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, May 20 & 27, 2:00--3:00 PM in NH 346

"From Newton's Laws to Variational Calculus"

Abstract.The principle of virtual work is a powerful tool in classical mechanics, often shrouded in mystery. According to one (quite good) physics text:"We have not given a ``proof'' of the principle of virtual work, but rather an indication of some types of situation in which the principle holds. Readers will have to judge from physical considerations whether and in what sense the principle holds for the particular physical systems they wish to consider." (In these talks we'll address the question:Lagrangian and Hamiltonian Mechanics, World Scientific, 1996, p. 31.)We’ll prove the principle and show how Newton's laws (about differentiation with respect to time) imply variational principles (about differentiation with respect to functions). All that’s required is a bit of undergraduate linear algebra and vector calculus.What on earth possessed Euler and Lagrange to think of variational methods in the first place?

Notes for this talk are here

Friday, May 6 & 13, 2:00--3:00 PM in NH 346

"A Review of the Operational Calculus"

Abstract.The Titchmarsh Convolution Theorem shows the algebra of continuous functions on the right half line (with convolution as multiplication) to be an integral domain. We can thus construct the field of fractions à la the rationals from the integers with many glorious consequences (Exercise: can you recognize the the function given by e^{x}- sin x - cos x divided by the function sin x in this field?).

Friday, April 22, 2:00--3:00 PM in NH 346

"Proof of The Titchmarsh Convolution Theorem"

Notes for this talk, as well as the two previous ones, are hereAbstract.The Titchmarsh Convolution Theorem states: If f and g are continuous on the interval [0,a], and their convolution vanishes identically on [0,a], then there exists a point c in [0,a] with f vanishing on [0,c] and g vanishing on [0,a-c]. The theorem implies that the ring C([0,∞], with convolution multiplication, is an integral domain. Previous talks reduced it to the "One-half Lemma," the special case $f=g$, i.e.: If f*f=0 on [0,a] then f=0 on [0,a/2].

This time I'll review the meaning of Titchmarsh's famous theorem, and then prove the One-half Lemma, thereby finishing the proof of the full theorem. The talk will be self-contained; the previous ones on this subject won't be necessary for understanding this one.

Friday, April 15, 2:00--3:00 PM in NH 346

"The Natural Equivalence of Two Functors in Analysis"

Abstract.The assertion that: "There exists a natural equivalence between two functors in analysis, namely the 'measurability' functor and the dual of the 'continuity' functor, both of which map the category of compact Hausdorff spaces and continuous maps to the category of normed linear spaces and contractive linear maps," is little more than a compressed restatement of two standard theorems taught in every beginning real analysis course.

The point of this talk will be to unpack the preceding categorical assertion in order to recognize the identity of the familiar theorems. The theorems themselves will not be proved.

Friday, April 1 & 8, 2:00--3:00 PM in NH 346

"The Titchmarsh Convolution Theorem"

Notes for these talks are hereAbstract.Last term we saw that the Titchmarsh Convolution plays an important role in proving the theorem that characterizes the invariant subspaces of the Volterra operator (in fact it is equivalent to this theorem). In these talks I'll further motivate the Titchmarsh Theorem, and prove the version of it that exhibits C(0, ∞), with convolution as multiplication, to be an integral domain.

Friday, February 26, 2:00--3:00 PM in NH 346

"The Chebyshev Condition"

Friday, February 12 & 19, 2:00--3:00 PM in NH 346Abstract.One of the open problems in Convex Analysis is determining for what spaces the Chebyshev Condition holds. It has been proven for certain normed spaces and counter examples have been found for others, but no one has been able to find a general rule for determining when the condition will hold. Throughout this lecture we will explore what a Chebyshev Set is, what it’s relationship is to convexity, and we will prove that the Chebyshev Condition holds for any finite dimensional Hilbert Space.

"Frobenius' Theorem via Differential Forms"

Notes for this talk are hereAbstract.Frobenius' theorem, a fundamental result in differential geometry, gives higher dimensional versions of the fact that smooth vector fields have integral curves. This lecture will show how one can use differential forms to understand this famous theorem.

Friday, January 29 & February 5, 2:00--3:00 PM in NH 346

"Differential Forms for Beginners"

Abstract.In these talks I will offer some motivation for introducing the language of differential forms into a beginning calculus course and some suggestions for presenting the requisite formalism in a relatively painless fashion. Then I will examine some standard results in multivariable calculus expressed in terms of differential forms. In particular, we will see that the fundamental theorem of calculus, Green's theorem, the fundamental theorem for line integrals, Stokes' theorem, and the divergence theorem are all special cases of a single, simply stated result.

Friday, January 15 & 22, 2:00--3:00 PM in NH 346

"The Volterra Operator---its invariant subspaces"

Notes for this talk are here and and here.Abstract.The Volterra operator V is the continuous linear transformation of L^2([0,1]) you get by taking a function to its integral over the subinterval [0,x]. As with every bounded operator on Hilbert space, we want to study its invariant subspaces; V is one of the few operators for which these can be described completely. In these talks I'll make the case for studying invariant subspaces, describe what they are for the Volterra operator, and show how one proves the result. The main tool in the argument is the famous Titchmarsh Convolution Theorem for which I'll give an easy Volterra-assisted proof of a special case (a full proof will be the subject of a later lecture).

Notes for the earlier Volterra talk are here

Monday, January 4, 2:00--3:00 PM in NH 346 (Please note change of day and time)

"Fejér monotone sequences and nonexpansive mappings"

Abstract.The notion of Fejér monotonicity has proven to be a fruitful concept in fixed point theory and optimization. In this work, we present new conditions sufficient for convergence of Fejér monotone sequences and we also provide applications to the study of nonexpansive mappings. Various examples illustrate our results.

Friday, Nov. 20 @ 2 PM in NH 346

George Nicol, will speak on:

"Ditronic Sets, delta-Dimensions, Mask Sets I-Spaces and: `What happened at t=0?' "

Abstract.Using simple properties of a nondescript group, a mathematical system is presented in an attempt to shed light on the question: What happened at time t=0 of the big bang? The system or model is also used to determine the structure of different delta-dimensional spaces. From this model, topological I-spaces emerge quite naturally. Time permitting, it will be shown that given a SLR (sensor-level resolution), there is an irreducible mask that when applied to the model matches the observable universe at that SLR. (This research is supported by Silicon Composers, Inc.)

Friday, Nov. 6 & 13 @ 2 PM in NH 346

Steve Silverman will speak on:

"Bell's Theorem and Quantum Mechanical Refutation"

Abstract.These talks will develop, in a manner accessible to college freshmen, a rigorous treatment of quantum measurements, tensor products, and entanglement, culminating in a new and simple version of Bell’s Inequality.

Friday, Oct 23 & 30 @ 2 PM in NH 346

Jim Rulla, James L. Rulla Consulting, will speak on:

"The Crandall-Liggett Generation Theorem"

Notes for this talk here.Abstract.The Crandall-Liggett generation theorem describes how to solve a broad class of differential equations by exponentiating operators. The technique is useful in both theory and practice, and illustrates the power of treating functions as points in a vector space.

The authors' proof of the paper's crucial lemma ends with the memorable sentence: "The (somewhat awkward) induction is left to the reader." I'll present a beautiful (induction-free) derivation using counting arguments from elementary probability theory.

Friday, Oct 16 @ 2 PM in NH 346

Joel Shapiro, PSU will speak on:

"A Volterra Adventure"

Abstract.This will be the first of an occasional series of talks on the Volterra operator--- the simplest nontrivial integral operator---which played a fundamental role in the birth of functional analysis during the latter part of the nineteenth century, and which to this day remains an object of intense interest. In this talk I'll focus on the connection between the Volterra operator and the Initial Value Problem of ordinary differential equations.

Notes for this talk are here here.

Friday, Oct 2 & 9 @ 2 PM in NH 346

Robert Lyons, PSU & Digimarc will speak on:

"Frobenius' Theorem, two ways"

Abstract.Frobenius' theorem, a fundamental result in differential geometry, gives higher dimensional versions of the fact that smooth vector fields have integral curves. These lectures will discuss Frobenius’ theorem via both vector fields and differential forms.