Visiting mathematicians from UC San Diego, Alina Bucur and Kiran Kedlaya, will give two number theory talks on Thursday, February 24 in Keller 301.

Schedule:

3-3:45 PM Kiran Kedlaya

The relative class number one problem for function fieldsAbstract: Gauss conjectured that there are nine imaginary quadratic fields of class number 1; this was resolved in the 20th century by work of Baker, Heegner, and Stark. In between, Artin had introduced the analogy between number fields and function fields, the latter being finite extensions of the field of rational functions over a finite field. In this realm, the class number 1 problem admits multiple analogues; we recall some of these, one of which was “resolved” in 1975 and then falsified (and corrected) in 2014, and another one of which is a brand-new theorem in which computer calculations (in SageMath and Magma) play a pivotal role.

3:45-4:15 PM

Q&A, break, refreshments

4:15-5 PM Alina Bucur Counting points on curves over finite fields

Abstract: A curve is a one dimensional space cut out by polynomial equations. In particular, one can consider curves over finite fields, which means the polynomial equations should have coefficients in some finite field and that points on the curve are given by values of the variables in the finite field that satisfy the given polynomials. A basic question is how many points such a curve has, and for a family of curves one can study the distribution of this statistic. We will give concrete examples of families in which this distribution is known or predicted, and give a sense of the different kinds of mathematics that are used to study different families. This is joint work with Chantal David, Brooke Feigon, Kiran S. Kedlaya, and Matilde Lalin.

Graduate school can be an alternating series of rewarding and challenging experiences. Piper Harron earned her PhD in mathematics from Princeton University in January 2016, after starting in 2003, leaving in 2009, having her first child in 2011, and her second child in 2014. She described her unique journey in a talk for the UH Manoa Women’s Studies Fall 2016 Colloquium Series.

On Friday Sept. 20, 2013 Prof. Jacek Brodzki of U. Southampton will give a colloquium lecture titled Subspaces of Groups and C^* Algebra Extensions.

The colloquium will take place at 3:30pm in Keller 401

While a great deal of information about the representation theory of a group is contained in its reduced C*-algebra, this is a very challenging object to study. One way to understand the structure of an operator algebra is to construct a C*-algebra extension, that is an exact sequence connecting the algebra under consideration to other algebras, the properties of which might already be known. In the case of groups, an ingenious way of constructing such extensions was proposed in the 1980s by Pimnser and Voiculescu, first for free groups, and then for groups acting on trees. This was a breakthrough result with many important consequences.

In this talk I will present a geometric picture that explains how extensions of this type arise, and how this unifying approach connects a number of important results of Lance, Pimsner and Voiculescu and others. A main ingredient in our construction is an operator algebra associated with a metric subspace of a discrete group, which plays the role of the reduced C*-algebra of a group. I will present several examples of how the interaction between the geometry of a subspace with that of the ambient group leads to interesting C*-algebra extensions. The talk will be aimed at non-specialists.