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.

Friday, June 8 in East Hall, Room 236: 2:00--3:00 PM

Strange Convex Sets

Abstract.Given a closed convex set Ω ∈ R^{n}, the metric projection of a given point x ∈ R^{n}is given by the unique point Π(x) ∈ Ω that minimizes the (Euclidean) distance {|y − x| | y ∈ Ω} between Ω and x.

Most mathematicians tend to think of convex sets in R^{n}as very tame objects. It is therefore surprising that it is easy to construct a compact convex set Ω in R2 with the following strange property [A. Shapiro, 1994]: There is a point x not in Ω and a vector v such that the directional derivative of Π in the direction of v fails to exist.

We revisit and modify that construction to obtain a convex curve in R^{2}that is C^{1,1}, i.e., differentiable with Lipschitz derivative, and that this curve bounds a convex set that has the property that the directional derivative of the projection is not defined. We also show how this construction can be made C^{n}for n ≥ 2 except at a single point, and such that directional differentiability still fails.

Friday, June 1 in East Hall, Room 236: 2:00--3:00 PM

The Radon-Nikodym Theorem ... Revisited

Friday, May 25 in East Hall, Room 236: 2:00--3:00 PMAbstract.This is a talk about ordered vector spaces, operators between them, and analogues of topological notions such as continuity and convergence that are present in the absence of a topology by virtue of the order structure. In this context, we'll get the Radon-Nikodym theorem as a corollary to a theorem about operators between ordered vector spaces.

Haar Measure

Abstract.I will describe the existence of a left-invariant measure on locally compact groups. Maybe also its uniqueness.

Friday, May 11 in East Hall, Room 236: 2:00--3:00 PM

Radon-Nikodym: "Made Easy"

Abstract.We'll discuss John von Neumann's beautiful (but mysterious) proof of the Radon-Nikodym Theorem. The setting will be Hilbert space, and the heavy lifting will be provided by the Riesz Representation Theorem.

Notes for this talk are available here.

Friday, May 4 in East Hall, Room 236: 2:00--3:00 PM

"Equidistribution of sequences on thep-adic unit ball"

Abstract.Techniques from harmonic analysis play a crucial role in understanding problems in analytic number theory. For example, in 1916 Hermann Weyl initiated the study of the equidistribution of sequences on the additive circle, connecting fourier analysis to number theoretic dynamics.

Such techniques can be extended to other locally compact abelian groups, leading to some interesting number theory. We look at the p-adic unit ball as one such example, and show how fourier analytic techniques can give us an understanding of the distribution of sequences.

This talk is primarily intended to give a general mathematical audience a flavor and appreciation of this type of mathematics, and only a basic knowledge of analysis will be assumed.

Friday, April 27 in East Hall, Room 236: 2:00--3:00 PM

" The Radon-Nikodym theorem and its applications"

Abstract.The Radon-Nikodym theorem is an important result in measure theory with applications to analysis and probability theory. We will outline a proof of the theorem and discuss its application to the concepts of conditional probability and conditional expectation.

Friday, April 20 in East Hall, Room 236: 2:00--3:00 PM

" On the generation of measures and basic probability"

Abstract.This talk has two parts. First we explore how to define an abstract measure following the same procedure used for the definition of Lebesgue measure as an extension of the notion of length of an interval. One application it to define Lebesgue measure in higher dimensions. Another is to define the Lebesgue-Stieltjes integral, which also has interesting applications in probability theory. This ties in to the second part of the talk where we explore in more detail the correspondence of several concepts of probability theory to those in measure theory.

Friday, April 13 in the

" Regularizing with Bregman-Moreau Envelopes"

Abstract.Moreau envelopes are useful for the regularization of convex functions in optimization. I will provide an introduction to Moreau envelopes and then go on to discuss Bregman envelopes, which are a useful generalization. I will then share some new results from joint work with Heinz H. Bauschke and Minh N. Dao.

Friday, March 2 in East Hall, Room 236: 2:00--3:00 PM

"The Kurzweil-Henstock Integral"

Abstract.We present the highlights of the Kurzweil-Henstock integral, also known by various other names: gauge integral, Denjoy-Perron integral, or (more modestly) the Generalized Riemann integral. This integral is an extension of both the Riemann and Lebesgue integrals, andits definition does not require measure theory.

Friday, February 9 & 26 in East Hall, Room 236: 2:00--3:00 PM

"An Hour of Noise"

Abstract.Noise is easy to observe but difficult to define precisely. We'll start our investigation with a look at a model of noise often used in finite discrete signal processing problems. This will motivate a discussion of Brownian motion. Using the discrete case as our guide we'll attempt to define a general stochastic process for white noise. Using the Wiener-Khinchine theorem we can show the relation between white noise and Brownian motion. Using the relationship to Brownian motion we can define the integral of white noise. We'll describe one of the generalizations of this integral and end with a description of Ito's stochastic integration.

Friday, January 26 & February 2 in Cramer Hall 124: 2:00--3:00 PM

"The SVD for Analysts"

Abstract.The Singular Value Decomposition represents linear operators by their ``stretching'' factors, calledsingular values. The largest singular value is well-known to analysts as the operator'snorm. ``Stretching'' factors are the analyst's bailiwick, and the SVD deserves to be in every analyst's toolbox.

We'll derive the SVD using simple analytic techniques, motivated by the operator norm. The derivation leads naturally to the notions of numerical rank, the pseudoinverse, and Jacobi's method for computing the SVD.

Friday, January 12 & 19, 2:00--3:00 PM in CH (Cramer Hall) Room 124

"Almost-everywhere convergence for Fourier series"

Notes for these talks are available here.Abstract.In what sense is an integrable function represented on a finite interval by its Fourier Series? Does the series converge to the function in some appropriate sense: in norm? Almost everywhere? We will investigate.

Friday, November 17 & December 1, 2:00--3:00 PM in NH 373

"Duality in Convex Optimization: Theory and Applications"

Abstract.A given optimization problem called the primal problem can be solved by formulating a new optimization problem called the dual problem. In this talk we present two duality approaches used broadly in convex optimization: Fenchel duality and Lagrange duality. Both theory and applications of Fenchel duality and Lagrange duality will be discussed in the talk.

Friday, October 27, 2:00--3:00 PM in NH 373

"Dirichlet Problem meets Maximal Function"

Notes for this talk are here.Abstract.We'll discuss the following Dirichlet problem: Given a Lebesgue integrable function f on the real line, find a harmonic function on the upper half-plane that "has f as its boundary values." How do you do this? What does it even mean? What does it have to do with maximal functions? We'll explore.

Revised notes for the "Lebesgue Differentiation/Maximal Function" talk are here.

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

"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

"Almost-Everywhere Convergence ... Done Right"

Notes for this talk are here.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.

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

"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, September 29, 2:00--3:00 PM in NH 373

"Variational Mechanics: Gradients in Function Space"

Notes for Jim's previous talks on variational mechanics are here.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 R^{n}can be reduced to a single gradient in function space. The talk is expository, requiring only a little calculus and linear algebra.