Speaker: Elizabeth Gross (San Jose State U.)

Title: Goodness of fit of statistical network models

Abstract: Exponential random graph models (ERGMs) are families of distributions defined by a set of network statistics and, thus, give rise to interesting graph theoretic questions. Indeed, goodness-of-fit testing for these models can be achieved if we know how to sample uniformly from the space of all graphs with the same network statistics as the observed network. Examples of commonly used network statistics include edge count, degree sequences, k-star counts, and triangle counts. In this talk, we will introduce exponential random graph models, discuss the geometry of these models, and show the role toric ideals play in determining the quality of model fit.

Speaker:
Leslie Hogben
Iowa State University and American Institute of Mathematics

Title:
Power domination and zero forcing: Using graphs to model real-world problems

Abstract:
A graph $G = (V, E)$ is a set of vertices $V = {1, dots , n}$ and set of edges $E$ of
two element sets of vertices. A graph can be used to model connections between
vertices, such as airline routes between cities, internet connections, a quantum
system, or an electric power network.
Power domination and zero forcing are related coloring processes on graphs.
We start with a set of vertices colored blue and the rest colored white. We apply
a color change rule to color the white vertices blue. A set of blue vertices that
can color all vertices blue by using the power domination color change rule (or
zero forcing color change rule) is called a power dominating set (or a zero forcing
set). Finding a such set allows us to solve various problems, and a minimum
such set can provide an optimal solution.
In an electric power network, a power dominating set (blue vertices) gives
a set of locations from which monitoring units can observe the entire network.
In a quantum system, a zero forcing set (blue vertices) gives a set of locations
from which the entire system can be controlled.
This talk will describe power domination and zero forcing processes on
graphs and some of their applications.

This semester the Logic Seminar continues at a new day and time, Fridays at 2:30 in Keller 314.

For the first meeting this Friday I will (probably) speak about _Skolem polynomials_:

Abstract:
Over 100 years ago Hardy proved that a certain large class of real functions
was linearly ordered by eventual domination. In 1956 Skolem asked
whether the subclass of integer exponential polynomials is *well*-ordered
by the Hardy ordering, and conjectured that its order type
is epsilon_0. (This class is the smallest containing 1, x, and closed
under +, x, and f^g.) In 1973 Ehrenfeucht proved that the class is
well-ordered, and since then there has been some progress on the order
type.

The proof of well-ordering is rather remarkable and very short, and I
will attempt to expose it (which is to say, cover it) in the hour.

David Ross

Mushfeq Khan will speak on amenability and symbolic dynamics.
As usual the seminar is in Keller 314.

A Talk Story in Number Theory.

There is a childish misconception that the occupation of a professional mathematicians
is to operate with very big numbers. That is presumably primarily applicable to those who
do Number Theory. In this talk, I will show that this sometimes may be not too far from truth.

The talk is supposed to be entertaining and is directed to grad students willing to get a rough idea
about what it takes (and what it may give) to choose Number Theory as a research speciality.

Speaker: Claus Sorensen (UCSD)
Title: Local Langlands in rigid families
Abstract: The local Langlands correspondence attaches a representation of GL(n,F) to an n-dimensional representation of the Galois group of F (a local field). In the talk I will report on joint work with Johansson and Newton, in which we interpolate the correspondence in a family across eigenvarieties for definite unitary groups U(n). The latter are certain rigid analytic varieties parametrizing Hecke eigensystems appearing in spaces of p-adic modular forms. These varieties carry a natural coherent sheaf and we show that its dual fibers are built from the local Langlands correspondence by taking successive extensions; even at the non-classical points. Our proof employs certain elements of the Bernstein center which occur in Scholze’s trace identity. The first half of the talk is intended for a general audience with a limited background in number theory.

Continuing the theme of symbolic dynamics, I will demonstrate a proof of Simpson’s result that “Entropy = Dimension” for N^d and Z^d, and discuss some of Adam Day’s work generalizing these results to amenable groups.