This is an archived page, from a previous year. Links may be broken.
The current schedule can be found here.

Notre Dame Math Graduate Student Seminar, 2009-2010

All talks are at 4:15 in HH231 unless otherwise noted.

Previous Semesters

To volunteer to give a talk, or for anything else regarding the seminar, contact Megan Patnott.

Schedule

Date Speaker Title
Monday, September 7 Liviu Nicolaescu Knots and their curvatures
Monday, September 21 Megan Patnott An Introduction to Using Beamer
Monday, October 5 Sarah Cotter Variations on a theme of Helly
Monday, October 26 David Karapetyan Cancelled due to illness
Monday, November 9 Bernadette Boyle Cayley-Bacharach Theorem through the ages
Monday, November 23 David Karapetyan On the Uniqueness of Solutions to the Burgers Equation in Sobolev Spaces
Monday, December 7 Brandon Rowekamp Uniform Distributions Modulo 1
Monday, February 1 MGSA meeting: discussion of departmental changes
Monday, February 15 Katie Grayshan General Abstract n-sense
Monday, March 1 Martha Precup Counting Nilpotent Ideals of a Borel Subalgebra
Monday, March 15 John Engbers Conway's Solitaire Army Problem
Monday, March 29 Jesse Johnson We're All Mad Here: The Mathematics of Lewis Carroll
Monday, April 12 Stephen Flood What can logic teach us about theorems of ordinary mathematics?
Monday, April 26 Don Brower Independence Relations

Abstracts

September 7, 2009

Speaker
Liviu Nicolaescu
Title
Knots and their curvatures
Abstract
I will discuss an old result of John Milnor stating roughly that if a closed curve in space is not too curved then it cannot be knotted.

September 21, 2009

Speaker
Megan Patnott
Title
An Introduction to Using Beamer
Abstract
Beamer is one of the more commonly used LaTeX packages for making presentations. We'll discuss the basics of using it, as well as a couple of useful tricks.

October 5, 2009

Speaker
Sarah Cotter
Title
Variations on a theme of Helly
Abstract
Helly's Theorem, proved by Eduard Helly in 1923, is a result in convex geometry dealing with the intersections of certain families of sets. While Helly's Theorem is rather useful on its own, by weakening its requirements we can broaden its applications to a surprising variety of problems. Topics covered may include (p, q)-properties, fractional Helly properties, VC dimension, the arrangement of sheep, art gallery problems, and robots.

November 9, 2009

Speaker
Bernadette Boyle
Title
Cayley-Bacharach Theorem through the ages
Abstract
Much of algebraic geometry focuses on the vanishing locus of systems of polynomials, as well as, the polynomials that vanish on a certain subspace. In this talk, I will discuss some of the major tools algebraic geometers use in studying these vanishing loci and polynomials while focusing on a classical result in algebraic geometry, the Cayley-Bacharach theorem. I will discuss several different formulations of the Cayley-Bacharach theorem from its earliest roots in Pappus' Theorem (4th century AD), through the twentieth century. This talk is based on the paper "Cayley-Bacharach Theorems and Conjectures" by D. Eisenbud, M.Green, and J. Harris.

November 23, 2009

Speaker
David Karapetyan
Title
On the Uniqueness of Solutions to the Burgers Equation in Sobolev Spaces
Abstract
It is shown that solutions to the Burgers initial value problem are unique in Sobolev Spaces $H^s$ for $s>3/2$. Using a modification of the Kato-Ponce commutator estimate, an energy estimate for the Burgers i.v.p is derived; an application of Gronwall's inequality then yields the desired result.
(See abstract as pdf.)

December 7, 2009

Speaker
Brandon Rowekamp
Title
Uniform Distributions Modulo 1
Abstract
A sequence of numbers is uniformly distributed modulo 1 when it "equally fills" every part of the unit interval. Sequences with uniform distribution, particular those that are well distributed in their early elements, are useful in a variety of contexts. In this talk I will define the concept of uniform distribution, and discuss some of its applications, such as in Diophantine approximation and numerical integration.

February 15

Speaker
Katie Grayshan
Title
General Abstract n-sense
Abstract
Since category theory was introduced in the 1940s by Eilenberg and MacLane, it has often been referred to as 'general abstract nonsense'. In this talk, we will delve into a deeper abstraction by looking at the idea of n-categories. We will examine some of the difficulties in defining a sufficiently general notion of an n-category and consider motivating examples in topology. In particular, we'll discuss how certain degenerate n-categories encode interesting structure and why Leinster says, "the world of n-categories is a mirror of the world of the homotopy groups of spheres."

March 1

Speaker
Martha Precup
Title
Counting Nilpotent Ideals of a Borel Subalgebra
Abstract
An important part of Lie theory is the study of the structure of Lie algebras. In this talk we will learn some of the basic facts about this structure. We will then count the number of nilpotent ideals of a Borel subalgebra of classical Lie algebras, $A_n$ and $C_n$. It turns out that counting these ideals can be done using combinatorial techniques, in particular, André's reflection. This simple combinatorial method can be used to simplify arguments that normally require a great deal of theory.
(See abstract as pdf.)

March 1

Speaker
John Engbers
Title
Conway's Solitaire Army Problem
Abstract
Peg solitaire is a common single-player game played on a board where every position but one starts occupied by pegs, and the goal is to use horizontal and vertical "checkers" style jumps to eliminate all but one of the pegs. John Conway's aggressive modification of the game asks the question: Given an army with soldiers on the integer lattice points of the lower half-plane (the new board), how far above the x-axis can the army place a soldier using these jumps? We present his surprising answer using the beautiful idea of pagoda functions, and then discuss one of many extensions to the problem.

Surprisingly, very little mathematics beyond geometric series is needed, which should be a welcome respite at the end of the first day post-Spring Break.

(See abstract as pdf.)
(See references.)

March 29

Speaker
Jesse Johnson
Title
We're All Mad Here: The Mathematics of Lewis Carroll
Abstract
Renowned author and freelance mathematician Charles Dodgson authored many books and poems beloved by children everywhere. While his works portray worlds of delightful fancy and humorous nonsense, his mathematical side runs deep in the core of all his literature, making these children's stories just as beloved by adults. During this talk, we will explore several examples of the mathematics in Dodgson's works. We will make sense of seemingly nonsensical arguments, demonstrating how interpretation of language affects arguments in symbolic logic. We will consider a method for computing determinants created by Dodgson using adjugate matrices. Finally, we will explore other paradoxes and syllogisms, making note of the historical impact they had in the logic and philosophy communities in the mid-nineteenth century.

April 12

Speaker
Steve Flood
Title
What can logic teach us about theorems of ordinary mathematics?
Abstract
Reverse mathematics is a sub-field of logic which seeks to classify theorems of ordinary mathematics according to the complexity of their proofs.

In algebra, we learn every field K has an algebraic closure F. Furthermore, we learn that K is isomorphic to a sub-field of F. Given a computable field, how complicated is F? How complicated is the isomorphic image I of K in F? If F is computable and I is not computable, then more proof-power is needed to produce I than to produce F. If we know that some I must exist, we can prove the existence of certain non-computable sets. In doing this, we have 'reversed' our proof, using the theorem of algebra to proving our set existence 'proof rule'.

In reverse mathematics, we prove and organize this type of result using subsystems of second order arithmetic. Second order arithmetic is a system of formal logic where we can quantify over numbers and sets and do basic arithmetic. By using the close connections that many subsystems of second order arithmetic have to computability theory, we learn more about the computational complexity of standard theorems. This talk will introduce the tools, goals, and practice of reverse mathematics, with examples.

No background in logic, algebra, or arithmetic is assumed.

April 12

Speaker
Don Brower
Title
Independence Relations
Abstract
Given two elements in some mathematical structure, what should it mean to say that they are ``independent''? Intutively this means the two elements are as free as possible. This turns out to be a combinatorial question, and it will turn out that not every structure admits such a property. No logic background will be assumed, and many examples will be provided.

Previous Years


MGSA - Math Department - University of Notre Dame