Graduate Student Advisory Committee (GSAC) Colloquium Schedule:

Welcome back!

There will be no colloquium this week.

The ADE Classification

This talk will explore a connection between a collection of seemingly unrelated mathematical objects. These objects come from diverse fields of mathematics and physics, ranging from linear algebra, singularity theory and Lie theory to conformal field theory and string theory. The common thread connecting these classes of objects is a collection of graphs called the simply laced Dynkin diagrams. It is possible to show bijections between the various classes of objects admitting this "ADE classification," but these constructions do not give satisfactory intuition as to why these objects are related. The deeper reason behind these connections remains mysterious, and motivates mathematicians to continue exploring this phenomenon. In this talk, I will discuss a handful of the classes of objects that fit into this classification and show how they all lead to the Dynkin diagrams.

An Introduction to the Schramm-Loewner Evolution

Probability theory is a very useful tool for modelling how simple interactions between objects at the molecular level produce the complicated phenomenon that we often observe in objects at the everyday, macroscopic level. An early example was the Ising model, which is an attempt to explain how the electrons on one side of a magnet know to spin in the same direction as the magnets on the other side, even though they can only "talk" to each other through their neighbors. A model of the interactions between molecular objects is usually easy to construct on a lattice, but to understand it on the macroscopic level requires taking a scaling limit; this is a version of the model when the spacing between lattice sites has gone to zero. Scaling limits for basic models such as the simple random walk have been well understood for over a century, but scaling limits for the Ising model and other related phenomenon were only discovered around 2000. This is the Schramm-Loewner evolution, which has lead to revolutionary new understanding in probability theory, complex analysis, combinatorics, and theoretical physics, and has led to the awarding of 2 Fields Medals in the last 10 years. This talk will give a simple and straightforward introduction to the Schramm-Loewner evolution.

Hipsters and Gillespie: Numerically Simulating a Stochastic Model of Real Life Trends

Last year, Jonathan Touboul published the paper "The Hipster Effect: When Anticonformists All Look the Same", a short but rather mathematically complex paper that investigated delay-induced Hopf bifurcations in a model of lifestyle trends sweeping across a population. Despite this complexity, the name and conclusion of the paper made it a viral sensation, sweeping across Facebook and Twitter feeds of non-mathematicians. We will take a closer look at the model presented by way of numerical simulations. This talk will introduce tools and topics seen throughout applied mathematics, including the Gillespie algorithm, bifurcations, and MATLAB coding. By the end, we will hopefully be able to answer the age old question: why do all hipsters look the same?

Special Department Colloquium

There will be no GSAC colloquium this week.

Trace Maps, Ramification, and Singularities

Suppose one has a finite extension of fields of rational functions K inside L. There is a canonical K-linear map from L back to K called the trace map (induced by taking the K-linear trace of multiplication by elements of L). This trace map tells you subtle relations between the two spaces that K and L are fields of rational functions on. For instance -- where one space is a covering space of the other. If these spaces are singular (i.e. not manifolds), even more interesting things can happen and these trace maps can detect that as well.

The Riemann-Roch Theorem for Fields of Characteristic p

The Riemann-Roch theorem is a critical tool in the fields of complex analysis and algebraic geometry. In this talk I will formulate the Riemann-Roch theorem in the case of fields of positive characteristic. I will then explore its connection with number theory, in particular with L-functions.

Computational Topology and Big Data Analysis

In this talk, we discuss the premise of topological data analysis, and introduce some tools from topology to try to answer the question, “what is the shape of my data?” We start by describing some topological spaces associated to “point cloud data,” indexed by an increasing sequence of parameters. We then apply tools from algebraic topology to compute some robust features of this collection of spaces that persist as our indexing set varies. Finally, we present a case study analyzing the local behavior of spaces of natural images.

An Invitation to Categorical Thinking

In this talk we will talk about categories. These objects are encountered naturally in various fields of mathematics, algebraic topology in particular. However categorical thinking is applied in computer science and computational fields of math such as topological data analysis. We will define categories, introduce functors as "structure preservers" and give examples. Then we describe how combinatorial models are built for simplicial complexes. If time permits, we will display a categorical proof as an example of effectiveness of category theory.

Spring break!

There will be no colloquium this week.

A Brief History of Rings

Commutative algebraists study abstract structures called rings, modules, and ideals. This talk will explore how these structures evolved by studying some conjectures of Fermat. Along the way we'll introduce quadratic forms, class numbers, and the prime decomposition of ideals. The only prerequisite for this talk is to know what a prime number is.

There will be no colloquium this week.

Graph Folding

Dirty Deeds Thunder Chief: The Mathematics of Music Recognition Software

Have you ever wondered how Shazam is able to identify a piece of music recorded on your phone in a noisy pub or how SoundHound is able to show you the lyrics in sync with the song you’re hearing? I will talk about how some simple mathematical ideas reduce the difficult problem of matching a music sample with an entry in a large database of song to the simple task of finding peaks in a distribution.

The Physics of Beer Tapping

Have you ever participated in the popular bar prank known colloquially as beer tapping? Did it cross your mind how this foaming over process occurs? I will talk about how this phenomenon can be divided into three stages involving different time scales and how understanding this subtle mechanism sheds insight into the dynamics of geological phenomenon.

Organizational Meeting

We will organize the GSAC subcommittees and establish contacts for prospective and incoming students. Please attend!