It is well known that the square root of any integer can be written as a linear combination of roots of unity. A generalization of this fact is the "Kronecker-Weber Theorem", which states that in fact any element which generates an abelian Galois extension of the field of rational numbers Q can also be written as such a linear combination. The roots of unity may by viewed as the special values of the analytic function e(x) = exp(2*pi*i*x) where x is taken to be a rational number. Broadly speaking, Hilbert's 12th problem is to find an analogous result when Q is replaced by a general algebraic number field F, and in particular to find the analytic functions which play the role of e(x) in this general setting. Hilbert's 12th problem has been solved in the case where F is an imaginary quadratic field, with the role of e(x) being played by certain modular forms. All other cases are, generally speaking, unresolved. In this talk I will discuss the case where F is a real quadratic field, and more generally, a totally real field. I will describe relevant conjectures of Stark and Gross, as well as current work using a p-adic approach and methods of Shintani. A proof of these conjectures would arguably provide a positive resolution of Hilbert's 12th problem in these cases.

We will be planning the preparatory talks for the Distinguished Women in Math colloquia. ALL students are invited to attend.

This semester we are studying topological quantum computation, roughly following http://arxiv.org/pdf/0707.1889?fname=cm&font=TypeI

We will cover some basics of algebraic surfaces, including intersection theory, adjunction formula, Hodge Index theorem and numerical criterion of ampleness.

One can ask whether two manifolds have a common finite sheeted cover. This property induces an equivalence relation on hyperbolic 3-manifolds. Hyperbolic knot complements appear to be fairly rare inside a fixed commensurablity class. In fact, there is a conjecture that states, there are at most three hyperbolic knot complements in a commensurability class. I will talk about how knot complements can appear in a commensurability class and some examples where the conjecture is realized. This is a practice talk for a candidacy talk.

I will describe some numerical methods based on topological degree and the Conley Index that can be used to analyze continuous discrete dynamical systems. Some applications will be given including computing fixed points, periodic orbits, and horse shoes in specific examples.

The three invariants of a quadratic form over are the rank, the discriminant, and the Hasse invariant. The two last ones can be obtained in terms of classes in the first and second cohomology groups , where for some separable closure of . In this talk we will see how Kummer theory provides the interpretation and we will try to unravel what is. The Brauer group will show up, and we will try to untangle it as well.

Abstract: I will briefly describe modular tensor categories and discuss a few of their properties in some detail, some open classification questions and some recent progress with Z. Wang and others. As time permits, I will describe a motivating application. References: 1) For a survey of MTCs from quantum groups see http://arxiv.org/abs/math/0503226 2) This talk will be based mostly on joint work with Wang and Stong http://arxiv.org/abs/0712.1377 3) Physical and computational motivation can be found in Freedman et al http://arxiv.org/abs/0707.1889

** PLEASE NOTE THAT TIME AND LOCATION HAVE CHANGED.

In this talk, I'll explain how one can construct a specific topological space from any category, called its classifying space. In the specific case that one starts with a group G (thought of as a category in a suitable way), this construction yields a space BG whose fundamental group is equal to G. In fact, we'll see that BG can be obtained as a quotient of a contractible space X via a free covering action of G on X.

One can ask whether two manifolds have a common finite sheeted cover. This property induces an equivalence relation on hyperbolic 3-manifolds. Hyperbolic knot complements appear to be fairly rare inside a fixed commensurablity class. In fact, there is a conjecture that states, there are at most three hyperbolic knot complements in a commensurability class. I will talk about how knot complements can appear in a commensurability class and some examples where the conjecture is realized. This is a candidacy talk.