Mathematics Department

# Operator Algebras and Dynamics Seminars

## Fall 2016

All talks are from 3:45-4:45 p.m. in the Seminar room, unless otherwise specified.

• Dec
06
• Algebraic information theory
Dr. Keye Martin
Naval Research Laboratory
Time: 12:00 PM

#### View Abstract

One goal of our current research is to define a new area called algebraic information theory. It began with the realization that many important classes of channels, both quantum and classical, possess the structure of a compact affine monoid. The idea is then to use this structure as the basis for a unified approach to information theory, one that can be used to solve optimization problems, derive useful inequalities and develop methods for interrupting and improving both classical and quantum communication.
• Dec
02
• The group of automorphisms of a zero entropy subshift
Van Cyr
Bucknell University
Time: 03:45 PM
• Nov
18
• Local limit theorem and mixing for certain hyperbolic flows
Peter Nandori
University of Maryland
Time: 04:00 PM

#### View Abstract

Let us consider a suspension (semi-)flow with roof function $\tau$ over some map $T: X \rightarrow X$. For both integrable and non-integrable $\tau$, we try to give some abstract conditions, under which the flow is mixing. In case of integrable $\tau$, some extra conditions on the observable imply a joint extension of mixing and the local (central) limit theorem (LLT) for the flow. The most important condition is the LLT for the map $T$. Examples include certain flows derived from the Liverani-Saussol-Vaienti map (infinite measure case) and some suspensions over systems possessing a Young tower with exponential return time (finite measure case). Joint work in progress with Dmitry Dolgopyat.
• Nov
07
• Random Walks and Harmonic Functions on Groups
Darren Creutz
USNA
Location: CH320
Time: 03:45 PM

#### View Abstract

A natural question in geometric group theory is to study the random walk of a finitely generated group. Specifically, for a probability distribution mu on a finite generating set S, one considers the behavior of the random walk on the Cayley graph built from S with law mu (meaning at each step in the walk, we choose which edge in S to follow according to mu). In particular, one considers the exit boundary of the walk--the space of all distinct paths to infinity. Another natural question is to study the space of bounded mu-harmonic functions on G: functions f : G --&gt; Reals such that for each g in G, Sum_{s in S} f(gs) mu(s) = f(g). The classical Dirichlet problem establishes a correspondence between bounded harmonic functions on SL_2 (the fractional linear transformations) and bounded measurable functions on the unit circle. I will present Furstenberg's Poisson Boundary construction which establishes that random walks on groups and harmonic functions are both determined by the bounded measurable functions on the boundary" of the random walk. In particular, the bounded harmonic functions are in one-one correspondence with the bounded measurable functions on the boundary. A concrete example of this is the free nonabelian group on two generators F_2: the Cayley graph (for the usual generating set) is the regular 4-tree and the natural weighing is to give all 4 directions equal weight; the boundary here is the big circle", the boundary of the 4-tree, and the harmonic functions on F_2 are in one-one correspondence with L^infinity of the big circle.
• Oct
28
• Unique Expectations and Pseudo-Expectations for Abelian C*-inclusions
Vrej Zarikian
Time: 03:45 PM
• Oct
21
• Pattern Formation and Fluctuations in Complex Networks
Jason Hindes
Naval Research Laboratory
Time: 03:45 PM

#### View Abstract

Networks form the backbone of complex communication systems ranging from computer and social networks to swarming sensor arrays. Much attention in recent years has focused on determining how the connectivity of such networks affects the types of behavior they can produce. However, many networks of interest operate in noisy environments and fluctuate due to random internal effects, both of which can cause sudden transitions from one network state to another. In this talk, I will survey the dynamics of several well known processes, focusing on swarm pattern formation and epidemic spreading, and discuss recent advances in understanding noise-induced pathways between distinct network states using large fluctuation theory.
• Oct
14
• Using Maths to Understand the Transmission of Infectious Diseases
Luis Mier-y-Terab
Johns Hopkings Bloomberg School of Public Health, Naval Research Lab
Time: 03:45 PM
• Sep
12
• Metric on regular languages using topological entropy
Kelly Yancey
Institute for Defense Analyses - Center for Computing Sciences
Time: 03:45 PM

#### View Abstract

A problem that has emerged in computer science is determining the similarity between regular languages. We will represent a regular language by a deterministic finite automata (a directed graph with some marked data) and then use ideas from symbolic dynamics to develop a metric between the languages. We will also discuss other distances based on the classical Jaccard distance and how they are related to the topological entropy of a regular language. There will be no prior knowledge of automata theory assumed.
• Aug
26
• Character Rigidity for Lattices in Lie Groups
Darren Creutz
USNA
Location: CH320
Time: 03:45 PM

#### View Abstract

Characters on groups (positive definite conjugation-invariant functions) arise naturally both from probability-preserving actions (the measure of the set of fixed points) and unitary representations on finite factors (the trace); the classical theory of characters is the first step in the classification of finite simple groups and culminates in the Peter-Weyl theorem for compact groups. I will present the results of J. Peterson and myself that the only characters on lattices in semisimple groups are the left-regular character and the classical characters. This is in actuality operator-algebraic superrigidity for lattices, answering a question of Connes. The main idea is to bring dynamics into the operator-algebraic picture; the second half of the talk will focus on the ergodic-theoretic ideas of contractiveness and the Poisson boundary and how these ideas lead to operator-algebraic results.