Portland State University

Our seminar features expository talks on topics in analysis that reflect the participants' interests, but often don't get into courses. In previous years we've heard talks on functional, complex, and harmonic analysis, probability, and dynamical systems. The atmosphere is relaxed and supportive; everyone is welcome to participate: give a talk, or just be part of the audience.



Lecture notes 2013-14

Joel Shapiro's lecture notes on:
Burnside's Theorem on Matrix Algebras

Joel Shapiro's lecture notes on:
The Invariant Subspace Problem

Blake Rector's lecture notes on:
Characterizations of differentiability in Rn

Cody Fuller's lecture notes on:
Quantum Nash Equilibrium

Steve Bleiler's slides from his lecture on Quantum Games:
A Formalism for Quantum Games and an Application

Steve Silverman's notes on Quantum Mechanics:
Local Realism
God Plays Dice

Nguyen Mau Nam's notes on Nash Equilibrium:
Nash Equilibrium via Convex Analysis

Joel Shapiro's notes on Nash Equilibrium via the Brouwer Fixed-Point Theorem
Equilibrium in non-cooperative games


Spring Term Schedule 2014:

Friday, May 9 @ 2 PM in NH 346
Mau Nam Nguyen will speak on: "Convex Analysis and some Applications."
Abstract. In this talk we present a simple path to understanding generalized differentiation of convex objects in finite dimensions. A geometric approach is employed to develop basic calculus rules for normal cones to convex sets and subgradients of convex functions that are in the mainstream of convex theory. We also present applications of convex analysis to some problems of convex optimization.
Friday, May 23 @ 2 PM in NH 346
George Nicol will speak on: "Closure Systems II."
Abstract. A closure system (X, k), made up of an non-empty set X and a closure function k, generates an Intersection Space (X, t), a topological space in which the arbitrary intersection of open sets is open.   An example will be presented that shows how to generate a non-discrete, non-trivial Intersection space for any group G, which then can be extended into a Boolean Algebra.

  In addition, a specific type of closure function called a cfilter and how it generates an Intersection space for any group will be shown.  How cfilters form a commutative, idempotent semi-group with identity will be presented.

Also to be given is a Minimal-Information Problem, the solution of which is found in a cfilter table which in turn generates an interesting class of numbers called S-Numbers.

Finally, time permitting, an example of an Automata that is both a group and an Intersection Space will be shown.  There seems to be no shortage of examples of Intersection Spaces.
Friday, May 16 @ 2 PM in NH 346
Steve Silverman will speak on: "A Minimal Information Problem."
Abstract. Sometime in the last 200 years a spaceship from an alien race landed on Earth in order to determine if it would make a nice way station to hunt and dine on humans. They did a quick but accurate survey and and found there were two types of humans, the edibles and the toxics; furthermore if two humans were selected at random from the entire population the probability that they would both be toxic was 1/2. This was deemed too high to make a go of the enterprise so the aliens left.

Question: Who was the US president when they landed?

Though the question may sound frivolous, this is a real mathematical problem; some of the techniques involved in the solution are sophisticated, however the main ideas are easy to follow.
Friday, May 9 @ 2 PM in NH 346
Joel Shapiro will speak on: "Burnside's Theorem on Matrix Algebras."
Abstract. Burnside's Theorem (1905) says that every proper subalgebra of n x n complex matrices has a nontrivial invariant subspace. I'll present Lomonosov and Rosenthal's recent (2004) short proof of this result and, if time permits, will show how Burnside's Theorem yields a generalization of Schur's triangularization theorem to commuting families of matrices.
Friday April 4 & 11 @ 2 PM in NH 346
Joel Shapiro will continue speaking on: "The Invariant Subspace Problem."

Winter Term Schedule 2014:

Friday, March 14 @ 2 PM in NH 346
Joel Shapiro will speak on: "The Invariant Subspace Problem."
Abstract. This famous open problem asks if every continuous linear operator on infinite dimensional Hilbert space has a nontrivial closed invariant subspace. I'll give some history and motivation and then present Victor Lomonsov's landmark result that gives an affirmative answer for any operator that commutes with a non-zero compact operator. Although over 40 years old, Lomonosov's methods continue to inspire present-day research on the problem.

Friday, February 14 @ 2 PM in NH 346
Blake Rector will speak on: "Characterizations of Differentiability in Rn."
Abstract. In this talk we discuss three notions of differentiability for functions defined on Rn (the Gâteaux Derivative, the Fréchet Derivative, and the directional derivative) and their relationship to the sub-differential of convex analysis.

Friday, January 24 @ 2 PM in NH 346
Cody Fuller will speak on: "Quantum Nash Equilibrium II."

Friday, January 17 @ 2 PM in NH 346
Cody Fuller will speak on: "Quantum Nash Equilibrium I."

Friday, January 10 @ 2 PM in NH 346
Steve Bleiler will speak on: "Quantum Multiplexers, Parrando Games, and proper quantization of Markov Processes "
Abstract. The multiplexing circuit lies at the heart of logic synthesis and their quantum analogues are of particular interest to the designers of the coming quantum computers. Traditional logic synthesis relates these circuits to a special type of game, originally analysed by Parrondo et al in the classical setting using the theory of Markov processes. Until this work, the proposed "quantizations" of the games and of the corresponding Markov processes have lacked a specific "game theoretic" property which has hindered both the synthesis and analysis of the corresponding quantum multiplexing circuits. Here that deficiency is repaired and new more accurate measures of circuit behavior are introduced.


Fall Term Schedule 2013:

Friday, December 6 @ 2 PM in NH 346
Steve Bleiler will speak on: "Introduction to Quantum Games"


Friday, November 15 & 22 @ 2 PM in NH 346
Steve Silverman will speak on: "An Easy Introduction to Quantum Mechanics"
Abstract. I'll cover the usual stuff in the lay literature: state, superposition, wave particle duality, interference, a new version of an uncertainty principle, entanglement, Bell Theorems, quantum computers, etc. It will be both rigorous and elementary.

Friday, November 8 @ 2 PM in NH 346
Mau Nam Nguyen will speak on: "Nash Equilibrium via convex analysis"
Abstract. I’ll present a simple proof of Nash’s famous theorem on the existence of Nash Equilibrium. The argument (due to R. T. Rockafellar) uses very basic elements of convex analysis, and the Brower Fixed-Point Theorem. The talk will be self-contained.

Friday, November 1 @ 2 PM in NH 346
Serge Preston will speak on: "How it was at the beginning. Newton, Hook and some others-- unknown history"


Friday, October 18 @ 2 PM in NH 346
Joel Shapiro will speak on: "How I learned to stop worrying and love Nash Equilibrium."
Abstract. I’ll describe John Nash’s notion of game-theoretic ``equilibrium’’ and show how Nash used the Brouwer Fixed Point Theorem to prove that such equilibria “always” exist.

This is a review and continuation of Steve Silverman's talk from last June, which featured Nash's short proof that based on Kakutani's Fixed-Point Theorem. I'll give another proof due to Nash that directly uses the Brouwer Fixed-Point Theorem. The talk will be self-contained.

Friday, October 11 @ 2 PM in NH 346
Peter Veerman will speak on: "Control of transients in the synchronization of oscillator networks."
Abstract. We develop a general quantitative theory for the growth of transients in large arrays of linear oscillators on the real line with decentralized nearest neighbor interaction. It turns out that in the best possible case, transients grow proportional to the number
of agents. The proportionality constant is related to the signal velocity in a related system, where it can be determined exactly. The numerics give excellent agreement with the theory. We apply the theory to various situations from the literature.


Seminar schedule 2012-2013

Lecture notes from previous year's seminars: