# Portland State University

## Analysis Seminar

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 listen.

Lecture Notes

### Spring Term Schedule 2021

All meetings this term will be conducted remotely using the conferencing app Zoom.

Friday, May 21, 28, & June 4

Professor J. J. P. Veerman (PSU) will speak on

The Prime Number Theorem

Abstract: The prime number theorem says that, roughly speaking, the nth prime in the natural numbers is found near $$n\log(n)$$. I will discuss a proof of that theorem that I adapted for a graduate class in number theory. The proof in the class notes (available through my webpage) does not require any knowledge beyond standard undergraduate classes. For the purposes of the seminar though, I will assume familiarity with the Cauchy integral formula.

In 1850, it seemed that Chebyshev was awfully close to proving the prime number theorem. But to bridge that last brook, a whole new approach to the problem was needed. That approach was the connection with analytic functions in the complex domain pioneered by Riemann in 1859. Even so, it would take another 37 years after Riemann’s monumental contribution before the result was finally proved by De La Vall\‘ee Poussin and Hadamard in 1896. My reasoning is a highly streamlined derivative of that proof, the last stage of which was achieved by Newman in 1982. We made heavy use of Zagier’s 1997 rendition of Newman’s proof and also of notes by Joel Shapiro.

Friday, May 7 & 14, 2–3 PM

Robert Lyons will speak on

The Routh-Hurwitz Stability Criterion

Abstract: The Routh-Hurwitz Stability Criterion is a test applied to a polynomial to determine the number of roots located in the left half of the complex plane. This problem comes up often in questions concerning the stability of dynamical systems. Although the Routh-Hurwitz recipe is well known, the proof of its veracity is not. In this presentation we will sketch a proof of the Routh-Hurwitz criteria. The outline includes a construction of a Sturm sequence from the polynomial and uses methods from complex analysis, calculus, and algebra.

Slides for this talk are here.

Friday, April 30, 2–3 PM

Logan Fox will speak on

Convexity in Metric Spaces of Nonpositive Curvature

Abstract: We will examine classes of metric spaces which serve as nonlinear generalizations of Hilbert spaces and strictly convex Banach spaces (known as Hadamard spaces and Busemann spaces, respectively). In these metric spaces, one can define convex sets and convex functions in a way that perfectly aligns with the linear notion of convexity. We will work through a few motivating examples to see how properties of linear convexity transfer to these possibly nonlinear spaces.

Slides for this talk are here.

Friday, April 23, 2–3 PM

Joel Shapiro will speak on

How the Cauchy-Riemann operator makes difficult constructions easy

Abstract: The Cauchy-Riemann operator $$\overline{\partial} = \frac{\partial}{\partial \overline{z}}$$ defined by: $\overline{\partial} ~:=~ \frac{1}{2}\left(\frac{\partial}{\partial x} + i\frac{\partial}{\partial y}\right)$ has appeared in this seminar on several recent occasions.

In this talk I’ll review this history and develop further the “d-bar method” wherein one constructs an analytic functions with certain prescribed behaviors (usually difficult) by:

(a) First constructing smooth “prototype’‘functions with the desired behavior (usually easy),

(b) Then creating analyticity by adding to the prototype a smooth correction function” that satisfies the non-homogeneous Cauchy-Riemann equation $$\overline{\partial}u=f$$ for an appropriate smooth $$f$$.

We’ll see how results obtained in this way can be used to study the ideal structure of rings of analytic functions on plane domains.

Slides for this talk are here.

D-bar Lecture notes are here.

Friday, April 9, 2–3 PM

Sheldon Axler (San Francisco State Univ.) will speak on

Applications of the Logarithmic Conjugation Theorem

Abstract: This talk gives some applications of the logarithmic conjugation theorem. The emphasis here will be on describing the behavior of a harmonic function near an isolated singularity or on an annulus.

Slides for this talk are here.

Friday, April 2, 2–3 PM

Sheldon Axler (San Francisco State Univ.) will speak on

A Logarithmic Conjugation Theorem

Abstract: If $$u$$ is a real­-valued harmonic function on a simply 􏰆connected domain in the complex􏱜 plane, then $$u$$ is the real part of some analy􏰆tic function on the domain. For finitely connected domains that are not simply􏰆 connected, this result fails … but not by􏰆 much. This talk gives a useful description of harmonic functions on finitely connected domains.

Slides for this talk are here.

### Winter Term Schedule 2021

Friday, March 5, 2–3 PM

Logan Fox will speak on

The No-Wandering-Domains Theorem

Abstract: For any rational function on the extended complex plane, the No-Wandering-Domains Theorem tells us that every component of the Fatou set is eventually periodic.

After reviewing the basic structures of complex dynamics, including the Julia and Fatou sets, we will examine how quasiconformal mappings are used to prove this theorem.

Slides for this talk are here.

Friday, February 19, 2–3 PM

Joel Shapiro will speak on:

How I learned to stop worrying and love the $$\overline{\partial}$$ operator

Abstract: I will show how the $$\overline{\partial}$$ (pronounced “dee-bar”) operator $\overline{\partial} = \frac{1}{2} \left(\frac{\partial}{\partial x} + i\frac{\partial}{\partial y}\right)$ introduces real-variable methods into complex analysis.

The $$\overline{\partial}$$ operator has already been introduced in the seminar by Logan Fox (February 5) in his talk about quasiconformal mappings.

In this talk: I’ll review in some detail what Logan showed us about the $$\overline{\partial}$$ operator’s connection with complex differentiability, after which we’ll see how it profoundly extends the classical Cauchy integral formula of complex analysis. Then—time permitting—we’ll see how this new integral formula solves the “$$\overline{\partial}$$ equation” $\overline{\partial} u = f,$ a PDE that introduces a new way of constructing analytic functions.

Slides for this talk are here.

Friday, February 5, 2-3 PM

Logan Fox (PSU) will speak on:

Quasiconformal Maps

Abstract: The goal for this talk is to provide an introductory approach to quasiconformal maps on the complex plane. We will begin by reviewing the Cauchy-Riemann equations and examining what it is that makes quasiconformal maps 'almost’ conformal. This is followed by some basic examples and properties of quasiconformal maps, concluding with how quasiconformal maps interact with Riemann surfaces.

Slides for this talk are here.

Friday, January 29, 2-3 PM

Kevin Vixie (Washington State Univ.) will speak on:

Derivatives Again, Measure Theoretically

Abstract: Surprisingly, many students of mathematics do not acquire an intuition for derivatives as linear approximations until graduate school, even though this instinct is key to the use of the the derivative in many areas of analysis. Further, the various generalizations of the derivative are often never encountered unless the student works in geometric analysis.

In this talk, after briefly reminding everyone of the linear approximation definition of the derivative, I explore three generalizations: tangent cones, approximate tangent cones and weak tangent planes. I will prioritize understanding/intuition and discussion and over getting through all the material.

A book chapter on which this talk is based is here

Friday, January 22, 2-3 PM

Joel Shapiro will speak on:

Liouville’s Theorem

Abstract: Liouville’s Theorem, in its simplest form, asserts that if a function is analytic on the whole complex plane and bounded there, then it must be constant. The theorem easily implies the Fundamental Theorem of Algebra: Every polynomial has a zero in the complex plane. In this talk we’ll further  develop the connection between Liouville’s Theorem and the zeros of analytic functions by generalizing the theorem to harmonic functions in a way that provides some basic information about the Riemann Hypothesis.

Slides for this talk are here.

Friday, January 15, 2-3 PM

Gary Sandine will speak on:

A theorem by Rockafellar on separation of convex sets

Abstract: I will provide background material and work through a constructive proof of a theorem by Rockafellar on proper separation of convex sets when one of the sets is polyhedral. In that case, a more restrictive proper separation of the sets is equivalent to a stronger intersection condition between the sets. I will discuss relevant convex analysis background material along the way.

Slides for this talk are here.

### Fall Term Schedule 2020

Friday, November 20, 2-3 PM

Logan Fox will speak on:

Complex Dynamics, Part II: Rational Functions on the Riemann Sphere

Abstract: The talk will give an overview of the dynamics of rational functions. We will begin with some historical background, briefly introduce Reimann surfaces (particularly the Riemann sphere), and cover many of the important results which help us determine the domains of normality.

Slides for this talk are here.

Friday, November 6, 2-3 PM

Logan Fox will speak on:

Complex Dynamics, Part I

Abstract: We explore the dynamics of entire functions on the complex plane. We will begin by defining the Fatou and Julia sets; then show how fixed points can be used to find components of the Fatou set; and finally give examples of transcendental entire functions which display a phenomenon known as wandering domains. My hope for this series of talks is to eventually reach Dennis Sullivan’s proof of the no wandering domains conjecture.

Slides for this talk are here.

Friday, October 30, 2-3 PM

Joel Shapiro will speak on:

The Riemann Hypothesis … Nontrivial zeros

Abstract: The nontrivial zeros of the Riemann zeta function lie on the critical line. This is the Riemann Hypothesis, arguably the most famous open problem in mathematics. In previous talks (October 2 & 23) we unpacked the statement of this celebrated problem, familiarizing ourselves with the zeta function, and Riemann’s completed zeta function, which uncovered zeta’s trivial zeros and which, thanks its symmetry about the critical line, showed us that any nontrivial zeros (if, indeed, they exist) had to lie in the critical strip.

In this talk we’ll show that nontrivial zeros actually exist.

Slides covering all three talks are here.

Friday, October 23, 2-3 PM

Joel Shapiro will speak on:

The Riemann Hypothesis … it’s all about zero!

Abstract: The nontrivial zeros of the Riemann zeta function lie on the critical line. This is the Riemann Hypothesis, arguably the most famous open problem in mathematics. In a previous talk (October 2) we began a quest to understand the statement of this celebrated problem.

In this talk I’ll review the highlights of that previous one, in particular the so-called trivial zeros. Then I’ll show why the zeta function has infinitely many nontrivial zeros, and we’ll see that these are located in a certain critical strip.

Slides for this talk are here.

Friday, October 16, 2-3 PM

Sheldon Axler (San Francisco State Univ.) will speak on:

Dirichlet and Neumann Problems

Abstract: The classical Dirichlet problem asks for a harmonic function that matches a specified function on the boundary of some region. The classical Neumann problem asks for a harmonic function whose normal derivative on the boundary of some region matches a specified function. This talk will discuss the interesting mathematics that goes into computing exact solutions to these problems when the regions involved are balls or ellipsoids and the specified boundary functions are polynomials.

This talk should be accessible to all mathematicians and math graduate students; expertise with harmonic functions is not required.

Slides for this talk are here.

Friday, October 9, 2-3 PM

Logan Fox will speak on:

The Embedding Theorems of Rådström and Hörmander

Abstract: We explore a not-so-surprising connection between the Hausdorff metric, the semigroup qualities of convex sets, and spaces of continuous functions. In particular, we will examine Hormander’s Embedding Theorem, which gives an isometric embedding of the hyperspace of closed bounded convex sets as a convex cone in a space of bounded continuous functions.

Slides for this talk are here. Notes for this talk are here

Friday, October 2, 2–3 PM

Joel Shapiro will speak on:

The Riemann Hypothesis: What is it?

Abstract: (Arguably) the most famous open problem in mathematics, the Riemann Hypothesis is the statement that:

“The non-trivial zeros of the Riemann zeta function lie on the critical line.”

In this talk we’ll begin to: unpack this cryptic statement, understand something of its importance, and introduce some of the beautiful classical mathematics on which it rests.

Slides for this talk are here