2026 CUMC

Title

An introduction to algebraic topology via vector calculus.

Abstract

I will give an introduction to homology theory and the fundamental group, motivated by familiar properties of integration in vector calculus. I will present these ideas informally by appealing to geometric intuition and offer some generalizations and applications.

Title

An introduction to algebraic topology via vector calculus.

Abstract

I will give an introduction to homology theory and the fundamental group, motivated by familiar properties of integration in vector calculus. I will present these ideas informally by appealing to geometric intuition and offer some generalizations and applications.

Title

Car Traffic on Knot Diagrams and Some Cool Knot Invariants

Abstract

We will study some strange traffic rules for cars driving through an interchange: When they enter via an underpass, they just go through. But when they enter via an overpass, they fall down to the underpass with some small probability p, and then keep going unharmed, down under. 

We will learn that to analyze this traffic game we need matrices and that to play it better we need probabilities that are not numbers between 0 and 1. We will also learn how this game can be used to define some knot invariants (a notion we will explain) which may be the best we presently have. 

Website

Title

Car Traffic on Knot Diagrams and Some Cool Knot Invariants

Abstract

We will study some strange traffic rules for cars driving through an interchange: When they enter via an underpass, they just go through. But when they enter via an overpass, they fall down to the underpass with some small probability p, and then keep going unharmed, down under. 

We will learn that to analyze this traffic game we need matrices and that to play it better we need probabilities that are not numbers between 0 and 1. We will also learn how this game can be used to define some knot invariants (a notion we will explain) which may be the best we presently have. 

Website

Title

Computational Enumerative Geometry

Abstract

Classically, an enumerative problem takes the form of a deceptively simple question: “How many geometric objects satisfy a given collection of geometric constraints, given by some input data?”. For example, there is exactly 1 circle through three points, 27 lines on a smooth cubic surface, and 3264 conics tangent to five conics in the plane.

But counting is only the beginning. Once we know there are finitely many solutions, we can ask richer questions:

(Monodromy) How do the solutions permute as the input data varies

(Reality) If the input is real, how many solutions can be real?

(Formula) Is there a formula for the solutions, analogous to the quadratic formula?

(Computation) How can we compute all of the solutions?

These questions connect classical enumerative geometry with other areas including group theory, field theory, computation, and topology.

In this talk, I will introduce ‘numerical algebraic geometry’ as a computational approach to these problems. Through many examples from both ‘pure’ and ‘applied’ mathematics, I will show how tools from numerical algebraic geometry allow us to explore enumerative problems through computation and experimentation. I will conclude by highlighting some of the projects currently being explored by our research group at Western University.

Title

Computational Enumerative Geometry

Abstract

Classically, an enumerative problem takes the form of a deceptively simple question: “How many geometric objects satisfy a given collection of geometric constraints, given by some input data?”. For example, there is exactly 1 circle through three points, 27 lines on a smooth cubic surface, and 3264 conics tangent to five conics in the plane.

But counting is only the beginning. Once we know there are finitely many solutions, we can ask richer questions:

(Monodromy) How do the solutions permute as the input data varies

(Reality) If the input is real, how many solutions can be real?

(Formula) Is there a formula for the solutions, analogous to the quadratic formula?

(Computation) How can we compute all of the solutions?

These questions connect classical enumerative geometry with other areas including group theory, field theory, computation, and topology.

In this talk, I will introduce ‘numerical algebraic geometry’ as a computational approach to these problems. Through many examples from both ‘pure’ and ‘applied’ mathematics, I will show how tools from numerical algebraic geometry allow us to explore enumerative problems through computation and experimentation. I will conclude by highlighting some of the projects currently being explored by our research group at Western University.

Title

Mathematical Modelling in Fluid Mechanics

Abstract

In this talk, we explore two examples of how mathematical modelling may be used to gain insight into physical phenomena observed in fluid mechanics. First, we consider pilot-wave hydrodynamics, a system in which a millimetric droplet bouncing on the surface of a vibrating liquid bath may self-propel through an interaction with its own wavefield. We explore how integro-differential equations may be used to predict the stability and dynamics of bound droplet states, including rings and lattices. Second, we turn to stratified turbulence, which plays a central role in governing ocean circulation. We focus on a statistical technique, involving dimensionality reduction and unsupervised clustering, to classify distinct turbulent regimes in experimental data. 

Title

Mathematical Modelling in Fluid Mechanics

Abstract

In this talk, we explore two examples of how mathematical modelling may be used to gain insight into physical phenomena observed in fluid mechanics. First, we consider pilot-wave hydrodynamics, a system in which a millimetric droplet bouncing on the surface of a vibrating liquid bath may self-propel through an interaction with its own wavefield. We explore how integro-differential equations may be used to predict the stability and dynamics of bound droplet states, including rings and lattices. Second, we turn to stratified turbulence, which plays a central role in governing ocean circulation. We focus on a statistical technique, involving dimensionality reduction and unsupervised clustering, to classify distinct turbulent regimes in experimental data. 

Title

An Introduction to Multivariate Cryptography

Abstract

Multivariate cryptography is a promising candidate for post-quantum digital signatures. In practice, a multivariate cryptographic instance corresponds to a system of multivariate polynomial equations, so the security of such protocols depends on the complexity of solving these systems. In this talk, I will discuss two key invariants that play a central role in this context. Finally, I will examine the notion of generic systems as it appears in the cryptographic literature.

Title

An Introduction to Multivariate Cryptography

Abstract

Multivariate cryptography is a promising candidate for post-quantum digital signatures. In practice, a multivariate cryptographic instance corresponds to a system of multivariate polynomial equations, so the security of such protocols depends on the complexity of solving these systems. In this talk, I will discuss two key invariants that play a central role in this context. Finally, I will examine the notion of generic systems as it appears in the cryptographic literature.

Title

What is mathematical modelling? And how to be great at it.

Abstract

During the COVID-19 pandemic, decision-makers and the public expected mathematical modelling to answer questions such as “How many new cases do we expect next week?” and “What is the value of social distancing measures?” The pandemic made mathematical modelling a conspicuous part of our lives, but what is mathematical modelling and how is it related to other disciplines such as statistics, data science, and artificial intelligence?  

In this talk, I will use the COVID-19 pandemic in Atlantic Canada and the territories as an example to discuss these issues. Prior to 2022, these jurisdictions experienced long periods with no COVID-19 infections, so the questions, modelling, and data needs of these jurisdictions were different. To date, there have been few discussions of these issues. The motivation for this talk is that by discussing the strengths and weaknesses of mathematical modeling and adjacent disciplines, we are better able to design methods to answer impactful research questions. 

Title

What is mathematical modelling? And how to be great at it.

Abstract

During the COVID-19 pandemic, decision-makers and the public expected mathematical modelling to answer questions such as “How many new cases do we expect next week?” and “What is the value of social distancing measures?” The pandemic made mathematical modelling a conspicuous part of our lives, but what is mathematical modelling and how is it related to other disciplines such as statistics, data science, and artificial intelligence?  

In this talk, I will use the COVID-19 pandemic in Atlantic Canada and the territories as an example to discuss these issues. Prior to 2022, these jurisdictions experienced long periods with no COVID-19 infections, so the questions, modelling, and data needs of these jurisdictions were different. To date, there have been few discussions of these issues. The motivation for this talk is that by discussing the strengths and weaknesses of mathematical modeling and adjacent disciplines, we are better able to design methods to answer impactful research questions. 

Title

Explicit Results in Number Theory and How to Find Them

Abstract

In this talk we will give a brief introduction to the study of certain weighted prime counting functions related to the Riemann zeta function. From here we will discuss what is known about these functions, how this connects to the famous Riemann Hypothesis and what information we are able to prove. There will be special attention paid to making the results explicit, the unique challenges this comes with and we aim to end on a discussion of some possible directions in the field.

Title

Explicit Results in Number Theory and How to Find Them

Abstract

In this talk we will give a brief introduction to the study of certain weighted prime counting functions related to the Riemann zeta function. From here we will discuss what is known about these functions, how this connects to the famous Riemann Hypothesis and what information we are able to prove. There will be special attention paid to making the results explicit, the unique challenges this comes with and we aim to end on a discussion of some possible directions in the field.

Title

Fun with Conditional Probability

Abstract

Conditional probability explains how the odds of an outcome change in the presence of partial information. Even very simple examples using coins and dice aren’t always obvious and can be surprising! They also have connections to classic questions such as Disease Testing, the Monty Hall Problem, the Sleeping Beauty Problem, Monte Carlo Simulations, and more. Time permitting, the extension to sub-sigma-algebras in continuous models will also be described. Best of all, this talk will use PollEverywhere software to let the audience guess the answers — can you get them right?

Title

Fun with Conditional Probability

Abstract

Conditional probability explains how the odds of an outcome change in the presence of partial information. Even very simple examples using coins and dice aren’t always obvious and can be surprising! They also have connections to classic questions such as Disease Testing, the Monty Hall Problem, the Sleeping Beauty Problem, Monte Carlo Simulations, and more. Time permitting, the extension to sub-sigma-algebras in continuous models will also be described. Best of all, this talk will use PollEverywhere software to let the audience guess the answers — can you get them right?

Title

On Gelfand spectra of Beurling algebras

Abstract

For an infinite abelian group, a Beurling algebra over it is a type of Banach algebra containing its complex group ring as a dense subalgebra. My recent student Z. Zhang and I became interested in determining exactly what sets comprise the Gelfand spectrum of some Beurling algebra. The answer uses very classical functional analysis, including the bipolar theorem in real vector spaces.

Title

On Gelfand spectra of Beurling algebras

Abstract

For an infinite abelian group, a Beurling algebra over it is a type of Banach algebra containing its complex group ring as a dense subalgebra. My recent student Z. Zhang and I became interested in determining exactly what sets comprise the Gelfand spectrum of some Beurling algebra. The answer uses very classical functional analysis, including the bipolar theorem in real vector spaces.

Title

You’re good at math. Help others be too.

Abstract

As future graduate students, teaching assistants, and professors, participants will have much influence over how hundreds- or thousands – of others experience math. The talk would be about the gap between how people think students learn math and what the evidence actually shows, drawing on topics from cognitive science, such as desirable difficulties (learning techniques that feel hard in the moment, but improve long-term learning), and designing student practice in an effective way. I would discuss what kind of math instruction leads to long-term success, and why mathematicians have a responsibility to get it right. 

Title

You’re good at math. Help others be too.

Abstract

As future graduate students, teaching assistants, and professors, participants will have much influence over how hundreds- or thousands – of others experience math. The talk would be about the gap between how people think students learn math and what the evidence actually shows, drawing on topics from cognitive science, such as desirable difficulties (learning techniques that feel hard in the moment, but improve long-term learning), and designing student practice in an effective way. I would discuss what kind of math instruction leads to long-term success, and why mathematicians have a responsibility to get it right.