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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.

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 (ºÚÁϳԹÏ×ÊÔ´)

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.