A fundamental observation in the study of one-dimensional complex dynamics is that the orbits of the (finite set of) critical points of a function f largely determine the dynamics of f on the whole projective line. So the study of PCF functions has a long history in complex dynamics, including Thurston's topological characterization of these maps in the 1980s and continuing to the present day. More recently, number theorists have advanced the idea of PCF functions as the right dynamical analog of elliptic curves with complex multiplication, suggesting that these functions may have special arithmetic interest as well. In this talk, I'll give an overview of some recent developments in the area of the arithmetic of PCF rational functions.
Some arithmetic properties of post-critically finite rational functions
Mar. 03, 2015 1pm (MATH 220)
Grad Algebra/Logic
Nikki Sanderson (CU Boulder)
X
In this talk, I will explain the connection between languages and automata. Focusing on regular languages and finite automata, I will present Brzozowski's algorithms to find canonical finite automata of interest - namely, the minimal DFA and the átomata. I will then briefly introduce a possible extension of Brzozowski's algorithm to the setting of stationary stochastic processes.
Languages and Automata
Mar. 03, 2015 2pm (MATH 350)
Lie Theory
Richard Green (CU)
X
Pancake sorting is the problem of sorting a disordered stack of pancakes in order of size, when a spatula can be inserted at any point in the stack and used to flip all pancakes above it. A pancake number is the maximum number of flips required for a given number of pancakes. I will survey some of what is known about these numbers and explain how they are connected to the problem of generating Coxeter groups of type A using longest elements of parabolic subgroups. I will also discuss a signed version of this problem, which relates to Coxeter groups of type B, as well as applications of these ideas to parallel processing and to genomics.
The rank of an elliptic curve , defined over Q, is the rank of the finitely generated abelian group E(Q) of rational points on E. This quantity is still rather mysterious in many ways. In the last five years, there has been significant progress on understanding the average rank of elliptic curves, led by work of Bhargava-Shankar. We will survey these results and the ideas behind them, as well as mention generalizations in various directions and some corollaries of these types of theorems. We will also describe recently collected data on ranks of elliptic curves (joint work with J. Balakrishnan, N. Kaplan, S. Spicer, W. Stein, and J. Weigandt).
This talk will be suitable for a general mathematical audience.
There are well-known explicit families of K3 surfaces equipped with a low degree polarization, e.g., quartic surfaces in P^3. What if one specifies multiple line bundles instead of a single one? We will discuss representation-theoretic constructions of such families, i.e., moduli spaces for K3 surfaces whose Neron-S\'everi groups contain specified lattices. These constructions, inspired by arithmetic considerations, involve explicit geometry and combinatorics. For some of these families, each K3 surface has many automorphisms with positive entropy.
This is joint work with Manjul Bhargava and Abhinav Kumar.
Families of lattice-polarized K3 surfaces Sponsored by the Meyer Fund
Some arithmetic properties of post-critically finite rational functions