Spring 2017 Colloquium

Talks (unless otherwise indicated) are in Davis 301 from 4–5 PM on Mondays. Refreshments begin at 3:30 on the second floor of Davis.

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Monday, January 30
Davis 201, 3:45
Diana Davis
Williams College

Three Flavors of Billiards

The mathematical study of billiards extends well beyond the popular table game. I’ll tell you about three different types of billiards (inner billiards, outer billiards and tiling billiards), and I’ll explain some of the research I’ve done in two of these areas.

Wednesday, February 1
Davis 201, 3:45
William R. Green
Rose-Hulman Institute of Technology

Time Decay for Schrödinger and Dirac Equations

Differential equations are ubiquitous in the sciences due to their ability to model real world phenomena. Unfortunately, many physically relevant differential equations are very difficult or impossible to solve in terms of elementary functions. The Schrödinger equation is a partial differential equation that models the behavior of sub-atomic particles. Despite the inherent difficulties in both solving this equation, and interpreting the meaning of these solutions, many advances have been made using quantum theory that arises from the Schrödinger equation.
In this talk, we will survey some mathematical ideas needed to study the behavior of sub-atomic particles. In particular, we will investigate what we can say about solutions to the underlying partial differential equations without solving them. We will also discuss what happens to and how to model a sub-atomic particle as it moves at relativistic speeds.

Friday, February 3
Davis 201, 3:45
Evan Randles
University of California Los Angeles

Convolution Powers of Complex-Valued Functions

Central to random walk theory is the study of iterative convolution powers of probability distributions on the square lattice. When such distributions are allowed to be complex-valued, their convolution powers exhibit surprising and disparate behaviors not seen in the probabilistic setting. In this talk, I will describe recent work on this topic, briefly discuss its connections to statistics and partial differential equations, and outline directions for future research. This is joint work with Laurent Saloff-Coste.

Monday, February 6
Davis 201, 3:45
Ekaterina Shemyakova
SUNY New Paltz

Differential Operators on the Superline, Berezinians and Darboux Transformations

Supermathematics is a standard tool for Quantum Physics that allows us to describe both fermions and bosons. The name comes from “supersymmetry.” Darboux transformations are a tool in integrable systems theory, which seeks exact solutions of nonlinear and linear partial and ordinary differential equations. In this talk I shall present my results on developing Darboux transformations in supergeometry. I shall touch on my results about superdeterminants, such as a non-trivial analogue of row (column) expansion. Overall I aim to introduce the audience to the world of supermathematics, show analogues of familiar concepts and give some idea of my work. I shall not be assuming any prior knowledge besides Linear Algebra and Calculus.

Thursday February 16
3:30 PM
Wes Viles
Dartmouth College

Network Data Analysis: Inference and Characterization

The analysis of network data is widespread across scientific disciplines. In applied network analysis, a common approach is to gather basic measurements on the interactions among elements in the system, construct a network-based representation of the system, and numerically summarize the structure of the resulting network graph. We address the empirically relevant problem that errors made in assigning links between nodes accumulate in the reporting of motif count statistics in characterizing the network as a whole. We demonstrate the utility of network characterization in two clinically relevant applications: (i) the high-order interactions among microbes constituting the ecological network of the human gut microbiome and (ii) the rate of percolation in the human brain network at the onset of an epileptic seizure.

February 27 Jan Holly
Colby College

Dizzy Patients and a Mathematical Playground of Probabilities on Spheres

Interesting mathematical topics often arise in real-life problems. This talk will begin with a description of research on vestibular patients at Baylor College of Medicine and a comparison with previously published data on eye movements, and then will dive into mathematical questions—and even some answers—that arose. Above all, this talk will point out some intriguing questions about probabilities on spheres that lead to a whole playground of mathematical (and statistical) questions.

March 13 Scott Taylor
Colby College

Constrained Flexibility: Some Surprising Connections between Geometry and Topology

Length, area, curvature are geometric properties. An object’s topological properties, on the other hand, are those which don’t change when the object is stretched, twisted, or bent. Curves in the plane or in 3-dimensional space exhibit an interesting variety of both geometric and topological properties. I’ll survey they ways these properties are related to each other, tracing a line of investigation which begins in the early 20th century and extends to very recent results concerning how twisted knots can be. (Image by Conan Wu)

March 27 Michael Barany
Dartmouth College

Inclusion, Exclusion, and the Theory and Practice of “Truly International” Mathematics

Group photo at the 1950 ICM
In theory, just about anyone can be a mathematician. Theorems and proofs, for the most part, don’t discriminate based on race, class, gender, disability, national origin, or anything else, at least in princniple. Historically, however, the field and profession has been open to very few. Mathematicians have grappled in many different ways with this gap between an ideal of openness and and a reality of exclusion and even outright discrimination. I will show how American mathematicians took leadership of the international mathematics community over the period between 1920, when they first proposed to host an International Congress of Mathematicians, and 1950, when they finally brought one to fruition. American mathematicians tried to reshape mathematics as a more interconnected and inclusive discipline, succeeding in some ways and failing in others. In particular, they tried to create what they called a “truly international” discipline. I will explain how this ambiguous and shifting phrase helped them navigate a wide range of political, financial, and other obstacles, while covering over persistent problems and blindspots.

April 3 Thomas Pietraho
Bowdoin College

Fast Matrix Multiplication: An Adventure in Machine Learning

Neural nets have been used to attack a variety of problems, including image classification, speech recognition, and autonomous driving. After detailing how neural nets can be used in settings of this sort, we will use them to solve a problem in algebra instead.

April 10 Tom Hulse
Colby College

Square Pegs in a Round Circle Problem

In this talk we’ll introduce Gauss’s Circle Problem, the centuries-old conjecture concerning estimates of the number of integer-coordinate points inside a circle as we vary the radius. We’ll also discuss variants on this problem in higher dimensions, and how we can get information from the Fourier coefficients of a particular theta function with interesting analytic properties. Some joint work with Chan Ieong Kuan, David Lowry-Duda, and Alexander Walker may also be presented.

April 17 Malia Kawamura ’14, Anne Crumlish ’98, Ben Hauptman ’09, Tom Hulse ’07

Mathematics and Statistics Alumni Panel


May 1 David Krumm
Colby College

Japanese Temple Geometry

During the period of Japanese history known as the Edo period, Japan followed a strict isolationist foreign policy in which travel outside the country was forbidden and cultural exchange with European nations was effectively cut off. Coupled with the peace and economic prosperity enjoyed during the Edo period, this “closed country” policy led to a flourishing of uniquely Japanese arts and culture. One of the curious practices that developed during this time was a tradition of hanging wooden tablets inscribed with geometry problems in Buddhist temples and Shinto shrines. In this talk we will discuss the Japanese mathematics of the Edo period generally and the sangaku tablets in particular, including the one problem from these tablets that remains unsolved.