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Math Club

Thursday, February 29th, 2024

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

Speaker: Ivan Dimitrov (Queen's University)

Title: Proof of the sensitivity conjecture

Abstract: IIn 2019 Hao Huang proved that, if $P$ is a set of $2^{n−1} + 1$ vertices of an $n$-dimensional cube, it contains a vertex with at least $\sqrt{n}$ neighbours in $P$.

This settled the nearly 30-year-old Sensitivity Conjecture.

We will discuss the conjecture, and see Huang’s proof.

Math & Stats Department Colloquium

Friday, March 1st, 2023

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

Speaker: Brian Hall (University of Notre Dame)

Title: Heat flow on polynomials with connections to random matrix theory

Abstract: Here is a simple question with a surprisingly rich answer: How do the roots of a polynomial change as the polynomial evolves according to the heat equation? The simplest case of this question concerns the backward heat flow on polynomials with all real roots. In this case, an old theorem of P´olya and Benz says that the roots will remain real. Much more recent results tell us how these real roots evolve in the high-degree limit—and there is a surprising connection to random matrix theory. Things become even more interesting when we study the heat flow (forward or backward) on polynomials with complex roots. I will present some recent conjectures and rigorous results on this question. This is joint work with Ching Wei Ho, Jonas Jalowy, and Zakhar Kabluchko. The talk will be self-contained and have lots of pictures and animations.

Bio: Brian Hall received his Ph.D. from Cornell University in 1993 under Leonard Gross and is currently a professor at the University of Notre Dame. He is the author of two textbooks in the Graduate Texts in Mathematics series, Quantum Theory for Mathematicians and Lie Groups, Lie Algebras and Representations. His recent interests concern random matrices, random polynomials, and free probability.

PDEs & Applications Seminar

Tuesday, February 13th, 2024

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

Speaker: Nicola de Nitti (EPFL)

Title: Sharp bounds on enstrophy growth for viscous scalar conservation laws

Abstract: We prove sharp bounds on the enstrophy growth in viscous scalar conservation laws. The upper bound is, up to a prefactor, the enstrophy created by the steepest viscous shock admissible by the $L^\infty$ and total variation bounds and viscosity. This answers a conjecture by D. Ayala and B. Protas (Physica D, 2011), based on numerical evidence, for the viscous Burgers equation. This talk is based on a joint work with D. Albritton [].

PDEs & Applications Seminar

Tuesday, February 27th, 2024

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

Speaker: Agnid Banerjee (ASU)

Title: Space like strong unique continuation for some fractional parabolic equations

Abstract: My lecture will focus on a very classical subject: when do the zeros of a solution to a PDE spread? I will start with a brief historic overview. Then I will talk on some recent work of mine on space like strong unique continuation for fractional heat type equations which is joint with Vedansh Arya, Donatella Danielli and Nicola Garofalo.

Number Theory Seminar

Monday, February 26th, 2024

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

Speaker: M. Ram Murty (Queen's University)

Title: THE WIENER-IKEHARA TAUBERIAN THEOREM REVISITED

Abstract: The celebrated Wiener-Ikehara Tauberian theorem is a versatile result used in analytic number theory to produce "instant asymptotic formulas" simply from knowing analytic continuation of associated Dirichlet series. We will give a new proof of this result using Fourier analysis. Our method also produces an error term which the classical approach does not. This is joint work with Jagannath Sahoo and Akshaa Vatwani at IIT Gandhinagar.

Math & Stats Department Colloquium

Friday, February 23rd, 2023

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

Speaker: Matilde Lalin (Universite de Montreal)

Title: Sums of the divisor function and random matrix distributions

Abstract: The divisor function gives the number of positive divisors of a natural number. How can we go about understanding the behavior of this function when going over the natural numbers? In this talk we will discuss strategies to better understand this function, issues related to the distribution of these values, and connections to the Riemann zeta function and some groups of random matrices. This talk includes joint work with Vivian Kuperberg.

Bio: Prof. Matilde Lal´ın received her PhD from the University of Texas in Austin. She then held post-doctoral positions at the IAS and UBC, and a tenure-track position at the University of Alberta before joining the Universit´e de Montr´eal in 2010. Prof. Lal´ın is currently a CRM distinguished research scholar, she is a fellow of the CMS, the AMS and the AWM. She received the Krieger-Nelson Prize in 2022 and a Liftoff Fellowship from the Clay Mathematics Institute in 2005.

Number Theory Seminar

Friday, February 23rd, 2024

Time: 4:00 p.m.  Place: Jeffery Hall, Room 422

Speaker: Karl Dilcher (Dalhousie University)

Title: Heronian triangles, Gauss primes, and some linear recurrences

Abstract: We will see that certain sequences of Heronian triangles, that is, triangles with sides of integer length and with integer area, occur in an unexpected way in the study of some specific factorials. In particular, we will consider the multiplicative order of ((p-1)/4)! modulo a prime p = 1 (mod 4). The question of when this order can be a power of 2 leads to the concept of a "Gauss prime". Apart from explaining these various connections, I will derive some divisibility properties of the sequences in question.

Time allowing, I will also discuss factorials ((p-1)/3)! modulo primes p = 1 (mod 6), and generalizations of such factorials. Quite recently, a close relationship between "exceptional primes" in this setting and Iwasawa theory was established by M. Stokes in his Ph.D. thesis. (Joint work with John Cosgrave.)

Curves Seminar

Thursday, February 15th, 2024

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

Speaker: Luke Steverango

Title: Folding

Abstract: Folding is a process that allows us to create new seed patterns from existing ones. We will construct a quotient object, which arises from considering the symmetries of the associated quiver to a cluster algebra. This technique will be useful when we attempt to prove that cluster algebras of type B,C,F,G are of finite type.

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.