurrence sequences form the mathematical basis for some of the most cutting edge modern cryptographic systems; for example, recurrence sequences are used to generate strings of uniformly distributed pseudo-random numbers."

Sum-preserving rearrangements of series - by Mclean Edwards.

To take a rearrangement of a series is to sum the terms in that series with a permutation in the ordering of the natural numbers. When dealing with series, it would be useful to know which rearrangements are sum preserving. For more than a quarter of a century we have known the necessary and sufficient conditions for permutations to be sum preserving, and I will present various characterizations of these results.

I will also demonstrate the result that the class of sum-preserving permutations contain a subclass of rearrangements that, while ensuring all convergent series remain convergent, allow some divergent series to converge under the new ordering.

This talk will be accesible to everyone and has the potential to benefit many areas of research. I will be providing a small handout summerizing the results of the first part of the talk, and will be recruiting research collaborators to help answer questions raised by the second part of this seminar.

The surreal numbers - by Alex Duncan.

The Surreal Numbers were constructed by J. H. Conway as a spin-off of his research in game theory. They are, in a certain sense, the "largest" totally ordered "field" and they subsume both the Real and Ordinal Numbers. I will outline their construction and discuss some of their strange and delightful properties. Familiarity with the ordinal numbers would be helpful but not essential.

Networks in large-scale epidemic simulations - by Ignacio Rozada.

Network theory has recently become a very important tool in large-scale epidemic simulations. It is now possible to do simulations in the individual level for large cities and even whole countries. The problem of efficiently partitioning a large network is important when performing simulations on very large networks in computer clusters. We present a fast multiscale algorithm based on an application of spectral theory to find the minimum cut of the network. We use a coarsening method specifically designed for networks with irregular degree distributions. The coarsening method preserves the degree distribution, and simple dynamical processes behave very similarly in both the coarse and the original networks.

Introduction to state space control theory - by Warren Code.

Control systems are dynamical systems with a control parameter that may be manipulated over time to guide the system to some desired behaviour. Applications abound; these are the basis of any kind of self-regulating system (thermostats, autopilots, cruise control, self-parking cars, noise cancellation).

We will see some of the basic objects of state space control theory (as opposed to the frequency domain approach typically favoured by engineers), including feedbacks, differential inclusions and Lyapunov functions, with an example or two to demonstrate their interplay. No special background will be required other than some elementary differential equations.

An introduction to p-adic fields and adeles - by Patrick Walls.

The field of p-adic numbers (where p is a prime integer) is a mysterious object for most of us but it does not have to be. In this discussion we will construct the p-adic numbers by completing the rational numbers with respect to an unusual metric motivated by congruence solutions to polynomial equations modulo p. This point of view will allow us to carry out simple arithmetic computations with p-adic numbers to characterize various algebraic, analytical and topological properties such as finite field extensions, integration and compactness. The strength of p-adic fields will become apparent and the construction of the adeles will be natural. The goal is to develop an overall intuition for p-adic fields and adeles by touching on several areas without much rigour.