This group will begin with a gentle introduction, constructing grassmannians as quotients of spaces of matrices, and looking at the three classical coordinate systems on them. This will lead to \bar M_{0,n}, the moduli space of n-pointed curves, and, in due course, to its higher-dimensional analogues, with many open research problems to study and make progress on. This will be a mixture of geometry, combinatorics, and tropical mathematics. The subject has plenty of questions to investigate, from the more elementary, to hard combinatorial, to very sophisticated.
We think this VRG group will combine well with either MATH 8300 "Introduction to algebraic geometry" which I propose to teach in the Fall, and the topics course MATH 8330 "\bar M_{0,n}" which Angela proposes to teach.
A conformal structure is more flexible than a Riemannian metric but more rigid than a topological structure. Conformal geometric methods have played important roles in PDE, topology and geometry. The focus of this VRG will be to develop the students intuition of this subject through specific topics and examples.
We will start with discrete surfaces. These are given as piecewise linear triangle meshes. First we will consider a discrete Ricci flow. The (hard) notions of Riemannian metric and Gaussian curvature are discretized as the edge lengths and the angle deficits, the discrete Ricci flow can be defined as the deformation of edge lengths driven by the discrete curvatures.
After getting an understanding of existence and uniqueness of the solution we will turn into convergence questions and connections to circle packing. prerequisites: It is recommended that you have taken an introductory course on: curves and surfaces and complex analysis. If not, you should be able to work assuming some facts. Also, to those who like programing (or wish to explore this), this VRG offers plenty of opportunities.
The Algebra VIGRE research group will focus on the topic of the cohomology of finite groups. This is a broad area with many connections to homological algebra, derived catergories and algebraic topology. We will introduce students and postdocs to the fundamentals of the subject in a one-month crash course at the beginning of the year.
We will take part of the time to finish up projects from the previous year. Also, we will look at new computations of cohomology rings, complexity of modules, and support varieties. For the calculations, we will introduce techniques from other areas and employ computer packages such as MAGMA.
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This VIGRE Seminar will be a year-long study of arithmetic dynamics, and is a precursor to a VIGRE Conference to be held at UGA in Summer 2011. Arithmetic dynamics concerns properties of the iterates f(n) = f o f ...o f of a rational function f(z) = P(z)/Q(z), under composition.
Arithmetic dynamics is a relatively young field, many of whose foundational results have only been established in the last 20 years. As frequently happens with topics in number theory, questions in arithmetic dynamics often break into local and global parts. "Local" questions concern rational functions defined over complete fields, such as R, C, and the p-adic fields. Here, topological and geometric methods come into play. The classical Mandelbrot and Julia sets (the fractal sets one often sees in computer-generated pictures) represent objects studied in the local theory of rational functions.
"Global" questions concern functions defined over the rational numbers or similar fields. Some questions which arise naturally concern the nature of the periodic points of f (points P for which f(n)(P) = P, for some n) and the preperiodic points (points Q for which f(m)(Q) is periodic, for some m). How many can there be? What galois groups can they generate? It is known that if f has coeficients in the rational numbers Q, then there are only finitely many preperiodic points defined over any finite extension K/Q.
A major open question is the Uniform Boundedness Conjecture: For all rational functions f E Q(z) with deg(f) < N, there is a uniform bound B = B(N, d) on the number of preperiodic points of f defined over extensions K/Q with [K : Q] < d.
A more specific conjecture by Bjorn Poonen says that if f(z) E Q[z] is a quadratic polynomial with rational coeficients, there can be at most 9 preperiodic points defined over Q. How many points can be defined over quadratic extensions K/Q? What if f is a cubic polynomial or a quotient of two quadratic polynomials? The answers are not known.
The basic reference for this seminar will be the excellent textbook by Joe Silverman, "The Arithmetic of Dynamical Systems". We will seek to understand the theory by working through major theorems, and by computer investigation of interesting examples, with the goal of shedding light on the questions above and related questions.
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