January 2006

• Date: Jan. 10, 2006
  Speaker: Jerry Hower, University of Georgia
  Title: An analogue of quadratic reciprocity.
  Abstract: We will discuss quadratic reciprocity over F[t] where F is a finite field of odd charteristic. We will aim to elucidate the similarities with quadratic reciprocity over Z.
  
  
• Date: Jan. 17, 2006
  Speaker: Jason Parsley
  Title: The Borromean Rings
  Abstract: The Borromean Rings are three circles* where no two of them can be pulled apart, but the three curves together cannot be separated. We will discuss their history and talk about ways to show they are in fact linked.
  * We will also show that you can't actually make the Borromean rings out of three circles but could use ellipses instead. We'll end with some connections to algebraic topology.
  
  
• Date: Jan. 24, 2006
  Speaker: Matt Hedden, Princeton University
  Title: Introduction to knot theory and knot invariants
  
  Abstract: I'll begin the talk by introducing what a knot is mathematically, and trying to motivate why someone (probably a topologist) might study them. I'll then discuss how one could go about studying knots through the use of invariants, and introduce two of the most famous invariants, the Alexander and Jones polynomials. I'll may try to conclude by speaking roughly about some beautiful modern generalizations of these polynomials which go by the names of Ozsvath-Szabo and Khovanov homology, respectively. The talk will be aimed at beginning graduate students or advanced undergraduates.
  
  
• Date: Jan. 31, 2006
  Speaker: Tatyana Sorokina
  Title: Multivariate Polynomial Spline Approximation
  
  Abstract: The talk is designed to provide an overview of multivariate polynomial splines. We will discuss basic concepts, main research directions, open problems, and applications.
  
  

February 2006

• Date: Feb. 7, 2006
  Speaker: Nathan Edington, University of Georgia
  Title: Computer Implementations of Five Important Approximations to Pi
  
  Abstract: We briefly introduce the historically significant and often surprisingly beautiful approximations to pi of Wallis, Newton, Gregory, Machin and Ramanujan. We then outline how these approximations were implemented in MATLAB and MathCAD in order to explore and compare the accuracy and rate of convergence of each approximation.
  
  
• Date: Feb. 14, 2006
  Speaker: Kenyon Platt, University of Georgia
  Title: Bird's-Eye View of Category O_S
  Abstract: Given a Lie algebra g over the complex numbers, a g-module V is a complex vector space on which g acts. I will discuss briefly a certain category of g-modules called category O_S.
  
  
• Date: Feb. 21, 2006
  Speaker: Robb Sinn, North Georgia College & State University
  Title: Interesting Applications of Game Theory: Law, Biology & Counter-Terrorism
  
  
• Date: Feb. 28, 2006
  Speaker: Rod Canfield, Department of Computer Science, University of Georgia
  Title: Introduction to the Circle Method
  Abstract: The circle method is a technique for applying Cauchy's integral theorem (from complex variables) to problems arising in combinatorics and number theory. We will provide an introduction, based on two examples.
  
  

March 2006

• Date: March 7, 2006
  Speaker: Michael Piatek, University of Washington
  Title: The ropelength problem and its applications
  Abstract: How much rope does it take to tie a particular knot? In this introductory talk, we will describe RidgeRunner, software developed at UGA to simulate the tightening process for arbitrary curves. Surprisingly, the algorithms and mathematical underpinnings of RidgeRunner have applications in computer science, physics, and biology. We will describe RidgeRunner's method before providing an overview of each of these, describing potential avenues of future work.
  
  
• Date: March 14, 2006
  No meeting due to spring break.
  
  
• Date: March 21, 2006
  Speaker: John Foley, Wake Forest
  Title: Nonlinear Difference Equations
  
  
• Date: March 28, 2006
  Speaker: Bobbe Cooper
  Title: Graphs and Root Systems of Type A
  
  

April 2006

• Date: April 4, 2006
  Speaker: Emille Davie, University of Georgia
  Title: Surface Topology, Geometry, and the Mapping Class Group
  Abstract: Do you believe that since compact surfaces are classified there is little to be studied? Do you believe that the geometry of all surfaces is Euclidean? If you do, then this talk will open your eyes to a hyperbolic universe where "straight" lines are curved, punctures are in a galaxy far, far away, and the 1-dimensional submanifold is king.
  
  
• Date: April 13, 2006 (Thursday), note: different day of the week
  Speaker: Tara Brendle, Louisiana State University
  Title: Mapping class groups and complexes of curves
  Abstract: We will give a brief introduction to some elementary combinatorial structures which arise naturally when studying curves on a surface. Examples include the "curve complex", the "pants complex", and the "cut-system complex". These simplicial complexes have recently become important in topology, since, although simple to define, they in fact encode a great deal of algebraic information, including the entire algebraic structure of the mapping class group of a surface.
  
  
• Date: April 18, 2006
  Speaker: Patrick Corn, University of Georgia
  Title: Conics and quaternion algebras
  Abstract: We will formulate an analogy between plane conic curves and quaternion algebras over number fields, by analyzing the field extensions over which they both become "trivial." This is the tip of a big iceberg--lurking underwater are such things as Brauer groups, Galois cohomology, twists and descent, and other beautiful concepts from algebraic geometry and number theory. We aim to give a gentle introduction to some of these ideas, assuming no prior knowledge of the subject.
  
  
• Date: April 25, 2006
  Speaker: Adrian Jenkins, Purdue University
  Title: Local classification problems in complex analysis
  Abstract: We will look at germs of holomorphic functions $f$ which fix the origin (and in fact, most of our interest will be in those functions so that $f'(0)=1$), and consider the question of local classification under certain changes of variable. The goal here will be to understand the local dynamics of such functions by relating them to certain "good" model functions (e.g. linear functions, Mobius transforms, and in general, functions which are easy to iterate). This talk will serve as an introduction to the analysis seminar to be given later in the day. All definitions will be given, and the talk should be of an introductory nature (really, the only prerequisite is a good first course in complex analysis).
  
  
• Date: May 2, 2006
  Speaker: Andrew Raich, Texas A&M
  Title: An introduction to the spectral theorem with an application to PDEs
  Abstact: I will begin the talk with a sketch of the ideas which comprise the spectral theorem and the functional calculus for self-adjoint operators on a Hilbert space. From there, I will introduce the spectral measures and give an application of the spectral theorem for unbounded operators to "solving" linear, elliptic PDE via heat semigroups. No knowledge of PDEs is required for the talk, but some knowledge of measure theory and basic functional analysis (Riesz Representation Theorem, self-adjoint operators, etc) will be helpful.
  
  This talk will serve as an introduction to the analysis seminar to be given later in the day.
  
  

August 2005

• Date: Thursday, Aug. 11, 2005, 11-12
  Speaker: Andrew Sornberger, University of Georgia
  Title: VIGRE group introduction: mathematical cardiac physiology (applied math)
  * note there are two Vigre talks scheduled today
  
  
• Date: Thursday, Aug. 11, 2005, 2-3pm
  Speaker: Elham Izadi, University of Georgia
  Title: VIGRE group introduction (algebraic geometry)
  
  
• Date: Tuesday, Aug. 16, 2005
  Speaker: Dan Nakano and Carrie Wright , University of Georgia
  Title: VIGRE group introduction: Lie algebra cohomology and its applications (algebra)
  
  
• Date: Thursday, Aug. 18, 2005
  Speaker: Jason Cantarella and Jason Parsley, University of Georgia
  Title: VIGRE group introduction (differential geometry)
  
  
• Date: Tuesday, Aug. 23, 2005
  Speaker: Robert Varley and Cal Burgoyne, University of Georgia
  Title: VIGRE group introduction (mathematical physics): Feynman Diagrams
  
  
• Date: Thursday, Aug. 25, 2005
  Speaker: Elizabeth Denne, Harvard University
  Title of talk: The distortion of a knotted curve
  Abstract: The distortion of a curve measures the maximum arc/chord length ratio. Gromov showed that any closed curve has distortion at least $\pi/2$ and asked about the distortion of knots. In this talk, I'll use the existence of an essential secant to show that a nontrivial knot of finite total curvature has distortion at least 4. No prior knowledge of knot theory will be assumed in this talk.
  
  
• Date: Tuesday, Aug. 30, 2005
  Speaker: Robert Rumely, University of Georgia
  Title: VIGRE group introduction (number theory)
  
  

September 2005

• Date: Sept. 1, 2005 (Thursday) note: this is our last Thursday talk for awhile
  Speaker: Ken Baker, University of Georgia
  Title: Braiding in lens spaces
  Abstract: A classical theorem of Alexander states that every oriented link in S^3 can be represented as a the closure of a braid. Observe how this suggests the decomposition of S^3 as a union of two solid tori along their boundaries. Viewing one of these solid tori as a neighborhood of the braid axis, Alexander's theorem says that any oriented link may be made to lie in the other solid torus in a well-groomed fashion.
  Lens spaces are also unions of two solid tori along their boundaries. In this VIGRE group our main goal is to study to what extent an analogue of Alexander's theorem holds for links in lens spaces. To this end we will address various topics in classical (i.e. S^3) knot theory and braid theory, some techniques of 3-manifold topology, and of course lens spaces.
  
  
• Date: Sept. 6, 2005 (Tuesday)
  Speaker: Joe Rusinko, University of Georgia
  Title: Introduction to tropical geometry
  Abstract: Tropical geometry is a developing tool with applications in both pure and applied mathematics. We will discuss what exactly tropical geometry is and explore its similiarities with regular geometry. Time permitting i will introduce some connections between tropical geometry and polytopes which are another common tool in algebraic geometry.
  
  
• Date: Sept. 13, 2005
  Speaker: Chad Mullikin, University of Georgia
  Title: Geodesics : An Introduction
  Abstract: What is the shortest path between two distinct points in the plane? Are you sure? Can you prove it? In this talk I will introduce the idea of the calculus of variations. This subject generalizes the idea of minimizing and maximizing functions (or in the language of Calculus of variations, functionals).
  
  
• Date: Sept. 20, 2005
  Speaker: John Etnyre, Georgia Tech
  Title: Knots, surfaces and contact geometry
  
  
• Date: Sept. 27, 2005
  Speaker: Ed Azoff, University of Georgia
  Title: Universal objects in analysis
  Abstract: We are used to gaining insight into complicated objects by resolving them into simpler components. For example, the Jordan canonical form theorem reduces the study of arbitrary n by n complex matrices to the study of direct sums of simple Jordan blocks. In this talk, we will discuss three "universal" constructions which go in the opposite direction: (1) an open set in the plane which is universal in the sense that every open subset of R occurs as one of its vertical sections, (2) a construction of C. Rota to the effect that there is a simplest operator (=linear transformation) acting on an infinite dimensional space in which all "reasonable" operators can be embedded, (3) a simple block matrix construction embeds every linear space S of operators in a commutative operator algebra A; a 1985 refinement due to W. Wogen tells us that we can even take A to be singly generated.
  These are "spoiler" results. For example, (1) can be used to produce a subset of R which is not the countable union of countable intersections of open sets. Similarly, we will bootstrap clever choices of S in (3) to produce some very pathological operators.
  
  

• Date: Oct. 4, 2005
  Speaker: Aaron Abrams, Emory University
  
  
• Date: Oct. 11, 2005
  Speaker: Amod Agashe, Florida State University
  Title: Diophantine equations
  Abstract: Consider a polynomial equation in two or more variables with coefficients that are rational numbers. Such equations are called Diophantine equations, and one can ask for a description of solutions to such equations whose coordinates are also rational numbers. This problem has been studied with interest for centuries. As an example, the rational solutions to the Diophantine equation x-squared + y-squared = 1 correspond to Pythagorean triples; for example, x=3/5, y=4/5 is a solution coming from the triple (3,4,5). In this talk we will study solutions to Diophantine equations in two variables, with emphasis on quadratic equations and certain cubic equations, where the local-to-global principle and the Birch and Swinnerton-Dyer conjecture provide a nice answer (respectively).
  
  
• Date: Oct. 18, 2005
  Speaker: Matt Boylan, University of South Carolina
  Title: Modular forms and partitions
  Abstract: Modular forms naturally occur as generating functions for many objects of arithmetic interest such as central critical values of L-functions associated to elliptic curves over the rationals, representation numbers of positive definite quadratic forms, and partitions. The partition function, p(n), counts the number of non-increasing sequences of positive integers whose sum is n. In this talk, we discuss recent results on the distribution and arithmetic of p(n) modulo a positive integer M and some of the ideas from the theory of modular forms used to prove these results.
  
  
• Date: Oct. 25, 2005
  Speaker: Erin McNelis, Western Carolina University
  Title: Sleep and the Shift Worker: A Mathematical Biology Approach to an Age-Old Problem
  Abstract: With continuing evidence of the difficulties experienced by shift workers, much focus has been placed on designing shift systems that minimize the adverse effects of shift work on human health and performance. As such, one goal is the development of schedules that require workers to be on duty during the times that they are most naturally alert and awake. Chronobiologists have developed circadian rhythm based guidelines intended to aid in designing such shift work schedules. Scientists have used these guidelines to develop automated scheduling tools and algorithms for the design of shift schedules that are less disruptive to a worker's natural biological rhythms. This research extends these empirical approaches by using a mathematical model of human circadian rhythms in developing ``optimal'' shift work schedules with respect to their compatibility with innate human biological rhythms.
  
  A differential equations model of the deep human circadian pacemaker developed by Kronauer is modified to include the influence of shift work on the circadian system. A numerical simulation of the pacemaker rhythm induced by a given shift work schedule is compared with simulated innate (benchmark) pacemaker rhythm. The degree of deviation between the work-influenced rhythm and benchmark rhythm serves as an indicator of how compatible a work schedule is with a worker's natural alertness level. By parameterizing the shift work model, the process of finding an optimal shift schedule is reduced to solving a non-linear optimization problem. Optimal one- and two-week shift schedules are developed subject to a few assumptions about the characteristics of the shift schedule itself. The results of these investigations are compared and optimal patterns and properties are identified.
  
  

• Date: Nov. 1, 2005
  The Vigre seminar will not meet this week; instead we recommend attending Serre's lecture at Emory.
  
  
• Date: Nov. 8, 2005
  Speaker: Patrick Bahls, UNC-Asheville
  Title: Groups and monoids and a trick of Magnus
  Abstract
  
  
• Date: Nov. 15, 2005
  Speaker: Ismar Volic, University of Virginia
  Title: An introduction to Vassiliev knot invariants
  Abstract: Vassiliev (or finite type) knot invariants have received much attention in the last ten years because of the many connections they have to physics and 3-manifold theory. They are also conjectured to ^Óseparate^Ô knots, meaning that any two different knots can be distinguished by a Vassiliev invariant. After stating the relatively simple definition of Vassiliev invariants, I will describe how they are related to a certain algebra of chord diagrams. This relationship, given by a celebrated integral due to Kontsevich, allows us to study Vassiliev invariants in a purely combinatorial fashion. However, the appearance of such an integral in knot theory is still mysterious and not well understood.
  
  
• Date: Nov. 22, 2005
  No meeting this week.
  
  
• Date: Nov. 29, 2005
  Speaker: Rafal Zbikowski, Cranfield University, UK
  Title: Some Mathematical Aspects of Reverse Engineering of Insect Flight Control
  Abstract: Insects exhibit highly maneuverable flight, and accomplish it with very limited computational power in their brains. On the other hand, they possess thousands of sensors, which allow them to acquire information on their motion relative to their surroundings by measurement rather than computation. In particular, insect vision is a highly capable measurement system producing a global vector field of optic flow. Optic flow is the apparent motion of surrounding objects, for example trees streaking past a car going forward. In the insect case, the global vector field of optic flow is projected on a 2-sphere. At the same time, insect flight control is governed by six coupled ordinary differential equations, connecting their inertial response with the action of the gravitational and aerodynamic forces. These differential equations also represent a vector field and the challenge is how to connect the global vector field of optic flow on the 2-sphere with the six differential equations of motion.
  
  

• Date: Dec. 6, 2005
  No meeting; the University considers this a Friday. The seminar will resume in the spring.
  
  

This schedule is subject to change. If you have questions or are interested in giving a talk, please contact the organizers, Nadia Mazza and Jason Parsley.