The Graduate Student Seminar is put on by the Mathematics Graduate Student Association . GSS meets approximately every other Monday.
All talks are at 4:15 in HH231 (Fall)/ HH229 (Spring) unless otherwise noted.
To volunteer to give a talk, or for anything else regarding the seminar, contact Justin Hilyard.
|Monday, September 5||Julia Knight||Comparing classes of countable structures|
|Monday, September 19||John Engbers||Graph theory — to the extreme!|
|Monday, October 3||Megan Patnott||A Story of a Theorem|
|Monday, October 24||Nicole Kroeger||Counting Abelian Ideals in a Borel Subalgebra|
|Monday, November 7||Stephen Flood||Ramsey's theorem: finding order in chaos.|
|Monday, November 21||Youngho Yoon||From morphisms to spectral sequences with schematic pictures|
|Monday, December 5||Jeffrey Madsen||Lattice polygons and the number 12|
|Monday, January 30||Victor OcasioGonzalez||Polygonal Rooms Not Illuminable From Every Point|
|Monday, February 13||Brian Shourd||Basic Representation Theory of Finite Groups|
|Monday, February 27||Martha Precup||Paving regular nilpotent varieties by affines|
|Monday, March 19||Ryan Grady||Manifolds and the craziness of dimension 4|
|Monday, April 2||Ryan Thompson||Blow-up Phenomena of the Camassa-Holm equation|
|Monday, April 16||Sarah Cotter||Directed families and definable sets|
|Monday, April 30||Douglas Smith||Sharkovsky's Theorem|
Classifying objects up to some important notion of sameness is important in all branches of mathematics. One important notion of sameness for algebraic structures is isomorphism. In mathematical logic, there are different approaches for comparing classes of countable structures and saying which is more difficult to classify, up to isomorphism. I will describe some of these approaches: cardinality, Borel cardinality, effective cardinality, and a restriction of effective cardinality to computable structures. I will give examples to show what the different approaches say about some familiar classes of structures.
Extremal graph theory is a extensive and largely accessible area of mathematics which asks the question "out of a certain family of graphs, which one has the largest or smallest _______?" We illustrate this field through the lens of finding the largest number of independent sets of a fixed size in different families of graphs, presenting both results and open questions. All graph theory terms will be explained with tender care.
We follow the development of a single theorem, beginning in the fourth century A.D. and ending in the twentieth century. Along the way, we'll pick up some classical tools and ideas of algebraic geometry.
This talk is based on the paper "Cayley-Bacharach Theorems and Conjectures" by David Eisenbud, Mark Green, and Joe Harris.
Studying the structure of Lie algebra is an important part of Lie theory. In this talk we will cover some of the basic facts about the structure of Lie algebra. We will then learn several methods that allow us to the number of abelian ideals of a Borel subalgebra for the Lie algebra sl(3,C). We will explore Peterson's theorem which gives a simple way to count the number of abelian ideals in a Borel subalgebra.
As T.S. Motzkin put it, the theme of Ramsey theory is that "complete disorder is impossible." More realistically, Ramsey theory gives us results like the following: any large enough graph must either contain a large complete subgraph (all vertices connected) or a large independent set (no vertices connected). In this talk, we will introduce Ramsey's theorem and describe one of the basic methods used to extract this type of order. We will also comment on some interesting connections between Ramsey theory and mathematical logic.
Homological methods are very nice machinery for generating invariants. Spectral sequences greatly enhance these methods. But it is not easy to understand these sequences only by reading standard texts. I will explain how spectral sequences naturally arise using intuitive pictures. If time allows me, I will show you how to use it.
We will see an elementary geometric proof of a neat little theorem about lattice polygons featuring a curious appearance of the number 12. We then explore the motivation for the theorem coming from the study of toric surfaces, which leads to an outline of another proof using an advanced theorem from algebraic geometry. Finally, we will see a topological proof that hinges on a basic theorem about modular forms. This talk is based on the paper "Lattice polygons and the number 12" by Bjorn Poonen and Fernando Rodriguez-Villegas.
Imagine two people in a (polygonal) room made out of reflective material; can one person light a match and still not be seen by the other? Spoiler Alert!! The answer is Yes. This problem was first introduced by Ernst Straus in the early 1950's and remained open over forty years until George Tokarsky published a solution in 1995. This talk will focused on providing the proof of the solution in all its mind-numbing simplicity and later extrapolating the results to three dimensions.
Representation theory is an approach to solving many types of problems, where an algebraic object (such as a group) is "represented" as a set of endomorphisms of a given vector space. In finite dimensions, it allows one to study the elements of the algebraic object as though they are matrices. One of the easiest examples of representation theory is representations of finite groups. This talk will give the basic definitions required to begin such a study, and show some initial results and proofs, so as to give a (very brief) introduction to the types of problems that can be solved. If there is time, I will also cover some more advanced results that can be proven simply using representation theory.
Regular nilpotent Hessenberg varieties are a family of subvarieties of the flag variety. We will describe a paving of these varieties by affines for the algebraic group GLn(ℂ). If you didn't understand the last two sentences, do not fear! Working with GLn(ℂ) will allow us to gain insight into the structure theory of algebraic groups without too much technicality. We'll define all the terms and review some well known results about the structure of the flag variety along the way.
We discuss some basic results about classification of manifolds. An interesting dimension is four and we illustrate the difference between topological and smooth structure. Fun will be had by all.
We consider the Cauchy problem for the Camassa-Holm equation and discuss the wave breaking property of this shallow water wave equation.
We'll look at two different ways of working with families of sets. Directed families provide a very visual way of looking at certain collections of sets, while definability gives us a very precise description of what goes into certain sets. We'll see some examples of both topics from around mathematics, and then look at a way to tie the two ideas together.
We define a point p to be a periodic point of least period n of a function f : ℝ → ℝ if p is fixed by f ○ f ○ … ○ f (n times), but not by f ○ f ○ … ○ f (m times) for any lesser positive m. It may not initially seem that the presence of a fixed point of a given least period would imply the existence of fixed points of other least periods, but Sharkovsky's Theorem states that this is in fact the case. We shall discuss this theorem and its proof.