WOMP 2022

10:00-11:00


Topics: characteristic and minimal polynomial, Cayley-Hamilton theorem. Jordan canonical form. Dual spaces, dual transformations, inner products, and the spectral theorem.

11:00-12:00


Topics: Measures and sigma-algebras, non-measureable sets and Cantor sets, Lebesgue measure, integration, Dominated convergence, and Fubini's theorem.

10:30-11:30


Topics: Maschke's theorem. Schur's Lemma and orthogonality of characters. Some examples of character tables.

11:30-12:30


Topics: Types of convergence and continuity, Stone-Weierstrass theorem, implicit function theorem, inverse function theorem, contraction mapping theorem

13:30-14:15


Title: Building "saturated" groups -- with divine intervention

Abstract: Most people, when presented with a proof that a problem cannot be solved, will just move on. Computability theorists, however, ask what would happen if we asked the Almighty for a little bit of help. I'll discuss what it means to pray to gods in the mathematical context, give an example of a problem from my research that can't be solved without divine intervention, and quantify just how much we really need from Above. (No logic background assumed)

14:15-15:00


Title: Classical mechanics and Hamiltonian reduction

Abstract: Classical mechanics allows us to predict the behavior of a system of particles in terms of differential equations. However, solving such equations explicitly isn't always possible except in some special cases, where one can exploit the symmetries of nature to make our job easier, or at least more accessible using the tools of linear algebra and algebraic geometry. This talk will be about one such example and how it fits in the general theory of tackling such problems.

10:00-11:00


Topics: Tangent and cotangent bundle. Vector fields and differential forms. Exterior derivative, which commutes with pullback. Integration and Stokes' theorem.

11:00-12:00


Topics: Rings and modules. Prime ideals and localization. Introduction to varieties and the Nullstellensatz, emphasizing the dictionary between algebra and geometry.

13:30-14:15


Title: The universe of LMFDB and Fermat's last theorem

Abstract: LMFDB is the database of L-functions, modular forms, and related objects. It has a universe as illustrated here: https://www.lmfdb.org/universe. I will first explain with a very specific example how to go through the path: motives -- Galois representations -- automorphic forms. Then I will explain how this path implies Fermat's last theorem. If there's time left, I will say a few words on my research.

14:15-15:00


Title: Percolation can help a feeble fish

Abstract: I'll discuss how results about percolation, a discrete process from statistical physics, can yield a quantitative rate of homogenization in a random environment for a model of combustion of a flammable gas. The presentation will be mostly self-contained, so you don't need to know what these words mean.

10:00-11:00

Room: Ryerson 251
Topics: Irreducible polynomials, field extensions, splitting fields, algebraic closure, normal extensions through examples -- with a goal of presenting the Galois correspondence theorem.

11:00-12:00

Room: Ryerson 251
Topics: Definition of homotopy and of the fundamental group. Some example computations. Covering spaces and deck transformations, towards the Galois correspondence for covers.

13:30-14:15

Room: Ryerson 251

Title: The Mathematician’s Terrible, No-Good, Very Bad Example

Abstract: The limiting behavior of a dynamical system is often completely divergent from approximations using finite time behavior— often in ways that are both surprising and computationally insurmountable. A classic example of such a system is the so-called “standard map,” a one-parameter family of diffeomorphisms of the torus that has eluded classification for decades. Here, we will review why it is so often difficult to understand the behavior of a dynamical system, what I mean by “behavior” in the first place, and present a criterion for eliminating this difficulty.

10:00-11:00


Topics: Curves on surfaces, First and second fundamental forms, the Gauss map, geodesics, principal curvatures-- with a goal of presenting the Gauss-Bonnet theorem.

11:00-12:00


Topics: Complex plane and Riemann sphere, Cauchy-Riemann equations, Laurent series, analytic functions-- with a goal of presenting the Cauchy integral formula and Liouville's theorem

13:30-14:15


Title: Cayley Bacharach sets lie on curves

Abstract: I will discuss a theorem that I proved showing that under some hypotheses Cayley Bacharach sets lie on low degree curves. I will try to keep this relatively self contained. However I will assume that the audience knows what projective space is and is familiar with the Nullstellensatz.

14:15-15:00


Title: Smooth Structures and Group Actions

Abstract: I will talk about smooth structures on manifolds and how we can construct exotic structures when we require compatibility with group actions.