Fall 2024
September 12 -- Rachmiel Klein
Title: The building blocks of spiderwebs
Abstract: Believe me when I say that there are some unforgiving infinite type surfaces out there, let alone locally finite graphs. In this talk I plan to share a compactified notation that I have been working on that describes all locally finite graphs up to proper homotopy. The same notation would work for infinite type surfaces. I then plan to share a few structure theorems I found and give examples of locally finite graphs with exotic properties. This talk relies on a bit of topology and includes some set theory as well!
September 19 -- Chengyang Wu
Title: Stable ergodicity and centralizers
Abstract: Our goal is to prove that a non stably ergodic affine automorphism on a compact homogeneous space admits an affine approximation with an infinite dimensional centralizer. In this talk we’ll try to explain a baby version of the aforementioned result.
September 26 -- Neha Goregaokar
Title: Characteristic polynomials for graphical arrangements via projection statistics and source components
Abstract: The characteristic polynomial is traditionally defined for hyperplane arrangements using the Mobius function of the intersection poset. Recently, Lofano-Paolini and Kabluchko gave an expression for the characteristic polynomial in terms of a projection statistic on the regions. In this talk, I will show that for hyperplane arrangements, this projection statistic has a combinatorial interpretation in terms of source components of acyclic orientations.
October 10 -- Kamryn Spinelli
Title: Period integrals and a Galois criterion for the Chevalley restriction property
Abstract: In this talk we will discuss the period integrals of two families of Calabi-Yau varieties and show how they naturally lead to questions in invariant theory. We will begin by motivating the investigation of period integrals and discuss some results so far in writing down explicit formulas for the periods. The contrasts between the pictures for the two families illustrate that there is some interesting invariant theory going on here related to some famous results from the latter half of the 20th century. We will explain how the ideas coming from period integral problems give rise to a new approach to such Chevalley restriction-type theorems. This is joint work with Bong Lian.
October 16 -- Sarah Dennis
Title: Numerical simulation of low Reynolds number fluids in cornered domains
Abstract: We consider models for a low Reynolds number (highly viscous) fluid flowing through a textured pipe. The pressure and velocity of a fluid are related by a system of partial differential equations known as Navier-Stokes equations. In general this system has no exact solution. I will present a finite difference iterative method used to solve the 2D biharmonic form of the Navier-Stokes equations. Under the additional assumption from lubrication theory that the fluid geometry is long and thin, the 2D Navier-Stokes equations can be reduced to the 1D Reynolds equation; I will briefly discuss the piecewise analytic solution for Reynolds equation.
There is noticeable discrepancy between flow structures predicted by the two models. Sudden changes in the surface gradient (sharp corners) produce vortical structures appearing in corners. These corner eddies are observed only with the Navier-Stokes model and not with Reynolds model. I will present some early results for error in fluid resistance (the ratio of global pressure drop to flux) between Reynolds and Navier-Stokes models. In general, Reynolds model underestimates the fluid resistance, and error in resistance is proportional to the magnitude and frequency of large surface gradients. This suggests raises concerns that geometries deemed optimal (resistance minimizing) via analysis of Reynolds model may not be optimal at all.
October 22 -- Kaitlin Ragosta
Title: Largest acylindrical actions and finite-type Artin groups
Abstract: The mapping class group of a surface has been an object of study for decades. It motivated the study of a property called acylindrical hyperbolicity. I will define acylindrical hyperbolicity in the talk, but roughly speaking, it refers to the existence of a nice enough group action on a hyperbolic metric space. An acylindrically hyperbolic group will generally admit lots of nice actions on hyperbolic spaces, but sometimes one of these actions will encode more information about the group than all the others. I will discuss some notable examples of this phenomenon and some interesting consequences. If time permits, I will introduce a result which suggests that finite-type Artin groups may also admit such an action.
October 31 -- Rachmiel Klein
Title: Group Conjugation in the Mapping Class Group of Graphs
Abstract: The mapping class group of a locally finite graph is the set of proper homotopy equivalences up to proper homotopy. It is meant to be the equivalent of the mapping class group of an infinite type surface one dimension lower, but it also mirrors the group of outer automorphisms of the free group, thus establishing a "Big Out(F_n)." In this talk, I plan to define the mapping class group for locally finite graphs while giving motivation. I will then give examples of proper homotopies, and introduce methods one can use to determine which mapping class groups contain dense conjugacy classes.
November 7 -- Vasiliy Nekrasov
Title: Geometry and Dynamics of Continued Fractions
Abstract: In this talk, we look at continued fractions from geometrical point of view. We start with two vectors, horizontal and vertical, and see how we can get the continued fraction expansion of any given numbers just through a sequence of pictures. Then, using even more pictures, we prove some classical algebraic facts almost without any calculations. If time permits, we will change gears, and look at the geometric means of the partial quotients and try to understand the reason why almost all of them behave in a very similar way.
December 5 -- Joshua Perlmutter
Title: Definition of a Hierarchically Hyperbolic Space
Abstract: In this talk I will define a hierarchically hyperbolic space. This will take the full hour.