Notre Dame Math Graduate Student Seminar, 2012-2013

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.

Previous Semesters

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

Schedule

Date Speaker Title
Monday, September 3 Juan Migliore A modern algebraic view of a classical geometric result
Monday, September 17 Martha Precup The Geometry of Hessenberg Varieties
Monday, October 1 Melissa Davidson Where did THAT come from?
Monday, October 22 Quinn Culver The Recursion Theorem
Monday, November 5 Joshua Lioi When Zombies Attack!
Monday, November 19 Victor OcasioGonzalez Classic Nintendo games are (NP-) Hard
Monday, December 3 N/A N/A
Monday, January 28 John Harvey The classification of G-spaces, with a new Slice Theorem
Monday, February 11 Brian Shourd Applications of Category Theory to Functional Programming
Monday, March 4 N/A N/A
Monday, March 18 Alexander Diaz Representations of Coxeter Groups and Hecke Algebras
Monday, April 8 Santosh Kandel Infinite determinants of Hilbert space operators
Monday, April 29 John Holmes N/A

Abstracts

September 3, 2012

Speaker
Juan Migliore
Title
A modern algebraic view of a classical geometric result
Abstract

Pascal's theorem says that if a hexagon is inscribed in a conic, the three points of intersection of pairs of opposite sides of the hexagon are collinear. This turns out, in the end, to be a simple but surprising application of a modern algebraic theory (liaison theory). We'll start off looking at how many points we expect two plane curves to meet in, look at a result about cubic curves commonly called the Cayley-Bacharach theorem but actually due to Chasles, and then apply this to Pascal's situation. Then we'll see how Chasles' theorem is a special case of "the Hilbert function for linked sets of points," and see that results of that sort are easy to obtain.

September 17, 2012

Speaker
Martha Precup
Title
The Geometry of Hessenberg Varieties
Abstract

Lie theoretic tools can be used to answer geometric questions about Hessenberg varieties, a family of subvarieties of the flag variety associated to an algebraic group G. We will show that structure of these varieties, including some open questions about connectedness, singularities, and irreducible components, is related to the root space decomposition of the Lie algebra 𝔤 corresponding to G.

October 1, 2012

Speaker
Melissa Davidson
Title
Where did THAT come from?
Abstract

Have you ever looked at a theorem or equation and wondered how on earth somebody ever thought of it? Today's your lucky day! We will trace the history of a specific solitary wave equation. Starting from the deep depths of time, we will travel through the eras and meet Pythagorus, Euler, and many others. Our journey will end with the Ostrovsky equation. This talk assumes you know how to spell your first name on a math exam. No other math is required.

October 22, 2012

Speaker
Quinn Culver
Title
The Recursion Theorem
Abstract

Behold ye! I am the self-referentialator,
Empowering would-be quine creators.
Proof simplificator, mystificator.
I am the virus propagator,

The vanguard of the supreme trick logical,
The point-fixer of all functions computable.
Lean upon me for intelligence artificial,
And as a bulwark against escapes diagonal.

I, too, charge you that you make preparation
to be worthy to meet me, by your creation
in any language of a computation
that outputs its own instructions.

Though a fine messenger Q. might be,
This lecture stems from me, is of me.
Nay! I'm neither Turing nor Kleene,
Though they are both patres mihi.

November 5, 2012

Speaker
Joshua Lioi
Title
When Zombies Attack!
Abstract

The impending zombie apocalypse has been well documented in film, television, books, and video games. If we are to survive, we will need to be prepared. How better to prepare ourselves than with mathematics? Differential equations are a useful tool in the realm of applied mathematics. They can be used to study the spread of disease in a population, and might be our best hope for determining a strategy for dealing with this undead threat.

November 19, 2012

Speaker
Victor OcasioGonzalez
Title
Classic Nintendo games are (NP-) Hard
Abstract

What does an mushroom-loving Italian plumber, a banana-hoarding ape, young swordsman clad in green tunic, a female space bounty hunter, and wandering teenager with a pocket full of monsters have in common? They've all had (NP-) hard lives! Join me as I navigate through computer science jargon and discover how to make our heroes lives just a little bit more difficult. My only assumption is that you're cool enough to know what the previous references are for!

January 28, 2013

Speaker
John Harvey
Title
The classification of G-spaces, with a new Slice Theorem
Abstract

The theory of classifying spaces for principal G-bundles is very widely known, but it seems that Palais' construction of classifying spaces for more general G-spaces is not so familiar to many mathematicians. In this talk, I will explain Palais' classifying theory, demonstrate a minor refinement, and apply it to prove a new and useful form of the Slice Theorem in the context of Alexandrov geometry.

February 11, 2013

Speaker
Brian Shourd
Title
Applications of Category Theory to Functional Programming
Abstract

In this talk, I will discuss some of the ways that category theory can be applied to programming, specifically to functional programming using the language Haskell. No prior knowledge of functional programming (or really of programming at all) is required or expected. We'll discuss how functors, monoids, and monads (what's a monad?) can be used to create and reason about purely functional code. "Abstract Nonsense?" - nonsense!

March 18, 2013

Speaker
Alexander Diaz
Title
Representations of Coxeter Groups and Hecke Algebras
Abstract

In the 1980's, David Kazhdan and George Lusztig were looking to decompose a complicated space of functions into irreducible representations of a group. In that quest, they found that this is equivalent to decomposing the regular representations of Hecke algebras. In this talk, I will construct such algebras and talk about all of the things they stumbled across in their mission (including the Kazhdan-Lusztig polynomials). My only assumption is that you took a basic algebra course.

April 8, 2013

Speaker
Santosh Kandel
Title
Infinite determinants of Hilbert space operators
Abstract

I'll talk about a way to generalize the determinant of operators in finite dimensional Hilbert spaces to the determinant of operator on infinite dimensional Hilbert spaces. I will also talk about a regularized determinant of a class of operators and depending on the time constraint I'll give a quick application these regularized determinants.


Previous Years


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