Posted November 1, 2017
Last modified January 7, 2025
Peter Nelson, University of Waterloo
Squaring the square
Is it possible to decompose a square into smaller squares of different sizes? The solution to this problem, which has surprising links to graph theory, linear algebra and even physics, was discovered by four undergraduate students at Cambridge University in the 1930's. I will tell the interesting mathematical story that led to this discovery.
This talk will be accessible to undergraduate students.
Posted November 1, 2017
Last modified January 7, 2025
Peter Nelson, University of Waterloo
How to draw a graph
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
Last modified January 7, 2025
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 March 14, 2018
Last modified January 7, 2025
Renling Jin, College of Charleston
A taste of Logic -- from the reasoning of a thief to a painless proof of the incompleteness theorem of Godel.
We will present some fun part of mathematical logic including a puzzle, a true paradox, and a fake paradox. The discussion will lead to Godel's Incompleteness Theorem. Godel's Incompleteness Theorem is well-known but difficult to proof. We will present a heuristic proof of the theorem which should be sufficient to understand the idea of the rigorous proof of the theorem.
This talk will be accessible to undergraduate students.
Refreshments will be served in the Keisler Lounge at 10am.
Posted April 6, 2018
Last modified January 7, 2025
Ken Goodearl, UCSB
How fast does a group or an algebra grow?
An algebraic object ``grows" from a set $X$ of generators as larger and larger combinations of those generators are taken. In the case of a group $G$, this means taking longer and longer products of generators and their inverses. For an algebra $A$ (a ring containing a field), it means taking linear combinations of longer and longer products of the generators. The growth rate of $G$ is the rate at which the number of elements that can be obtained as products of at most $n$ generators and their inverses grows with increasing $n$. The growth rate of $A$ amounts to counting dimensions of subspaces spanned by products of at most $n$ generators. These rates of growth provide important measures for the ``complexities" of $G$ and $A$, respectively. They may be given by a polynomial function or an exponential function, but there are quite a few surprises -- rates like a polynomial with degree $\sqrt 5$ can occur, or rates in between polynomial and exponential functions, whereas some other potential rates are ruled out. We will discuss the basic ideas of growth for groups and algebras; the distillation of growth rate into a ``dimension" for algebras; and the values that this dimension can take.
This talk will be accessible to undergraduate students.
Refreshments will be served in the Keisler Lounge at 3:00 pm.
Posted April 6, 2018
Last modified January 7, 2025
Ken Goodearl, UCSB
From dimension to Grothendieck groups and monoids
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
Last modified January 7, 2025
Joeseph E. Bonin, George Washington University
What do lattice paths have to do with matrices, and what is beyond both?
A lattice path is a sequence of east and north steps, each of unit length, that describes a walk in the plane between points with integer coordinates. While such walks are geometric objects, there is a subtler geometry that we can associate with certain sets of lattice paths. Considering such sets of lattice paths will lead us to examine set systems and transversals, their matrix representations, and geometric configurations in which we put points freely in the faces of a simplex (e.g., a triangle or a tetrahedron). Matroid theory treats these and other abstract geometric configurations. We will use concrete examples from lattice paths to explore some basic ideas in matroid theory and some of the many intriguing problems in this field.
This talk will be accessible to undergraduate students.
Posted November 13, 2018
Last modified January 7, 2025
Joeseph E. Bonin, George Washington University
Old and New Connections Between Matroids and Codes: A Short Introduction to Two Field
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 naturally 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 January 7, 2025
Gilles Francfort, Université Paris XIII and New York University
Spring Brake
I wish to demonstrate that minimization is a natural notion when dealing with even simple mechanical systems. The talk will revolve mainly around a simple spring brake combination which will in turn illustrate how the search for minimizers tells us things are never as simple as first thought. All that will be needed for a correct understanding of the material are basic notions of convexity, continuity as well as some familiarity with integration by parts.
This talk will be accessible to undergraduate students.
Refreshments will be served at 3:00PM in the Keisler lounge.
Posted March 21, 2019
Last modified January 7, 2025
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
Last modified January 7, 2025
Peter Jorgensen, Newcastle University
Knots
Abstract: Knots are everyday objects, but they are also studied in mathematics. They were originally envisaged as models for atoms by Lord Kelvin, and have been studied by increasingly sophisticated mathematical methods for more than 100 years. Two knots are considered to be "the same" if one can be manipulated to give the other without breaking the string. The natural question of whether two given knots are the same turns out to be highly non-trivial; indeed, this is the central question of Knot Theory.
The talk is a walk through some aspects of this fascinating area of pure mathematics and will be accessible to undergraduate students.
Refreshments will be served in the Keisler Lounge from 10 to 10:30 am.
Posted April 10, 2019
Last modified January 7, 2025
Peter Jorgensen, Newcastle University
Quiver representations and homological algebra
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 January 7, 2025
Keith Conrad, University of Connecticut
Heuristics for Statistics in Number Theory
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 January 7, 2025
Keith Conrad, University of Connecticut
Applications of Divergence of the Harmonic Series
The harmonic series is the sum of all reciprocals $1 + 1/2 + 1/3 + 1/4 + ...$, and a famous counterintuitive result in calculus is that the harmonic series diverges even though its general term tends to 0. This role for the harmonic series is often the only way students see the harmonic series appear in math classes. However, the divergence of the harmonic series turns out to have applications to topics in math besides calculus and to events in your daily experience. By the end of this talk you will see several reasons that the divergence of the harmonic series should be intuitively reasonable.
This talk will be accessible to undergraduate students.
Posted October 15, 2019
Last modified January 7, 2025
Steven Leth, University of Northern Colorado
Fixed Points of Continuous Functions in the plane
If we crumple up a map of Colorado, then place that crumpled map on top of an identical map, some point on the top map is directly over the same location on the lower map. This might not be true if we use a map of Louisiana or Michigan, which have "disconnected" portions. Also, we must be careful to not tear the map while we are crumpling it. This is an example of a consequence of the famous Brouwer Fixed Point Theorem. We will examine this beautiful mathematical result, and demonstrate how it follows from a simple combinatorial theorem about coloring vertices of triangles. The extent to which the Fixed Point Theorem can be generalized is still unsolved, and we will briefly discuss that as well.
This talk will be accessible to undergraduate students.
Posted October 15, 2019
Last modified January 7, 2025
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 January 7, 2025
Hal Schenck, Auburn University
Combinatorics and Commutative Algebra
This talk will give an overview of the spectacular success of algebraic methods in studying problems in discrete geometry and combinatorics. First we'll discuss the face vector (number of vertices, edges, etc.) of a convex polytope and recall Euler's famous formula for polytopes of dimension 3. Then we'll discuss graded rings, focusing on polynomial rings and quotients. Associated to a simplicial polytope P (every face is "like" a triangle) is a graded ring called the Stanley-Reisner ring, which "remembers" everything about P, and gives a beautiful algebra/combinatorics dictionary. I will sketch Stanley's solution to a famous conjecture using this machinery, and also touch on connections between P and objects from algebraic geometry (toric varieties).
This talk will be accessible to undergraduates. No prior knowledge of any of the terms above will be assumed or needed for the talk.
Posted October 24, 2022
Last modified January 7, 2025
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
Last modified January 7, 2025
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 January 7, 2025
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 January 7, 2025
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 January 7, 2025
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.
Posted September 18, 2024
Last modified January 7, 2025
Ian Tobasco, Rutgers University
Rigidity and Elasticity
This talk will introduce elasticity theory from the geometric point of view for students from mathematics and related disciplines. Our basic objects of study will be (nearly) length preserving maps that arise from (nearly) minimizing an energy functional having to do with the amount of work required to deform a body. After defining the basic quantities of interest, we will discuss Fritz John's seminal study of small strain maps, along with his counterexample to rigidity and its ultimate resolution in the early 2000s by Friesecke, James, and Müller. Time permitting, we will discuss a bit about elastic patterns --- fine structures that occur in naturally wrinkled or crumpled sheets that show us what we do not yet understand about the rigidity of thin elastic domains.
Posted November 27, 2024
Last modified January 7, 2025
Dave Auckly, Kansas State University
TBD
Posted November 12, 2024
Last modified February 3, 2025
Dave Auckly, Kansas State University
TBD