January 21, 2022: Grad Coordination Meeting

January 28, 2022: Ethan Farber

Title: Inverting the universe with everyone’s favorite awful person

Abstract: What do arithmetic functions, polynomial rings, finite fields, and Tony Stark have in common? The answer is a neat trick called Möbius inversion, which has wide-ranging implications. In this talk I’ll elucidate this trick using the language of abelian groups, and along the way we’ll learn about Tony’s greatest discovery.

February 4, 2022: Siddharth Mahendraker

Title: Expander graphs and connections to number theory

Abstract: Expanders are special families of graphs with almost contradictory properties: they are simultaneously highly connected and very sparse. Although we can prove expanders are plentiful, it is very challenging to explicitly construct a single example. Curiously, many important constructions of expanders exploit deep and subtle connections to number theory, harmonic analysis and representation theory. In this talk we'll define expanders, and discuss some of the surprising links between explicit constructions of expanders and number theory.

February 11, 2022: Braeden Reinoso

Title: Milnor Fibrations and Vanishing Cycles

Abstract: Milnor fibrations appear naturally in several areas of topology and geometry (algebraic or not). For example, in algebraic geometry and number theory, they were (supposedly) a key motivation for the construction of nearby and vanishing cycles which led to the proof of the Weil conjectures. In low-dimensional topology, they offer a natural source of fibered knots, and of slice surfaces for those knots. The theory of fibered knots coming from Milnor fibrations is intimately tied to the algebraic constructions they motivated, but this history may be easily glanced over (I at least didn't know this until very recently). I'll present one view of Milnor fibrations from the topologist's perspective, and try my best to draw connections to objects an algebraist might care about, when possible.

February 18, 2022: Eric Moss

Title: Computing Modular Forms

Abstract: Classical modular forms—highly symmetric holomorphic functions on the upper half plane—have all sorts of interesting arithmetic connections. Computing examples of these functions is fairly straightforward, and I’ll go over the construction of the Eisenstein series for those who haven’t seen it before. But, take one step past classical forms and you basically enter the shadow realm if you want to actually write something down. I’ll show one method of doing this, called Hejhal’s Algorithm, and if we have time I’ll say a word about how I am trying to use it to compute the first Hilbert-Maass form.

February 25, 2022: Miguel Prado Godoy

Title: A problem on abelian differentials and intersection theory in the moduli space of rational curves

Abstract: How many possibly meromorphic abelian differentials over a smooth rational curve exist if we fix the order of their zeroes, their poles, and their residues? This question is hard to answer for a general number of zeroes, but I will show some progress on the case of having a single zero and how it quickly complicates when passing to two zeros. We will translate the problem into a top intersection of divisors inside a particular compactification of the moduli space of rational curves with marked points.

March 4, 2022: Mujie Wang

Title: The space of closed subgroups of \mathbb{R}^2

Abstract: In this talk, I will gently introduce Chabauty topology. It is the topology used in geometric convergence, in contrast to algebraic convergence, if you have heard of these words. John Hubbard and Ibrahim Pourezza proved that the space of closed subgroups of \mathbb{R}^2 is homeomorphic to S^4 with this topology. I will talk about a proof given by Kloeckner. There will be a knot, the fundamental domain for SL(2,Z), and Seifert fiber space.

March 11, 2022: Spring Break

March 18, 2022: Siddharth Mahendraker

Title: The Polynomial Method

Abstract: The polynomial method is an elementary but powerful tool that has been used in the past 10 years to solve several outstanding problems in combinatorics. The goal of this talk is to explain the key ideas of the polynomial method, and demonstrate an application of these techniques to a concrete problem.

March 25, 2022: Fraser Binns

Title: (2,2n)-cables of L space knots

Abstract: I will discuss knots, knot Floer homology, and a process called cabling. This talk is based on joint work in progress with Subhankar Dey.

April 1, 2022: Jeeuhn Kim

Title: Legendrian knots from Stokes data

Abstract: I will talk about how Legendrian knots encode the growth rates of solutions to ordinary differential equations(i.e. Stokes data) If time permits, I will discuss how Legendrian knots can be used to formulate irregular Riemann-Hilbert correspondence.

April 8, 2022: Grad Student Union Healthcare Rally

April 15, 2022: Good Friday

April 22, 2022: Laura Seaberg

Title: Pisot tilings of the plane

Abstract: Consider the question of expanding a real number in a non-integer base with a set of specified digits as coefficients. For some cases of real numbers and bases, there exist expansions with desirable properties--in particular, using a Pisot number beta (a real algebraic integer with all Galois conjugates inside the open unit disc) as a base permits interesting and 'nice' expansions. We exposit work by Thurston, Akiyama, and others, which tiles the plane using a projection of the lattice embedding of the ring of integers of the number field generated by a cubic Pisot beta. These tilings have prototiles with fractal boundary and have a number of properties in common with 'simpler' classes of substitution tilings constructed in other ways. We will basically spend our whole time understanding how to make some cool-looking pictures in this talk, and hopefully stuff of interest to different subfields of math will surface!

April 29, 2022: Zachary Gardner

Title: Some curiosities on divided powers

Abstract: Divided powers are an interesting algebraic construction that make cameo appearances in arithmetic geometry, representation theory, and algebraic topology. In this talk we will define divided powers and use them to play some fun algebraic games. Get ready to think about Taylor series, tensor algebras, and Legendre's formula, all in one fell swoop!

May 6, 2022: Career Panel