Tuesday, September 1, 2009
Alex Rice, University of Georgia
Title: An introduction to elliptic curves
Abstract: Elliptic curves have been of extreme interdisciplinary interest in mathematics for over a century, lying predominantly in the intersection of algebraic geometry and algebraic number theory. They are occasionally introduced to undergraduates as the solution set in a plane to a very specific type of cubic equation, with a mysterious group law and "point at infinity". We will begin with a more complete definition and derive the simplified version (if you saw Maxim Arap's excellent talk over the summer, there will be some reduncance). After that, time permitting, we will discuss in little detail some of the myriad of applications, including some results and open problems in number theory, the method of elliptic curve cryptography, and the role of elliptic curves in the proof of Fermat's Last Theorem. Included will be direct mention of the work of at least 3 of our faculty members' thesis advisors.
About 90% of the talk will be accessible to anyone with a knowledge of elementary concepts like complex numbers and equivalence relations. There will be a very brief discussion of topology, with terminology such as 'manifold' and 'codimension', but intuitive explanations will be given and will definitely suffice. It may be useful to know the definition of 'number field', which is just any finite degree field extension of the rational numbers Q, so just anything that looks like Q(alpha) for some algebraic number alpha. The speaker can personally guarantee that no previous knowledge of algebraic geometry will be required, since the speaker doesn't have any!
Tuesday, September 8, 2009
Danny Krashen, University of Georgia
Title : Local-global principles and the Mayer-Vietoris sequence
Abstract : In this talk I will discuss a speculative conceptual approach relating the Mayer-Vietoris sequence in algebraic topology (for computing singular cohomology) and local-global principles in number theory (wherein one tries to prove the existence of a solution to a particular equation over the rationals by first trying to solve the equation for each $p$-adic field). If time permits, I will also describe how this idea is influencing my current research on the topic of field patching.
Tuesday, September 15, 2009
Niles Johnson, University of Georgia
Title: Clifford Algebras and their modules
Abstract: Clifford algebras are a generalization of the complex numbers and the quaternions. We'll explain how to classify modules over Clifford algebras, and the surprising periodicity for these algebras (and hence their modules). This is the same periodicity that Bott observed for the homotopy groups of the infinite orthogonal group, and as we explain this connection we'll also sketch a connection with astronomy. If time allows, we'll mention how the classification of Clifford modules applies to the study of vector fields on spheres and normed algebras over the real numbers (and why these are basically the same thing).
Tuesday, September 29, 2009
Jason Cantarella, University of Georgia
Title: Tight Knots and Ropelength
Tuesday, October 13, 2009
Matt Mastin, University of Georgia
Title: Introduction to Knot Theory
Abstract: The main purpose of the talk will be to give an introduction to some basic proof techniques in the field. After a few definitions and examples we will describe two fundamental knot invariants. The first is tricolorability which we will use to show that the "trefoil" knot is indeed knotted. We will then develop the Jones polynomial which is a much stronger knot invariant able to distinguish the trefoil from its mirror image. Time permitting we will discuss the topological symmetries of knots and describe some current work.
Tuesday, October 20, 2009
Angela Gibney, University of Georgia
Title: Introduction to the moduli space of curves
Abstract: In algebraic geometry, the most basic objects of study are varieties, which can often be thought of as the set of points that satisfy a collection of algebraic equations. By varying the the equations, the zero sets vary too and by thinking of a particular variety as a member of a family of varieties parametrized by these varying polynomial equations, one can often learn a great deal. I will talk about this notion of studying a variety as an element of a moduli space. The main example will be the moduli space of curves.
Tuesday, October 27, 2009
Dave Swinarski, University of Georgia
Title: Four mathematical shorts Abstract: I will give four short talks on topics from biology, commutative algebra, linear algebra, and probability.
Tuesday, November 3, 2009
Jim Stankewicz, University of Georgia
Title: M.C. Escher and maps between elliptic curves
Abstract: In 1956 Escher made a lithograph called 'Prentententoonstelling' or 'The Print Gallery' which had a very special "doubly periodic" symmetry but a mysterious hole in the middle. In 1999 Dutch Mathematician Hendrik Lenstra noted this and was able to use the theory of elliptic curves to 'fill the hole'. We will show how one parametrizes isomorphism classes of elliptic curves over the complex numbers and use this knowledge to see how Lenstra did it.
Tuesday, November 10, 2009
Chris Drupieski, University of Georgia
Title: Cohomology and Support Varieties
Abstract: Studying cohomology is hard, but the problem can sometimes be made easier through the introduction of a geometric tool called support varieties. In this talk we'll discuss the support varieties that arise while studying the cohomology of restricted Lie algebras and quantized enveloping algebras (also called quantum groups). It is a somewhat amazing fact that the geometric objects that arise in these contexts admit explicit descriptions in terms of nilpotent matrices. We will look at some easy examples, discuss some recent progress on computing cohomology and support varieties, and mention some open questions. This talk is designed to be accessible to a wide mathematical audience.
Tuesday, November 17, 2009
Robert Rumely and Dino Lorenzini, University of Georgia
Title: 150th Birthday of Riemann Hypothesis: What is the Riemann zeta function?
Abstract: The purpose of this talk is to give an explanation of what the Riemann Hypothesis is, and why it is so important. I will define Riemann Zeta function and discuss some of its properties (in particular its analytic continuation and Euler product). I will then explain the Riemann Hypothesis and its relation to the prime number theorem.
Tuesday, December 1, 2009
Kate Ponto, Notre Dame
Title: The Brouwer Fixed Point Theorem with Applications to the Jordan Curve Theorem
The Jordan curve theorem says that any simple closed curve in the plane divides the plane into two parts. This is one of those theorems that seems completely obvious, but is actually really difficult to prove. I'll describe one of the more approachable proofs of this theorem and its connection to another important theorem in topology, the Brouwer fixed point theorem.
Tuesday, January 19, 2010
Whitney George, University of Georgia
Title: Invariants of Legendrian Knots using Chekanov's Differential Graded Algebra
Abstract: The topology seminar is doing a "working seminar" this semester that will use this talk as background information. If enough graduate students are interested in this subject, we could extend this subject into a summer research/reading group in the summer.
Tuesday, January 26, 2010
Michael Ching, University of Georgia
Title: Framed Cobordism, Stable Homotopy, and the Kervaire Invariant Problem
Tuesday, February 2, 2010
Sybilla Beckman, University of Georgia
Title: Why should you care about math courses for prospective elementary and middle grades teachers?
Abstract: In this presentation I will discuss some of the major problems with preparing math teachers in the US today. I will then describe our math courses for prospective elementary and middle grades teachers and explain why you should learn about these courses.
Tuesday, February 9, 2010
Justin Noel, IRMA
Title: Vector bundles and homotopy theory
Abstract: In this talk I will discuss the Hopf invariant one problem. The Hopf invariant one problem is tied to the existence of real division algebras, multiplicative structures on spheres, and framings on the tangent bundles of spheres. I will talk sketch a proof to the Hopf invariant one problem using topological K-theory.
Tuesday, February 16, 2010
Pete Clark, University of Georgia
Title: Covering Numbers of Vector Spaces, Groups and Topological Spaces
Abstract: Let F be a field and V a vector space over F of dimension at least 2. Define the _covering number_ of V to be the least cardinality of a covering of V by proper linear subspaces. For instance, if V is R^n, then it is intuitively clear that the covering number is infinite, but we are after a more specific answer: countable? uncountable? equal to the cardinality of the set of all subspaces of R^n? On the other hand, when V is a finite-dimensional vector space over a finite field, the covering number is clearly finite, and we are looking for the exact, finite number.
I will solve this problem and also call your attention to an "irredundancy paradox". If time premits, I will mention some other interesting covering problems: groups by subgroups, groups by cosets, topological spaces by closed subspaces, etc.
Tuesday, February 23, 2010
John Doyle, University of Georgia
Title: Self-similar Apollonian Circle Packings
Abstract: I will introduce the notion of an Apollonian circle packing and describe how the group of similarities of the Euclidean plane acts on the set of such packings. The main result will be a classification of all self-similar Apollonian circle packings that contain a line, i.e., a circle of zero curvature. At the end of the talk I hope to mention some connections with algebraic number theory.
Tuesday, March 16, 2010
Andrew Sornborger, University of Georgia
Title: Summer Internships
Tuesday, March 23, 2010
Alex Brown and Jim Stankewicz, University of Georgia
Title: On a Generalization of the Frobenius Number
Abstract: Alex Brown will discuss results obtained in the Fall of 2008 in an IVRG directed by Prof. Lorenzini. The resulting paper is now published in the Journal of Integer Sequences. This is joint work with Eleanor Dannenberg, Jennifer Fox, Joshua Hanna, Katherine Keck, Alexander Moore, Zachary Robbins, Brandon Samples, and James Stankewicz. Jim Stankewicz will discuss a recent sequel to the above paper that he co-authored with Prof. J. Shallit (U. Waterloo)
Tuesday, March 30, 2010
Neil Lyall, University of Georgia
Title: Looking out for number one
Abstract: Consider the powers of 2: 2, 4, 8, 16, 32, 64, 128, 256,... Their right-most digits follow a simple repetitive pattern: 2, 4, 8, 6, 2, 4, 8, 6,...
But what about their left-most digits? The sequence of the first digits of the first 40 powers of 2 is:
2, 4, 8, 1, 3, 6, 1, 2, 5, 1,
2, 4, 8, 1, 3, 6, 1, 2, 5, 1,
2, 4, 8, 1, 3, 6, 1, 2, 5, 1,
2, 4, 8, 1, 3, 6, 1, 2, 5, 1.
Do we ever see a 7? The answer is yes (2^46 starts with a 7). We will show that there are in fact infinitely many integers n such that 2^n starts with a 7 and that they (surprisingly!) have a well-defined frequency. The existence of this frequency follows from the uniform distribution of multiples of an irrational number modulo 1.
Tuesday, April 6, 2010
Clay Shonkwiler, Haverford
Title: The search for higher helicities
Tuesday, April 13, 2010
Noah Giansiracusa, Brown University
Title: Classical Invariant Theory with a View Toward Moduli Spaces
Abstract: We will discuss some basic notions of invariant theory, mostly by way of examples, and show how these may be used to describe some basic yet fundamentally important moduli spaces such as Grassmannians and the space of plane curves. If time permits we will also discuss Molien's formula for counting invariants of a finite group.
Tuesday, April 27, 2010
Ben Jones, University of Georgia
Title: Sage and @interact for calculus
Abstract: Sage is free and open source mathematical software developed by working mathematicians. The goal of the Sage project is to provide software which is an open and "peer reviewed" alternative to closed and proprietary systems such as Mathematica, Maple, and Magma. I'll give an informal introduction to Sage, describe how I've used it in my own research, and give a demonstration of it's @interact feature which is perfect for creating online (and in-class) interactive demonstrations.