Math Club

Thursday, February 15th, 2024

Time: 5:30 p.m.  Place: Jeffery Hall, Room 118

Speaker: Mike Roth (Queen's University)

Title: What is the average area of the shadow of a cube?

Abstract: Imagine a cube in space, with a light shining on it so that it casts a shadow. As we move the cube around, the shadow changes. What is the average area of the shadow?

The talk will answer this and related geometric questions. The method is a bit surprising : we will generalize the problem until it solves itself!

Math & Stats Department Colloquium

Friday, February 16th, 2023

Time: 2:30 p.m.  Place: Jeffery Hall, Room 234

Speaker: Nicole Looper (UIC)

Title: Arakelov–Green’s functions in dynamics and number theory

Abstract: This talk will discuss how Green's functions, as well as their dynamical adaptation by Baker and Rumely, have been leveraged in studying arithmetic dynamical systems. Key to the utility of these functions is their connection to the equidistribution of points of small canonical height, along with the fact that Elkies-type lower bounds on their averages may be proven in a way that is nicely uniform across the spaces and dynamical systems in question. After sketching the background, I will talk about recent progress in adapting these ideas to the higher-dimensional setting, which has been far less explored up to this point. As time allows, I will close with natural questions for further exploration and an application to the Lehmer conjecture for abelian varieties.

Bio: Prof. Nicole Looper received her PhD from Northwestern University. She then held post-doctoral positions at Cambridge and Brown University before joining the University of Illinois Chicago as an assistant professor. Prof. Looper received the 2020 AWM Dissertation Prize for her thesis work, the 2022 Brin Dynamical Systems Prize for Young Mathematicians and a Sloan fellowship in 2023. Her research lies at the intersection of number theory and dynamical systems.

Number Theory Seminar

Monday, February 12th, 2024

Time: 2:30 p.m.  Place: Jeffery Hall, Room 202

Speaker: Sunil Naik (Queen's university)

Title: A note on Matsuda monoids

Abstract: A commutative, torsion-free, cancellative monoid M is called a Matsuda monoid if for every indivisible element $\alpha$ in M, the polynomial $X^{\alpha} - 1$ is irreducible in F[X; M] for any field F, where F[X; M] denotes the ring of all polynomials with coefficients in F and exponents in M. In this talk, we will discuss recent work on Matsuda monoids that leads to questions in analytic number theory.

Curves Seminar

Thursday, February 8th, 2024

Time: 4:00 p.m.  Place: Jeffery Hall, Room 319

Speaker: Alexandre (Sasha) Zotine

Title: Seed Patterns of Type D_n are Finite Cont.

Abstract: We'll continue our discussion of the argument for why seed patterns of type D_n are of finite type. Namely, we'll construct a modified seed pattern from tagged triangulations, and use this to construct a full rank exchange matrix.

Math & Stats Department Colloquium

Friday, February 9th, 2023

Time: 2:30 p.m.  Place: Jeffery Hall, Room 234

Speaker: Kathryn Mann (Cornell)

Title: Anosov flows on 3-manifolds

Abstract: Anosov flows are a beautiful class of dynamical systems, generalizing and including geodesic flows on manifolds of negative curvature. These systems exhibit "local chaos but global stability" - individual orbits diverge wildly, but the systems as a whole are stable under perturbation. This stability means there is some hope to classify them by discrete, algebraic invariants. Even on 3-dimensional spaces, this is an interesting and challenging problem. In this talk, I will describe some of the history and motivation for classification (dating back to work of Anosov, Smale and others in the 60s), connections with low-dimensional geometric topology, and will describe recent joint work with Barthelmé, Bowden, Frankel and Fenley (in various combinations) answering one thread of the classification problem in dimension 3.

Bio: Professor Kathryn Mann received her PhD from the University of Chicago in 2014, she then held positions at MSRI, UC Berkeley, the Institut de math´ematiques de Jussieu and Brown University. In 2019, she joined Cornell University where she is now an Associate Professor and Rosevear Faculty Leadership Fellow. Prof. Mann was an invited speaker at the 2022 ICM, she also received a Sloan Fellowship, an NSF Career Award, the AWM-Birman Research Prize in Topology and Geometry, the Kamil Duszenko Award and the Mary Ellen Rudin young researcher award.

Math Club

Thursday, February 8th, 2024

Time: 5:30 p.m.  Place: Jeffery Hall, Room 118

Speaker: Thomas Barthelmé (Queen’s University)

Title: Bifoliations of the plane to prelaminations of the circle and back again

Abstract: Think of the plane as an (infinite) plate. Then a foliation of the plane is like (infinite) spaghettis on your plate but arranged so that every point of the plate is covered by one and only one spaghetto.This is joint work with Kathyrn Mann and Christian Bonatti.

An old (1940s) result of Kaplan, improved by Mather (1982), is that given your plate of spaghettis, you can make a border of your plate so that each spaghetto is attached at exactly two points of the border of the plate.This is joint work with Kathyrn Mann and Christian Bonatti.

Now a bifoliation of the plane is a plate with two types of spaghetti (some reds and some greens) such that through every point of the plate passes two spaghetti (one red and one green) and such that all the red spaghettis are transverse (ie crosses) the green spaghettis. Similarly, one can add a border to the plate so that each spaghetto ends at exactly two distinct points of the boundary (in that generality, this result is due to Bonatti, in 2023). This set of ends is an example of a prelamination of the circle.This is joint work with Kathyrn Mann and Christian Bonatti.

After describing more precisely the results above, and how one can obtain them, I'll discuss the opposite direction: Given a set of pairs of points on the border of the plate, what are sufficient conditions so that these are the ends of a platter of green/red spaghettis?This is joint work with Kathyrn Mann and Christian Bonatti.

This is joint work with Kathyrn Mann and Christian Bonatti.

PDEs & Applications Seminar

Tuesday, February 6th, 2024

Time: 9:30 a.m.  Place: Jeffery Hall, Room 319 (Via Zoom)

Speaker: Jameson Graber (Baylor University)

Title: The Master Equation in Mean Field Game Theory

Abstract: Mean field game theory was developed to analyze Nash games with large numbers of players in the continuum limit. The master equation, which can be seen as the limit of an N-player Nash system of PDEs, is a nonlinear PDE equation over time, space, and measure variables that formally gives the Nash equilibrium for a given population distribution. In this talk, I will emphasize the fact that the master equation can be seen as a nonlinear transport equation. In particular, the Nash equilibrium is unique if and only if the characteristics do not cross, and when they do cross, we are faced with the question of making a rational selection among multiple equilibria. I will provide some examples to show how subtle this problem is, and in particular, I will show that the usual theory of entropy solutions is in general not sufficient for the purposes of equilibrium selection.

Math & Stats Department Colloquium

Friday, February 2nd, 2023

Time: 2:30 p.m.  Place: Jeffery Hall, Room 234

Speaker: Regina Rotman (UofT)

Title: Length of a shortest closed geodesic on a closed Riemannian 3-manifold with positive scalar curvature

Abstract: Let $M$ be a closed Riemannian manifold with the scalar curvature bounded below by some positive constant $\kappa$. We will prove that there exists a closed non-trivial geodesic on $M$ of length at most $\frac{c}{\sqrt{\kappa}}$. (Joint with Y. Liokumovich, D. Maximo.)

Bio: Prof. Regina Rotman obtained her Ph.D. from Stony Brook University in 1998 and is now a Professor of Mathematics at the University of Toronto. Her research interests include geometric inequalities, periodic geodesics and minimal surfaces on compact and non-compact manifolds, geodesic nets, and width of homotopies.

Curves Seminar

Thursday, February 1st, 2024

Time: 4:00 p.m.  Place: Jeffery Hall, Room 319

Speaker: Calvin Fletcher

Title: Seed patterns of type D_m

Abstract: In this talk we will demonstrate that seed patterns of type D_m are of finite type. To this end, we will introduce a new combinatorial construction: tagged arcs on a punctured disc. This construction will allow us to use the "full Z-rank" argument to generalize a particular case to the more general case. Finally, we will return to some examples from the opening weeks of this seminar to see this result in practice.

Math Club

Thursday, February 1st, 2024

Time: 5:30 p.m.  Place: Jeffery Hall, Room 118

Speaker: David Wehlau (Queen’s University and Royal Military College)

Title: Pascal's Hexagrammum Mysticum: Solving a 400-Year-Old Geometry Problem

Abstract: In 1639, the 16 year old Blaise Pascal proved his Hexagrammum Mysticum Theorem which provides a straightedge construction to test whether 6 points in the plane lie on a conic. This naturally led to the question of whether there is a straightedge construction which tests whether there is a cubic curve through 10 given points in the plane. I will discuss my joint solution with former Queen's mathematics undergraduate Dr. Will Traves (US Naval Academy) to this problem explaining some of the history.