Graduate Student Advisory Committee (GSAC) Colloquium Schedule:

Title:Connections Between Geometry and Group Theory

In this talk, we will explore some ways that geometry can help us to understand the algebra in certain groups. In particular, we will explore the examples of the integers and the group SL2(Z). Our goal will be to use geometry to prove that SL2(Z) is a free product with amalgamation (don't worry if you don't know what this means!). The only prerequisites for the talk are knowing how to add and how to multiply matrices.

Archimedes' Cattle Problem

In 1773, a problem concerning cattle was discovered that Archimedes had posed to Eratosthenes. The exact solution was found in 1880 by Amthor, but it wasn't until the advent of computers that the number of cattle could actually be calculated. Spoiler alert: it has 206,545 digits. In this talk, the solution will be given with Pell equations and continued fractions discussed along the way.

Symmetry and Representations

Symmetry appears everywhere in mathematics and mathematical applications. Representation theory evolved as a branch of mathematics to systematically exploit such symmetries. The purpose of this talk is to explain a little bit about representations, concentrating mostly on applications to interesting physical problems.

Introduction to Spatial Statistics

Given observations of quantities that vary over space, such as temperature, pollutant concentration, or wind speed, we'll talk about three different methods of estimating the quantity at unobserved locations: 1. Kernel-based interpolation, 2. Kriging using the Matheron estimator, and 3. Kriging using maximum likelihood estimation.

Platonic Solids and their Symmetry (or Polytopes Gone Wild)

The sequence \infty,5,6,3,3,3.... tells us how many regular polytopes there are in each dimension. Dimensions 2, 3, and 4 are exceptional. Thinking of the real plane as the complex numbers, one can realize the symmetries of a 2-dimensional polytope as a two dimensional polytope itself! For 3-dimensional polytopes (also known as platonic solids) we can realize the symmetries as unit quaternions and plot them on the unit sphere in 4-space. We will see in the talk which objects arise this way, and find an unusual irregularity.

The Hilbert Scheme

Polynomials (in several variables) are graded by degree. Homogeneous polynomials of a fixed degree define hypersurfaces in projective space as their locus of zeroes. Sets (or, more properly, ideals) of polynomials define the subsets of projective space that are the objects of study of algebraic geometry. A collection of theorems by Hilbert establish some basic information about such subsets (subvarieties) of projective space and the ideals of polynomials that cut them out. A natural and audacious question arises. Can we classify ALL such ideals/subsets at once? This is what the Hilbert scheme does. I'll talk about this, focusing on the example of the twisted cubic in projective threespace.

What do 3-manifolds look like and what does geometric group theory have to do with it?

Abstract: Classification of surfaces has been known for a long time. Remarkably, every closed connected surface comes with one of 3 standard geometries, and can be obtained from just two building blocks by a "sum" operation. How much of this can be generalized to 3-manifolds? The answer is "a lot", and the last piece of the puzzle came into place just last year, and it uses geometric group theory.

Bounds on the Volume Fraction of an Inclusion in a Body

In this talk, I will describe an application of the splitting method introduced by Milton and Nguyen. In the splitting method, we use measurements of the voltage and current on the boundary of a region \Omega to estimate the volume fraction of an inclusion inside \Omega. (The volume fraction of an inclusion is defined as the volume of the inclusion divided by the volume of \Omega). For example, one may be able to use this method to estimate the volume fraction of dead cells inside an organ before it is transplanted into a patient.

Going, Going.... Gone?

What happens to the information that falls into black holes? Is it lost forever? In this non-technical talk, we'll see how Stephen Hawking's (incorrect) answer to these questions led to a decades-long debate between some of the world's best physicists, and to the seemingly outrageous discovery that our observable universe is really just a hologram. Moreover, we'll learn what the ever-mysterious String Theory has to do with the solution, and how the Universe may, in fact, have many more dimensions than most people think! Absolutely no mathematics or physics background required!

Two Puzzles

Abstract: A lion is chasing a gladiator in a closed circular arena. The lion is past its prime, and it soon transpires that the lion and the gladiator have the same top speeds. Can the man continue running without being caught, or does the lion always catch him after some time? In this talk we will consider two famous mathematical puzzles. You should be able to follow everything if you know what a convergent series is.

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