Title: An Introduction to the Weil Representation and Theta Functions
Abstract: A large part of what makes automorphic forms difficult to study is that they are very difficult to write down. Every automorphic form is a small miracle. I know of only two ways of writing down explicit examples of automorphic forms: using Eisenstein series (which are the subject of Dubi’s class) and using the Weil representation. I’ll give an introduction to the Weil representation, how it can be used to systematically produce examples of automorphic forms called theta functions, and time permitting, discuss some applications of theta functions in the Langlands program.
Title: The Cyclic Peg Problem
Abstract: Does every continuous simple closed curve in the plane inscribe a square? This is the famous square peg problem and it is still open! If one assumes the curve is smooth, then we can show it inscribes every cyclic quadrilateral up to similarity; I will talk about how one can prove this using tools from symplectic geometry. If time permits, I will also talk about how we can generalize the cyclic peg problem to non-Euclidean geometries on other surfaces.
Title: We Are Mortal and Number Theory is Futile
Abstract: We go over a proof of the undecidability of Hilbert’s 10th, with great humility at our folly.
Title: Math 1005: Countable Probability and Applications
Abstract: Theorem: A drunk man will find his way home (as long as he doesn't take any stairs), but a drunk bird may get lost forever. We will take a not-so-random walk through the proof of this theorem. The computations will be a bit gnarly, and I'm not Pól(ing)ya leg.
Title: Coxeter groups, Tits cones, and K(❤️, 1)
Abstract: This Valentine's I'm hoping to steal your fundamental aortic chambers with an introduction to the history of the K(\pi, 1) conjecture. Let us braid some memories!
Title: Braid Groups
Abstract: If you ask for the definition of a braid group, the answer will likely depend on whom you ask. It’s not immediately obvious, though, that these different definitions are actually the same. In this talk, we’ll explore several ways to define braid groups, convince ourselves that they are indeed equivalent, and get a sense for how we can derive interesting properties of braid groups that have applications to other related groups in low-dimensional topology.
Title: Abelian Varieties and Finiteness Theorems
Abstract: In this talk, I will present a survey on some of the important ideas and perspectives as it pertains to Abelian Varieties, a generalization of elliptic curves, and in so doing briefly touching upon its intersections with various fields of math. Finally, I will cover Falting's Theorem, which appertains to counting points on curves over number fields, where I will attempt to cover in very broad terms some of the ideas behind its proof.
Title: Homotopy Groups of Spheres
Abstract: Homotopy theory is a beautiful and rich area of mathematics. Part of what makes it so interesting is that one of its most central objects, the homotopy groups of spheres, are incredibly difficult to compute. By the indomitable human spirit, and the help of Serre and Freudenthal, we will compute infinitely many of them in this talk. Along the way, we will pick up some tools and motivation for studying *stable* homotopy theory.
Title: Kakeya needle problem
Abstract: Kakeya needle problem is to find the least area to rotate a needle around in the plane. We’ll do a brief survey of the Kakeya needle problem and related topics. If time permits, we’ll talk about an elegant short proof of finite field Kakeya conjecture, which is combinatorial in nature.
Title: "Why, a child could do that!": proving Belyi's theorem
Abstract: Belyi's theorem is a lovely result which characterizes projective curves defined over the algebraic numbers. Specifically, that any such curve admits a ramified covering of the sphere branched over at most three points. Perhaps more astonishing than the result itself is the fact that its proof is so elementary! We will go over this proof together, and hopefully also discuss the concept of "Dessins D'enfants" which are a beautiful combinatorial construction born out of Belyi's theorem.
Title: Fast & Curious - Math Drift
Abstract: You probably already know what's coming: more problems!
This is the sequel of the problem-solving session "Sitting is the Opposite of Standing II- Revenge of the Sit" from last semester. The format is identical: We'll work in small groups to solve some fun math problems (which are very similar to olympiad problems).
Again, only high school mathematics will be assumed.
Title: Build Your Own Knot Invariant!
Abstract: Quandles are algebraic devices first devised by Wraith and Conway in the 60s, and nowadays have found some use in forming invariants of knots. In this talk we will learn about what Quandles are, and about the fundamental Quandle of a knot/link and how we can obtain an infinitude of knot invariants from it!