The Geometry-Topology Seminar has two speakers this week: Tim DeVries from the University of Pennsylvania and Elena Fuchs from of the Institute for Advanced Study. Talks will be held in Wachman 617, starting at 3:30 (Fuchs) and 5:00 (DeVries).
Fuchs will speak on
Title: Counting in Apollonian circle packings
Abstract: Apollonian circle packings are constructed by continuously inscribing circles into the curvilinear triangles formed in a Descartes configuration of mutually tangent circles. An observation of F. Soddy in 1937 is that if any four mutually tangent circles in the packing have integer curvature, then in fact all of the curvatures in the packing will be integers. In the past few years, this observation has led to several developments regarding the number theory of such integer Apollonian packings. In this talk, I will discuss a very generalizable approach to counting integers appearing as curvatures in integer Apollonian packings. I will also discuss some natural questions to consider along these lines. This is joint work with J. Bourgain.
DeVries will speak on:
Title: An Algorithm for Bivariate Singularity Analysis
Abstract: How do you count? Of primary interest to enumerative combinatorists is obtaining counting formulas for various discrete, mathematical objects. For instance, what is the nth Fibonacci number? What is the n,mth Delannoy number? A common technique is to embed the sequence as the coefficients of a formal power series, known as a generating function. When this function is locally analytic, we hope that its analytic properties may help us to extract asymptotic formulas for its coefficients. We will explore this technique, known as singularity analysis, in the case that the generating function is bivariate rational. We then sketch an algorithm that, for many such generating functions, automatically produces these asymptotic formulas. Despite its combinatorial origins, this algorithm is quite geometric in nature (touching on topics from homology theory, Morse theory, and computational algebraic geometry).