GIST will be held in Maloney 560. Contact Matt or Siddharth if you want to speak or attend!
Title: Linear Algebra is the Only Thing I Know
Abstract: Modular forms are fascinating highly-symmetric functions on the complex upper-half-plane that play an important role in much of number theory. In the classical case, these functions are not hard to explicitly write down, and the coefficients of their Fourier series often contain arithmetic data. However, as soon as we generalize a bit, knowing how to write down an explicit one is a mysterious task. Hejhal’s algorithm is a method that leverages linear algebra to help us explicitly compute examples of Maass wave forms (these are slight generalizations of classical modular forms). I’ll give the big ideas of Hejhal’s algorithm while hopefully sparing you from too many gory details.
Title: Center of big mapping class group and Tits alternative
Abstract: Big mapping class group is the mapping class group of a surface that is super BIG, i.e. has infinite punctures and/or genus. We will explore the center of those big mapping class groups, and prove/show counterexamples for classic/strong Tits alternative.
Title: Social Choice and Impossibility
Abstract: We will discuss various social choice procedures, including methods of ranked choice voting. We'll talk about some properties that any reasonable social choice procedure should have, and conclude by looking at Arrow's Impossibility Theorem, which will say that there is no such procedure that satisfies all of these properties.
Title: How to draw continuously, but not smoothly
Abstract: Here's a philosophical challenge: use chalk and a chalkboard to draw a continuous, non-smooth curve. I claim this is impossible. If you disagree, then accept my challenge and come to the talk to prove me wrong. If you agree, then come to the talk to learn about exotic 4-manifolds, and the important role played by Stein structures in constructing them. If you don't care either way, that's understandable, and I hope you have a nice day. In any case, the main objects of our study (Stein structures) lie at the intersection of topology, complex analysis, and Kahler geometry. Stein structures have connections to a wide range of fields, from mapping class groups to differential equations. Our goal will be to understand why they're so important in dimension 4.
Title: Enumerative geometry and String theory
Abstract: I'll explain the connection between algebraic geometry and string theory via the quintic threefold in P^4.
Title: Modal Logic and Gödel's Proof for the Existence of “God"
Abstract: Gödel is known mainly for his incompleteness theorems. It is perhaps lesser known to us mathematicians that he once sketched an ontological proof, a proof for the existence of God, an argument that can ultimately be traced back to that of Saint Anselm and Gottfried Leibniz. For this talk, I will introduce the main language with which Gödel devised his argument, that of modal logic, which is ordinary propositional logic with some additional ingredients to allow us to speak about possibilities and necessities. I will then proceed to present Gödel's argument and discuss some of its philosophical aspects. If time permits, I will talk about some criticisms of this argument, and the recent attempts to computationally verify his proof. (Disclaimer: this talk is not intended to be in any shape or form religious or spiritual in nature, it is a talk about mathematical logic; and I am personally not religious, although it may be relevant to mention that Gödel was a deeply religious person. In any case, one is free to interpret the word "God" and related philosophical and spiritual issues in their own way, independently of the contents that I shall present.)
Title: The specter of hyperbolicity
Preliminaries: In each free homotopy class of a manifold of (strictly) negative sectional curvature, there exists a unique geodesic representative.
Abstract: As everyone knows, in each free homotopy class of a manifold of (strictly) negative curvature there exists a unique geodesic representative. We can define a function called the marked length spectrum of a metric which eats free homotopy classes and spits the length of that unique geodesic representative.
I will try to motivate the following (open!) conjecture: if two metrics of negative sectional curvature have the same marked length spectrum then they're isometric with an isometry isotopic to the identity. After that I'll try to show you as much as I can about a beautiful proof due to Otal of the conjecture in the case of surfaces. This proof has it all guys! Coarse arguments, clever tricks and best of all, dynamics.
Disclaimer: hopefully the talk won't be spooky.
Title: Any gamers in the chat?
Abstract: Have you ever wanted to be better at picking up beans than your peers? We will start off by playing a two-player bean-taking game and trying to determine its win conditions. A precise characterization of the winning positions will transition into a discussion of partitions of the natural numbers and the ways they can arise within the area of symbolic sequences. If time, we might even consider an instance of partitions in far-fetched linguistics.
Title: Gromov volume detects hyperbolic ingredients of 3-manifolds
Abstract: Gromov volume of a manifold is the coefficient infimum of cycles representing the fundamental class. If you have been to Matt’s TRG talk on Gromov volume, you’d probably remember that for hyperbolic manifolds the Gromov volume is proportional to the hyperbolic volume. In this talk, we hope to show that for a general 3-manifold with toric boundary, the Gromov volume is in fact the volume sum of hyperbolic pieces in the geometric decomposition of that manifold. In particular, the Gromov volume vanishes iff the manifold is a graph manifold.
Title: Decolonizing mathematics, a starting point
Abstract: Inspired by a discussion in the math discord, I have prepared a talk loosely based on critiquing the usage and meanings of "exotic" as an adjective in math lexicon. While deconstructing the "exotic" in mathematics was the starting point of my inquiry, this line quickly devolved into a reading rabbit hole in which I found myself asking broader questions about mathematics as a cultural institution. Needless to say, I don't have good answers to any of my questions, nor do I even feel prepared to give this talk, given its scope. However, I am excited for a lively discussion.
Title: Ethan, what do you think about?
Abstract: I’ll tell you! More precisely, I’ll give for you a variation of a research talk I gave a few weeks ago. My goal is to provide a peek at (1) the questions do researchers in my field ask, (2) the connections between my field and others, and (3) one way of preparing a research talk.
Title: How to understand Fermat's infinite descent trick?
Abstract: There was an ancient problem in number theory: How to show that there is no right triangle with area 1 and rational number sides? This was proved by Fermat using an infinite descent trick. In this talk, we will see how to understand this trick in the context of 2-descent on elliptic curves and how to use the idea to prove Mordell's theorem (Rational points on elliptic curves form a finitely generated abelian group). Finally, just for fun, we will end up with one line proof of the ancient problem.
Title: An introduction to Gauge Theory.
Abstract: This talk is going to be a summary of the stuff I have learnt this semester. I will start with connections and curvature and how one might try to relate these geometric objects to topology. Then I will tell you about the connections with "minimal" curvature and how we can use these objects to understand the underlying topology. In particular, I will talk about how we can prove the diagonalization theorem using this theory.