In 1955 Serre asked whether finitely generated projective modules over k[x_1, ... ,x_n], where k is a field, are free. This question, which became known as "Serre's Conjecture," was proved independently by D. Quillen and A Suslin in 1976. We will discuss the proof given by Seshadri in 1958 for the case of two variables. We will also consider some geometric applications and possible motivation for the original problem.
An algebra B over a ring R may be viewed simply as an R-module by forgetting about the multiplication. It's Jacobson radical is a submodule, and if B/J is projective as an R-module, then B as an R-module is a direct sum of J and B/J. With just two extra conditions, this actually turns out to be a direct sum decomposition *of R-algebras*. We will present the proof of this fact, introducing such interesting characters as A-B bimodules and Hochschild cohomology along the way.
The physical mathematics of the quantum theory of fields is awash with mathematical problems. In this talk, I'll try to give you a whirlwind tour of physics and exhibit an appealing geometric framework for quantum field theory established by Segal. Time permitting, I'll exhibit Segal's correspondence for a basic sigma-model with non-linear target S^1. Come to this talk if you want to know what: integrals over infinite dimensional spaces, zeta-function determinants of Laplacians on differential forms over genus 1 elliptic curves, eta-functions and Poisson Summation has to do with physics.
In 1965 Bruno Buchberger devised a solution to the problem of ideal membership in polynomial rings known as Gröbner bases. In 1978 Bergman extended the notion of a Gröbner basis to the case where the variables in our polynomial ring do not commute. A discussion of the noncommutative algorithms, differences from the commutative setting, and some examples and applications will be given.
Let S be a closed oriented surface and G be the group of holomorphic automorphisms of the unit disk - PSU(1,1). We want to look at the space of homomorphisms from (S) into G. What is the topology of this space? What is its geometric significance? Starting from the definition of the group PSU(1,1), we are going to look at the conjugacy classes in PSU(1,1), construct its universal covering space and look at product of commutators. Finally, we will try to give some answers to the above questions.
I will present some results for the class of finite groups in which every two subgroups having the same order are necessarily isomorphic, denote this class by ISO. Of course, there are many examples of such groups: cyclic groups, elementary abelian p-groups, Sym(3). In particular, we will show that if the group G is element of ISO is nonabelian then G is simple iff G is isomorphic to SL(2, 2^k) for some k greater or equal to 2.
In this talk I will explain how to derive the general equations of motion for the n-dimensional free rigid body using the underlying symplectic structure and results from the theory of Lie groups and Lie algebras. I'll show that the n-dimensional system is Hamiltonian and integrable. In particular I will use the more concrete 3 dimensional case as a motivating example and discuss the stability of equilibrium solutions to the 3D case. If time permits, I'll add some reflections on merry-go-rounds, figure skating, lakes, and the stability of equilibrium solutions in the general n dimensional case.
Curvature flows are a means of deforming geometric objects, such as curves, into regular geometric objects, such as round circles, only using local information. Flows such as Ricci flow, mean curvature flow, and inverse mean curvature flow have been important tools in understanding and solving geometric problems such as the Poincare conjecture and the Penrose inequality. In this talk we will analyze an analogue of curve shortening flow on polygons and see that almost any curve will deform to an affine transformation of the regular polygon. If time permits we will also look at a curvature flow on triangulated surfaces which can be used to discover circle packings.
The goal of this talk is to show a method of algorithmically constructing a projective R/I-resolution of a right R/I-module M. A basic outline of the construction and an example will be given.
The Shafarevich-Tate group of an elliptic curve defined over a number field plays an important role in the arithmetic of that curve. Our goal will be to define this group and then give a pleasant geometric characterization of its elements.
Do you know what you get if you intersect a polynomial in two complex variables with a small ball centered on an isolated singularity of the polynomial? The answer is a very nice knot. These knots classify plane curve singularities up to topological equivalence. I will start with basic definitions, describe the knots for irreducible curves and talk about the reducible case.
Wanna win a quick buck or two? Challenge your friends to a game of Mogul, a vast generalization of the children's game of "matches," all the while using algebraic coding theory to mentally calculate the optimal strategy. In addition to this game-theoretic application, we'll see tie-ins between Golay codes and number theory, group theory, algebraic geometry, and even sphere-packing, all (theoretically) within an hour!
I will discuss the intimate relationship between Lie groups, Lie algebras and their intermediary, Formal groups.
Let U be an s x r matrix (s \leq r) whose elements are indeterminates. Consider the ideal in the polynomial ring over R with sr indeterminates, which is generated by subdeterminants of order s. In this talk, I will discuss some properties of this ideal.
I will discuss the bound states and scattering states for the two-dimensional Dirac Operator by looking at the spectrum of the one dimensional differential operator.