Mathematics

Logarithmic Corrections to Black Hole Entropy

Finn Larsen (Michigan Physics)

Abstract: Logarithmic corrections to supersymmetric black holes offer a unique window into the precistion black hole entropy. Related considerations play a role in several other current directions of research. We present a selfcontained and elementary on-shell computation of these corrections that takes advantage of the symmetries in the near horizon geometry. For bulk modes interactions are incorporated using group theory alone. The spectrum of boundary states is identified explicitly. The final result is the sum of elementary contributions in 4D, 2D, and 0D.

Homology and dynamics of pseudo-Anosovs

Chris Leininger (UIUC Math)

Abstract: I'll explain a connection between a pseudo-Anosov homeomorphism's stretch factor, and its action on homology. This provides a kind of interpolation between a result of Penner and our prior work with Farb and Margalit, and answers a question of Ellenberg. This is joint work with Agol and Margalit.

Uniformization of metric surfaces

Kai Rajala (University of Jyväskylä)

Abstract: We discuss uniformization results for metric spaces that are homeomorphic to the euclidean plane or sphere and have locally finite Hausdorff 2-measure. We give a necessary and sufficient condition for such spaces to be quasiconformally equivalent to a euclidean space. We also discuss connections to quasisymmetric parametrization problems.

Zeros of families of L- functions

Junxian Li (UIUC Math)

Abstract: In order to study the zeros of the Riemann zeta function, Spira considered a family of approximations of the Riemann zeta function defined as $ \zeta_N(s)=\sum_{n \leq N} n^{-s}+\chi(s)\sum_{n\leq N} n^{1-s} $. These functions approximate to the Riemann zeta function as $N$ goes to $\infty$. He also proved all the zeros of $\zeta_1(s)$ and $\zeta_2(s)$ lie on the critical line, and suggested infinitely many zeros off the critical line. Montgomery and Gonek studied the zeros of this family and obtained asymptotic formulas for the number of zeros up to height $T$ within a critical strip and on the critical line, which implies that 100% of the complex zeros lie on the critical line as $T$ goes to $\infty$ provided $N$ is not too large compared to $T$. We investigated the zeros of approximations of Hecke L-functions associated to cusp forms and Dedekind zeta functions and discovered similar behaviors. This is a joint work with A. Roy and A. Zaharescu.

Uniformization of metric surfaces

Kai Rajala (University of Jyväskylä)

Abstract: We discuss uniformization results for metric spaces that are homeomorphic to the euclidean plane or sphere and have locally finite Hausdorff 2-measure. We give a necessary and sufficient condition for such spaces to be quasiconformally equivalent to a euclidean space. We also discuss connections to quasisymmetric parametrization problems.

Cubic surfaces and error-control codes

John Little (College of the Holy Cross)

Abstract: In a very interesting application of algebraic geometry, Goppa's construction from the 1980's has led to some very good error-control codes. Goppa's original method started from an algebraic curve over a finite field and produces codes by two dual constructions. Recently, there has been interest in extending this construction to make use of higher-dimensional varieties over finite fields in similar ways. We will discuss how some ideas of M. Zarzar can be applied to the test case of codes from cubic surfaces over a finite field. The classical geometry of the 27 lines on a smooth cubic surface, in combination with their arithmetic properties determine the parameters of the corresponding codes.

The Magic of Prime Characteristic

Karen Smith (Michigan)

Abstract: Many a calculus student has used the trick (x+y)^p = x^p + y^p to dramatically simplify calculations and sometimes even prove remarkable statements unbelievable to their professors. In this lecture, I hope to show you a context where this trick is valid: the world of "characteristic p." This trick has been used to understand complex varieties better---for example, to see that certain cohomology groups vanish or that certain kinds of differential forms exist. It has been used to show that rings of invariants for nice group actions has a particularly nice structure. Most recently, it has been used to show that a natural class of combinatorial algebras called ``cluster algebras" arising in many contexts have especially nice properties. The latter work is joint with Angelica Benito, Greg Muller and Jenna Rajchgot.