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.

This week Umar Gaffar will give Shelah’s proof of the following result:

Let $\lambda$ be the cardinality of an ultraproduct of finite sets. If $\lambda$ is infinite then $\lambda=\lambda^{\aleph_0}$.

Speaker: Rohit Nagpal

Title: Stability in the high dimensional cohomology of certain arithmetic groups

Abstract: Borel-Serre duality relates high dimensional cohomology of arithmetic groups to the low dimensional homology of these groups with coefficients in the Steinberg representation. We recall Bykovskii’s presentation for the Steinberg representation and explain its connection to modular symbols. Next, we describe the Steinberg representation as an object in a symmetric monoidal category, and use its presentation to describe an action of the free skew commutative algebra. Finally, we perform a Gröbner-theoretic analysis of this action to obtain new information on the homology of certain arithmetic groups with coefficients in the Steinberg representation. For example, we show that the sequence of homology groups H_1(Gamma_n(3), St_n) exhibit representation stability. This is an ongoing project with Jeremy Miller and Peter Patzt.