Posted November 1, 2017

1:30 pm – 2:30 pm Lockett 284
Peter Nelson, University of Waterloo

How to draw a graph

Abstract: Given a network of points and edges that can be drawn in the plane without crossing edges, what is the best way to actually draw it? Can such a network always be drawn with just straight lines? I will discuss and (mostly) prove a beautiful theorem of William Tutte that answers this question using intuitive ideas from physics.

Posted March 14, 2018

1:30 pm – 2:20 pm tba
Renling Jin, College of Charleston

Can nonstandard analysis produce new standard theorems?

The answer is yes. Nonstandard analysis which was created by A. Robinson in 1963 incorporates infinitely large numbers and infinitesimally small positive numbers consistently in our real number system. But the strength of nonstandard analysis in the research of standard mathematics has not seemed to be sufficiently appreciated by mathematical community. In the talk, we will introduce two parts of the work done by the speaker and his collaborators on the standard combinatorial number theory using nonstandard analysis. In each of these two parts new standard theorems that were proved by nonstandard methods will be presented.

The audience is not assumed to have prior knowledge of nonstandard analysis. Refreshments will be served in the Keisler lounge at 1:00 pm Posted April 6, 2018

Last modified April 30, 2020

Ken Goodearl, UCSB

From dimension to Grothendieck groups and monoids

Abstract: In trying to generalize the concept of ``dimension'''' from finite dimensional vector spaces to structural size measures for other classes of mathematical objects, one quickly arrives at the idea that such ``sizes'''' should be elements of some abelian group, so that (at the very least) sizes can be added. The natural group to use in linear algebra is $\bf Z$, but in general there is no obvious group at hand. Grothendieck pointed out how to construct an appropriate group as one satisfying a certain universal property. Typically, one wants to not only add but compare ``sizes'''', in the sense of inequalities. To accommodate comparisons, a combined structure is needed -- an abelian group which is equipped with a (compatible) partial order relation. On the other hand, demanding subtraction for ``sizes'''' is sometimes asking too much, and ``sizes'''' should take values in a monoid rather than a group.

We will introduce the above concepts and constructions in the context of modules over a ring, and we will discuss various examples.

Refreshments will be served in the Keisler lounge at 3:00 pm.

Posted November 13, 2018

1:30 pm – 2:20 pm Lockett 235
Joeseph E. Bonin, George Washington University

Old and New Connections Between Matroids and Codes: A Short Introduction to Two Field

Abstract: The theory of error-correcting codes addresses the practical problem of enabling accurate transmission of information through potentially noisy channels. The wealth of applications includes getting information to and from space probes, reliably accessing information from (perhaps scratched or dirty) compact discs, and storage in the cloud. There are many equivalent definitions of a matroid, each conveying a different perspective. Matroids generalize the ideas of linear independence, subspace, and dimension in linear algebra, and cycles and bonds in graph theory, and much, much more. The wealth of perspectives reflects how basic and pervasive matroids are. Matrodis arise naturallly in many applications, including in coding theory. The aim of this talk is to give a glimpse of both of these fields, with an emphasis on several ways in which matroid theory sheds light on coding theory. One of these applications of matroid theory dates back to the 1970's; another is a relatively new development that is motivated by applications, such as the cloud, that require locally-repairable codes.

Posted March 21, 2019

Last modified March 2, 2021

Gilles Francfort, Université Paris XIII and New York University

The mysterious role of stability in defective solids

Adjudicating the correct model for the behavior of solids in the presence of defects is not straightforward. In this, solid mechanics lags way behind its more popular and at- tractive sibling, fluid mechanics. I propose to describe the ambiguity created by the on- set and growth of material defects in solids. Then, I will put forth a notion of structural stability that helps in securing meaningful evolutions. I will illustrate how such a notion leads us from the good to the bad, and then to the ugly when going from plasticity to fracture, and then damage. The only conclusion to be drawn is that much of the mystery remains.

Refreshments will be served at 3:00PM in the Keisler lounge.

Posted April 10, 2019

3:30 pm – 4:30 pm Lockett 277
Peter Jorgensen, Newcastle University

Quiver representations and homological algebra

Abstract: The word "quiver" means oriented graph: A graph where each edge has an orientation, i.e. is an arrow from one vertex to another. A representation of a quiver Q associates a vector space to each vertex of Q and a linear map to each arrow of Q. The representations of Q form a so-called abelian category. It is also possible to construct triangulated categories of quiver representations, and abelian and triangulated categories are the basic objects of homological algebra. The talk will consider a simple example and present a number of fundamental properties of abelian and triangulated categories of quiver representations. Refreshments will be served in the Keisler lounge from 3:00 to 3:30pm

Posted October 15, 2019

Last modified October 16, 2019

Keith Conrad, University of Connecticut

Heuristics for Statistics in Number Theory

Abstract: Last month the sum of three cubes was in the news: mathematicians discovered with a computer how to write 42 as a sum of three cubes and then how to write 3 as a sum of three cubes in a new way; it's in fact expected that both 42 and 3 are a sum of three cubes in infinitely many ways. There are many other patterns in number theory that are expected to occur infinitely often: infinitely many twin primes, infinitely many primes of the form $x^2 + 1$, and so on. The basis for these beliefs is a heuristic way of applying probabilistic ideas to number theory, even though there is nothing probabilistic about perfect cubes or prime numbers. The goal of this talk is to show how such heuristics work and, time permitting, to see a situation where such heuristics break down.

Posted October 15, 2019

Last modified March 2, 2021

Steven Leth, University of Northern Colorado

An introduction to the use of nonstandard methods

Nonstandard methods utilize the technique of viewing relatively simple structures that we wish to study inside much richer structures with idealized properties. Most famously, we might look at nonstandard models of the real numbers that contain actual "infinitesimal" elements. The existence of the idealized structures gives us access to powerful tools that can often support more intuitive proofs than standard methods allow. We will look at examples of simple nonstandard arguments in several different settings, as well as a few recent results obtained using these methods.

Posted October 24, 2022

Last modified October 31, 2022

Hal Schenck, Auburn University

Numerical Analysis meets Topology

One of the fundamental tools in numerical analysis and PDE is the finite element method (FEM). A main ingredient in FEM are splines: piecewise polynomial functions on a mesh. Even for a fixed mesh in the plane, there are many open questions about splines: for a triangular mesh T and smoothness order one, the dimension of the vector space $C^1_3(T)$ of splines of polynomial degree at most three is unknown. In 1973, Gil Strang conjectured a formula for the dimension of the space $C^1_2(T)$ in terms of the combinatorics and geometry of the mesh T, and in 1987 Lou Billera used algebraic topology to prove the conjecture (and win the Fulkerson prize). I'll describe recent progress on the study of spline spaces, including a quick and self contained introduction to some basic but quite useful tools from topology, as well as interesting open problems.

Posted August 21, 2023

1:30 pm Lockett 138
John Etnyre, Georgia Institute of Technology

Invariants of embeddings and immersions via contact geometry

There is a beautiful idea that one can study topological spaces by studying associated geometric objects. In this talk I will begin by reviewing the Whitney-Graustein theorem that tells you precisely when two immersed curves in the plane can be deformed into each other. We will then see how this result can be interpreted in terms of contact geometry and Legendrian knots, so we see how one can turn a topological problem (deforming immersed curves) into a geometric one (isotoping Legendrian knots). Along the way I will give a brief introduction to contact geometry and end by discussing how one can try to study immersions and embeddings in all dimensions using contact geometry.

Posted July 31, 2023

Last modified August 21, 2023

John Etnyre, Georgia Institute of Technology

The Job Process

In this talk I will discuss academic jobs for people with a math PhD, focusing on postdoctoral positions and beginning tenure track jobs. We will discuss what these jobs entail and how to apply for them, and most importantly, what you can be doing now to maximize your chance at getting such a position.

Posted September 11, 2023

Last modified October 1, 2023

Allison Miller, Swarthmore

Algebra and topology in dimension four.

In order to understand and distinguish complicated topological spaces, we often compute algebraic invariants: if two spaces have different invariants, then they are certainly different themselves. (For example, for those who recognize them: the Euler characteristic of a surface, the Alexander polynomial of a knot, the fundamental group of a manifold.) But one might also wonder about the converse: are there algebraic invariants that completely determine something about the topological structure of a space? We will talk about this question in dimension four, where the answer is a resounding "Sometimes!". Knots, surfaces, and 4-dimensional spaces will all play important roles.

Posted March 27, 2024

Last modified April 2, 2024

Dr. Lisa Kuhn, Southeastern LA University

PDE Structures: Finite Elements, Data Science and the Search for Efficient Solutions

Recent advancements in smart materials have significantly influenced the complexity of partial differential equation (PDE) structures, which frequently exhibit material discontinuities and intricate boundary conditions, especially with PDE systems. As we transition further into the age of artificial intelligence, researchers are increasingly exploring machine and deep learning methodologies to derive PDE solutions. However, success has been limited when considering control of distributed parameter systems which is supported by finite element theory. This presentation will present recent findings in generating PDE solutions utilizing both finite elements with adapted bases and hybrid techniques while striving to uphold infinite-dimensional distributed parameter control theory. The discussion will include results of one and two-dimensional clamped structures employing Euler-Bernoulli beams and isotropic plates. Computational methodologies such as modified higher-order bases and neural finite elements will be elaborated upon.