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.
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.
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.
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
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.
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)
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.
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