¶¡ÏãÔ°AV

(University of Richmond)

The search for difference sets in groups of order 256

Difference sets have played an important role in theoretical and practical problems for almost a century. I will give a gentle introduction to difference sets, and I will give several examples why you ought to care about them. I will then describe a decades long search for difference sets in the 56,092 nonisomorphic groups of order 256. I will also describe the next size up, groups of order 1024, which will require us to search 49,487,367,289 nonisomoprhic groups. I will provide a possible road map for working on this case.

(¶¡ÏãÔ°AV)

Biodiversity Mathematics: 100 years of modelling diversity dynamics

A fundamental aim of evolutionary biology is to describe and explain biodiversity patterns; this aim centers around questions of how many "species" exist, where they are most/least abundant, how this distribution is changing over time, and why. Practically speaking, deciphering biodiversity trends and understanding their underlying ecological and evolutionary drivers is important for monitoring and managing both the biodiversity crisis and emergent epidemics. In this colloquium I will discuss 100 years of biodiversity mathematics, beginning with Yule's 1924 foundational work on the model that now bears his name. Despite the twists and turns of the intervening years, I will then introduce recent work from my group with direct connections to Yule's. Throughout, I will highlight the importance of using math and models to clarify biological thinking and will argue that a fully interdisciplinary approach that integrates math, biology, and statistics is necessary to understand biodiversity, be it at the macroevolutionary or epidemiological scale.

Dr. Nilima Nigam (¶¡ÏãÔ°AV)

Whimsical excursions in spectral geometry

Most humans encounter instances of eigenvalues and eigenfunctions - we can tell apart the tones of a small bird from those of a large mammal; we've probably watched standing waves develop on clamped strings. Engineers compute resonances for all manners of applications. In our own courses, we extensively talk about the spectra of familiar mathematical objects - Hermitian matrices, differential operators, graphs and networks. The spectra of differential operators depend intricately on the geometry of the domains the operators are defined on, and this is the focus of spectral geometry.  

In this talk, I'll describe some of the classical questions which arise - can you hear the shape of a drum? Can you design a drum to have a given tone? How do we compute these spectra?  

I'll provide a whimsical collection of examples showing how numerical analysis and spectral geometry interact and enrich each other.  And I'll describe some open problems in the field, which are hopefully of interest to some.

(University of Victoria)

Bootstrap percolation and related graph processes

Bootstrap percolation processes are a family of cellular automata that were originally introduced in 1979 by the physicists Chalupa, Leith and Reich as a model of ferromagnetism. Since then, they have been used throughout a variety of disciplines to model real world phenomena such as the spread of influence in a social network, information processing in neural networks or the spread of a computer virus.

The r-neighbour bootstrap percolation process on a graph G starts with an initial set A of infected vertices and, at each step of the process, a healthy vertex becomes infected if it has at least r infected neighbours (once a vertex becomes infected, it remains infected forever). If every vertex of the graph eventually becomes infected, we say that A percolates.
   
In this talk I will discuss results concerning this and other related processes, including graph saturation, weak saturation, and rainbow saturation.

(University of Victoria)

Mathematical models for the neutral genetics of populations under climate change

In this talk I will discuss the genetic structure of populations subject to climate change and undergoing range expansion. The models and analyses are based on reaction diffusion and integrodifference equations for the asymptotic neutral genetic structure of populations. We decompose solutions into neutral genetic components called neutral fractions. The "inside dynamics" then describe the spatiotemporal evolution of these fractions and can be used to predict changes in genetic diversity. Results are compared with small-scale experimental systems that have been developed to test the mathematical theory.

Dr. Nadish de Silva (¶¡ÏãÔ°AV)

Efficiently achieving fault-tolerant quantum computation via teleportation

Quantum computers operate by manipulating quantum systems that are particularly susceptible to noise.  Classical redundancy-based error correction schemes cannot be applied as quantum data cannot be copied.  These challenges can be overcome by using a variation of the 'quantum teleportation' protocol to implement those operations which cannot be easily done fault-tolerantly.

This process consumes expensive resources called 'magic states'.  The vast quantity of these resources states required for achieving fault-tolerance is a significant bottleneck for experimental implementations of universal quantum computers.

I will discuss a program of finding and classifying those quantum operations which can be performed with efficient use of magic state resources.  This is motivated by both practical reasons and for the resulting theoretical insights into the ultimate origin of quantum computational advantages.  Research into these quantum operations has remained active from their discovery twenty-five years ago to the present.

The results in this talk will include joint work with Chen, Lautsch, and Bampounis-Barbosa.

Dr. Caroline Colijn (¶¡ÏãÔ°AV)

Mathematical method and models to learn about virus transmission from sequence data

The capacity to read the RNA or DNA from large numbers of viral or bacterial genomes in principle offers a lot of information about how these pathogens spread and how to control them. But there is a wide gap between genetic sequences and interpretable information that can be used for onward modelling, to understand pathogen evolution and for public health interventions. In this talk, I will describe how new mathematical tools help fill this gap, using a combination of discrete structures, estimation and dynamic modelling. Mathematical innovations offer new ways to describe and summarize the information in genetic data, new methods to use those data to learn how pathogens move from person to person and around the world, and new ways to learn where the highest levels of transmission are occuring.

(¶¡ÏãÔ°AV)

Kinetic Inverse Problems

This talk aims at a general audience and it consists of two parts: the first part is an overview of my research program which is roughly divided into four subjects. The second part will focus on one of the subjects on kinetic inverse problems. In particular, we will explain the application of the singular decomposition method. This method has been successfully applied to the linear transport equations to recover their absorption and scattering coefficients. Our interest is to extend this method to more general kinetic inverse problems. As examples, we will show various reconstructions of parameters in nonlinear transport equations, linear transport with an interface and the nonlinear Vlasov-Poisson equation.

(¶¡ÏãÔ°AV)

Towards a combinatorial understanding of transcendental functions

Mathematical transcendance refers to objects (usually numbers or functions) that do not satisfy any polynomial equation, that is, they are not algebraic. The numbers $\pi$ and $e$ are famous transcendental numbers, and $e^x$ and the Gamma function are examples of transcendental functions. The problem of understanding the structure of transcendental objects has fascinated mathematicians for well over a century.  Combinatorics provides an intuitive framework to study power series. A combinatorial family is associated to a power series in $\mathbb{R}[[x]]$ via its enumerative generating function wherein the number of objects of size $n$ is the coefficient of $x^n$.  Twentieth century combinatorics and theoretical computer science provided characterizations of classes with rational and algebraic generating functions. Finding natural extensions of these correspondences has been a motivating goal of enumerative combinatorics for several decades.

This talk will focus on two well studied classes of transcendental functions: the differentiably finite and differentially algebraic. We will illustrate how a geometrical approach allows us to classify the generating functions of families of walks on lattices. Lattice path  and random walk models are in bijection with a striking number of classes with transcendental generating functions: from pattern avoiding permutations, to Young tableaux of bounded height and so this program and has led to progress in characterizing differential transcendence of other combinatorial generating functions arising in the literature, and indeed generally.

Dr. Nathan Ilten (¶¡ÏãÔ°AV)

Polynomials, Polytopes, and Geometry

Algebraic geometry is the systematic study of solution sets of systems of polynomials equations. Questions in this field range from enumeration (how many solutions are there?) to classification (what geometric objects arise this way?). In this talk, I will give a brief introduction to algebraic geometry in general, and then focus on a special class of equations that arise by considering relations between integral points in lattice polytopes. The corresponding solution sets, toric varieties, form an important bridge between algebraic geometry and combinatorics. I will discuss several questions involving toric varieties, and how toric varieties can be used to study more general systems of equations.