First-year seminar

September 16 John Dukes: Twists and Turns: A (very) Gentle Introduction to the Mapping Class Group

What happens if you take a surface, like a donut or a rubber band, and look at all the ways you can stretch and twist it without tearing? The collection of these “symmetries up to wiggling” is called the mapping class group. In this talk, we’ll explore what that means in a picture-driven way. We’ll start with the annulus, where the only motion is twisting, and then move to the torus, where the symmetries secretly form a very classic group. No technical background will be needed — just an openness to seeing how curves, twists, and simple pictures can reveal deep mathematical structure.

Friday September 26 1pm in B01/401 Tuong Le: An introduction to matroid theory

Matroids are structures that abstract the notion of independence from linear algebra and graph theory. In a vector space, all bases have the same size; in a connected graph, all spanning trees have the same size—these are both special cases of the fact that all bases of a matroid have the same size. This talk will go over the various definitions of matroids, which are miraculously all cryptomorphic, and discuss the realization problem of matroids.

Friday October 3 1:00pm Franciszek Knyszewski: Ultraproducts

I will explain how to put together fields of positive characteristic to obtain the field of complex numbers, and show how this newfound presentation of $\mathbf{C}$ can be used to give a straightforward proof of a curious theorem in complex algebraic geometry.

October 7 Quanlin Chen: Arithmetic subgroups and rigidity

I’ll explain how to hunt lattices inside Lie groups that are of arithmetic nature and explain Margulis’ extraordinary results on how essentially every lattices are arithmetic and maps between them are “rigid.” Time permits I’ll prove a funny little fact that an arithmetic subgroup of a semisimple Lie group comes from “a unique Q-structure.”

October 14: Fall break

October 21 Kenta Suzuki: BunG jumping

Let $X$ be a compact Riemann surface of genus $g$. I will study the geometry of $\operatorname{Bun}_n$, the space of rank n vector bundles on $X$. We will see that $\operatorname{Bun}_1$, the space of line bundles on $X$, is (roughly) the "Jacobian," a $2g$-dimensional torus. However, it turns out that when $n>1$, naively thinking about $\operatorname{Bun}_n$ as a "manifold" (or scheme) does not work well, and we instead think of it as an "orbifold" (or stack). For example, even though $Bun_n$ is always zero-dimensional for $X=CP^1,$ it can still have an interesting geometry! With this correction, I will explain the geometry of $\operatorname{Bun}_2$ in detail. $\operatorname{Bun}_2$ will have a cusp, analogous to the cusp of $SL(2,Z)\\H$ in the theory of modular forms.

October 28 Jessica Zhang: The Thom Conjecture

Let $h$ be the homology class of $\mathbb{CP}^1$ in $\mathbb{CP}^2$, i.e., a generator of $H_2(\mathbb{CP}^2; \mathbb{Z})$. The Thom conjecture, first proved by Kronheimer and Mrowka in 1994, says that a smooth surface in class $dh$ has genus at least $(d-1)(d-2)/2.$ In particular, complex submanifolds minimize genus within their homology class. We prove this for $d=3$.

November 4 Ankit Bisain: Transcendence of $\pi$

We give an overview of major results in transcendental number theory and prove (via Lindemann–Weierstrass) that $\pi$ is transcendental.

November 11 Vadym Koval: Fair division algorithms

This time we will finally discuss something that actually matters: how to fairly divide seminar pizza between people. We will first have to answer some philosophical questions like "What is fair?" and "What is pizza?". Once we settle those, we will discuss algorithms of fair division in various settings.

November 18 Zachary Lihn: Ricci Flow

The Ricci flow is a certain nonlinear "heat flow" for a Riemannian metric introduced by Hamilton in the 1980s. Its study has been extremely influential in geometric analysis and has found applications across topology and differential and algebraic geometry. I'll introduce this PDE, give some examples, and describe some of its basic properties, including why it exists and why it usually blows up in finite time. Then I'll give an idea of some applications.

November 25 Zahir Ahmed: Tempered Distributions

One of the most common ways to solve partial differential equations is to determine the existence of "weak" solutions, then to promote the regularity of such solutions using the structure of the equation to make them classical. To illustrate the first step of this procedure, we consider the Poisson equation $\Delta u = f$ on $\R^3$, where physically-motivated (though classically nonsensical!) computations yield hints towards solutions. The properties that need to hold for these calculations to go through naturally give rise to the space of tempered distributions $\mathscr{S}'(\R^3)$, which can be fruitfully interpreted as a "generalized function space" where partial differential equations can be posed. Time-permitting, we will also discuss the structure theory of tempered distributions.

December 2 Dhruv Goel: The Weil Conjectures

What do Poincaré duality in algebraic topology, the functional equation for the Riemann zeta function in complex analysis, and the Riemann-Roch theorem for curves in algebraic geometry have in common? What does this have to do with the brother of one of the great women philosophers of the twentieth century, a guy who was imprisoned because he refused to fight for France in WWII? Come find out! In this talk, I'll present the story behind one of the greatest mathematical hits of the twentieth century: the Weil Conjectures. The cast includes ideas from the theory of differential equations, complex analysis, number theory, algebraic and differential topology, and algebraic geometry, with some guest appearances (if time permits) by ideas from combinatorics and modular forms. Alternative talk titles: why I love algebraic geometry, why algebraic geometry is for everyone, the sea is risen, etc.