Portland State University

This 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.

Fall Term Schedule 2017:

Friday, October 20, 2:00--3:00 PM in NH 373

George Nicol will speak on:
"Relative-Strength Theorems for Q-Systems"
Abstract. A Q-system consists of four elements: a non-empty set X, two binary operations on X2, and a function of one variable defined on X. While the binary functions provide a way to combine elements of X algebraically, the function of one variable provides a way to create a topological or geometric structure for X. For a given Q-system, say Q, there is an inherent way to combine the binary functions of Q in such a manner as to give rise to pre-closure function on X and a method to determine the "relative strength" of the binary functions of Q. A proof for a one-dimensional relative-strength theorem is given for the natural numbers to show the relative-strength of addition and multiplication. There is also a two-dimensional relative-strength theorem for pairs of natural numbers as well.

Friday, October 13, 2:00--3:00 PM in NH 373

Joel Shapiro will speak on:
"Almost-Everywhere Convergence ... Done Right"
Abstract. In this talk I'll discuss the Lebesgue Differentiation Theorem ("Differentiation undoes integration a.e."), and show how it follows from the famous Hardy-Littlewood Maximal Theorem. I'll prove the Maximal Theorem and, if time permits, will discuss its connection with the "cosmic truth" about almost-everywhere convergence.
Notes for this talk are  here.

Friday, October 6, 2:00--3:00 PM in NH 373

Jim Rulla will speak on:
"Variational Mechanics II: Lagrange Multipliers"
Abstract. Lagrange multipliers are more meaningful than a mere ``trick pulled out of a hat''. The derivation, which requires only a little calculus and linear algebra, is an application of the ``fundamental subspaces'' of a linear operator. (The derivation is independent of last week's talk). Lagrange used his multipliers to solve variational problems.

The final example of the talk will use the gradient found last week to solve Dido's problem: "What is the largest area that can be bounded by a loop of string of fixed length?"

The talk is expository, requiring only a little calculus and linear algebra.


Friday, October 6, 2:00--3:00 PM in NH 373

Jim Rulla will speak on:
"Variational Mechanics: Gradients in Function Space"
Abstract. Lagrangian mechanics replaces Newton's (vector) momenta with (scalar) kinetic energies. Gradients turn the scalars into vectors, but there is a twist: the gradients with respect to position are treated differently from the gradients with respect to velocity. In this talk we'll show that in Lagrange's variational formulation, this "weird" combination of two different gradients in Rn can be reduced to a single gradient in function space. The talk is expository, requiring only a little calculus and linear algebra.
Notes for Jim's previous talks on variational mechanics are  here.

Seminar schedule and lecture notes 2016-2017

Seminar schedule and lecture notes 2015-2016

Seminar schedule and lecture notes 2014-2015

Seminar schedule and lecture notes 2013-2014

Seminar schedule and lecture notes 2012-2013