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.

For Zoom login information, email: joels314@gmail.com

Lecture Notes

Spring Term Schedule 2024

Thursday, June 6 @ 2 PM in FMH 462 and remotely via Zoom

Sam Reynolds (PSU) will speak on:

\(~~~~~\) (Semi-)Inner Products of Harmonic Functions in Connected Planar Domains

Abstract: We consider the problem of finding the H1 semi-inner product and L2 inner product of two harmonic functions, given only their Dirichlet data. For the former, it suffices to reconstruct the normal derivative, which can be accomplished in simply connected domains by determining a harmonic conjugate. In multiply connected domains, we employ the Logarithmic Conjugation Theorem. To compute the L2 inner product, we construct a biharmonic function with a given Laplacian, which has an elegant characterization in the language of complex analysis.

Thursday, May 30 @ 2 PM in FMH 462 and remotely via Zoom (note change to usual time)

Robert Lyons (Transformative Optics Corp.) will speak on:

\(~~~~~\) SU(2) and SO(3): A Taste of Lie Groups

Abstract: A Lie Group is a differentiable manifold that is also a group with smooth group operations. We start with a quick discussion of Lie Groups and their Lie Algebras. Then we introduce our main examples: SU(2) and SO(3). SU(2) and SO(3) have isomorphic Lie Algebras and so are locally isomorphic. We can make this isomorphism concrete by constructing a homomorphism from SU(2) to SO(3) using the Adjoint Representation. This mapping forms the basis of the Hopf Fibration. Along the way we introduce a variety of concepts that are applicable to any Lie Group.

In this brief presentation we will not include complex proofs but will instead try to get a taste of the modern language of Lie Groups. The talk should be understandable to advanced undergraduates.

Thursday, May 23 @ 1 PM remotely via Zoom (note change from usual time)

Gary Sandine will speak on:

\(~~~~~\) An extreme value type theorem for convex functions on reflexive Banach spaces

Abstract: We will review the interplay between weak topologies, convexity of sets and functions, and lower semicontinuity and will apply the results to optimization and the existence of minimizers in an infinite dimensional setting.

Thursday, April 25 @ 2 PM in FMH 462

Serge Phanzu (PSU) will speak on:

\(~~~~~\) Nuclear operators on Hilbert space are trace class

Abstract: An operator \(T\) on a Hilbert space \(H\) is said to be of trace class if it has the form \[ Tx = \sum_n s_n\langle x,f_n\rangle g_n \qquad (x\in H) \] where \((f_n)\) and \((g_n)\) are orthonormal bases in \(H\), and \((s_n)\) is a summable decreasing sequence of non-negative real numbers.

An operator \(T\) on \(H\) is said to be nuclear if, in (*) above we merely require \((f_n)\) and \((g_n)\) to be unit-vector sequences. In this talk we’ll show that every nuclear operator on Hilbert space is of trace class. Consequently every trace class operator is compact, and is the product of two Hilbert-Schmidt operators.

Thursday, April 18 @ 2 PM in FMH 462 and via Zoom

Prof. Peter Veerman (PSU) will speak on:

\(~~~~~~~~~~\)Birkhoff Sums, A survey of some recent research.

Abstract: We will define Birkhoff Sums and explain their connections with various subbranches of mathematics: numerical analysis, number theory, ergodic theory, and dynamical systems.

Thursday, April 11 @ 2PM in FMH 462 and via Zoom

Serge Phanzu (PSU) will speak on: Hilbert-Schmidt operators are compact.

Abstract: For \(T\) a continuous linear transformation on a complex Hilbert space \(H\), and \(\{e_n\}\) an orthonormal basis for \(H\), we define the Hilbert-Schmidt norm \(\|T\|_2\) of \(T\) to be: \[ \|T\|_2 : = \left(\sum_{n=1}^\infty\|Te_n\|^2 \right)^{½} \] a (possibly infinite) quantity that turns out to be independent of the basis \(\{e_n\}\).

We call \(T\) a Hilbert-Schmidt operator whenever its Hilbert-Schmidt norm is finite. In this talk we’ll prove that every Hilbert-Schmidt operator is compact, and will discuss important examples of such operators.

Winter Term Schedule 2024

Thursday, January 25 and February 1 in FMH 462 and via Zoom.

Robert Lyons (Transformative Optics) will speak on:

\(~~~~~~~~~~\)Separation Properties for Topological Groups

Abstract: A topological group is group with a topology for which the group operations are continuous. Using group operations we can construct special open sets that restrict the types of topologies allowed. We shall show that many of the Tychonoff separation axioms are, in fact, equivalent for topological groups. For example, we shall show that if a topology is \(T_0\) then it must be Hausdorff. We also outline Kolmogorov’s construction showing how to generate a Hausdorff topological group from a group that is not \(T_0\).

In the second talk we’ll show that all topological groups are regular, but only topological groups that are second countable are normal. We’ll show how to take a topological group that is not T0 and create a related group that is T0 and so Hausdorff. We’ll finish with a short discussion of Hilbert’s Fifth problem.

Fall Term Schedule 2023

Thursday, November 16 @ 2 PM Via Zoom

Professor Hung Phan (UMass Lowell) will speak on

\(~~~~~~~~~~\)Conical averagedness and convergence analysis of fixed point algorithms

Abstract: In this talk, we discuss a conical extension of averaged non-expansive operators and its role in convergence analysis of fixed point algorithms. In particular, we study various properties, for example, stability under relaxations, convex combinations and compositions of conically averaged operators. We then utilize such properties in order to analyze the convergence of proximal point algorithm, forward-backward algorithm, and the adaptive Douglas-Rachford algorithm. This talk is based on joint work with Sedi Bartz (University of Massachusetts Lowell) and Minh Dao (RMIT, Australia).

Thursday, November 9 @ 2 PM in FMH 462 and via Zoom

Pieter VandenBerge (PSU) will speak on:

Orthogonality of eigenfunctions of the Helmholtz eigenproblem.

Abstract: The 1-d Helmholtz eigenproblem

\[u"+~k^2(x) = \beta^2 u,\]

with piecewise linear \(k(x)\), arose my October 26 talk to describe the \(x\) dependent portion of product solutions for fields of the slab waveguide. This equation has both a discrete and continuous spectrum. Because of this, the general form of the full solutions will contain both a discrete sum and an integral

\[f(x,z) = \sum_{j=1}^n \alpha_j u_j(x)e^{i\beta_j z} + \int_{\beta=0} ^\infty \alpha(\beta) u(x,\beta) e^{i\beta z} \; d\beta.\]

In order to find the values of the unknown coefficients, we require some sort of orthogonality among the eigenfunctions. In this talk we derive the orthogonality properties of these functions.

Thursday, November 2 @ 2PM in FMH 462 and via Zoom

Serge Phanzu (PSU) will speak on:

Nuclear Operators in Banach Spaces

Abstract: A bounded operator on a Banach space is said to be nuclear if it is expressible as a possibly infinite sum, convergent in the strongest possible way, of rank-one operators.

More precisely: For Banach spaces \(X\) and \(Y\), to say a linear transformation \(T\colon X\to Y\) is nuclear means that \[ Tx = \sum_n f_n(x)y_n \qquad (x\in X).\] where \((f_n)\) is a sequence of bounded linear functionals on \(X\) and \((y_n)\) a sequence of vectors in \(Y\), with \[ \sum_n\|f_n\|\,\|y_n\| <\infty~. \] We show that every finite rank operator is nuclear, and that nuclear operators are compact.

Slides for this talk are here.

Thursday, October 26 @ 2PM in FMH 462 and via Zoom

Peter VandenBerge (PSU) will speak on:

Deriving the Fields of the Slab Waveguide I

Abstract: One of the fundamental examples in mathematical optics is the slab waveguide: a device consisting of layered slabs of dielectric material, which, when properly constructed, may be used to guide electromagnetic waves. After adopting a reasonable ansatz and applying appropriate conditions on the fields at the material interfaces, the total fields can be expressed as a superposition of eigenfunctions, allowing us to determine the propagation of an arbitrary (within reason) input field.

This superposition will generally require both a discrete sum and an integral over a continuous spectrum. The discrete sum corresponds to the portion of the field that is propagated without loss, the guided portion, while the integral over the continuous spectrum corresponds to the portion of the input field that is lost to radiation. Recent developments in waveguides have focused interest on structures with no guided portion. Because of this, methods for computing the integral over the continuous spectrum have become more important.

In this talk, we first derive the necessary equations governing the fields from Maxwell’s equations and determine the eigensolutions, also called modes of the system. We discuss normalization of the modes of the continuous spectrum and perform several integrals over this spectrum, effectively propagating an input field through several types of slab waveguides.

In a second talk (TBA) we will use the method of steepest descent in complex analysis to study asymptotic approximations to the propagating fields.

Spring Term Schedule 2023

Friday, April 28 @ 1 PM. In-person in FMH 462 and remotely via Zoom.

Prof. J.J.P. Veerman (PSU) will speak on:

Insolvability by radicals of the quintic without Galois

Abstract: In 1963, V. I. Arnold proposed a proof of the insolvability by radicals of the quintic that was (supposed to be) accessible for high school students. The proof is notable in that it usesmuch less algebra than the traditional proof based on Galois theory. It also has the advantage that it is conceptually much clearer.

The proof made the press only in the last 20 years or so. It was the subject of a paper in the Monthly as recently as 2022. I will try to give a proof based on these recent publications that is accessible to graduate students in mathematics (though probably not to most high school students).

Winter Term Schedule 2023

Friday, March 10, In-person in FMH 462 and remotely via Zoom

Prof. J.J.P. Veerman (PSU) will speak on: Primes!

Abstract: A non-technical review of some classical results in number theory. We identify some of the currents in number theory (analytic, algebraic, and ergodic), and informally discuss some results that had great impact on mathematical (and physical) thought. On the analytic side, we look at the prime number theorem (or PNT), which tells us how dense primes are in the natural numbers. While its proof starts with some combinatorial estimates, it turns out, very surprisingly, that the full proof makes essential use of complex analysis (the Cauchy integral formula). For that reason, this branch is now called analytic number theory. We will touch on algebraic number theory to extend the PNT to arithmetic progressions. This theorem gives the density of primes in sequences of the form {a+ib} where a and b are fixed integers. Time permitting we will very briefly mention ergodic theory.

Slides for this talk are here.

Friday, March 3, In-person in FMH 462 and remotely via Zoom

Prof. J.J.P. Veerman (PSU) will speak on:

The Bak-Sneppen Model of Evolution

Abstract: We investigate a class of models related to the Bak-Sneppen (BS) model, initially proposed to study evolution. In this model, random fitnesses in [0, 1] are associated to N agents located at the vertices of a graph G, in our case a cycle. Their fitnesses are ranked from worst (0) to best (1). At every time-step the agent with the lowest fitness and its neighbors on the graph G are replaced by new agents with random fitnesses. This simple model after 30+ years still defies exact solution, but captures some forms of complex behavior observed in physical and biological systems.

We use order statistics to define a dynamical system on the set of cumulative distribution functions R : [0,1] → [0,1] that mimics the evolution of the distribution of the fitnesses in these models. We then show that this dynamical system reduces to a 1-dimensional polynomial map. Using an additional conjecture we can then find the limiting distribution as a function of the initial conditions. Roughly speaking, this ansatz says that the bulk of the replacements in the Bak-Sneppen model occur in a decreasing fraction of the population as the number N of agents tends to infinity. Agreement with experimental results of the BS model is excellent.

Friday, February 24, remotely via Zoom and in-person in FMH 462

Prof. J.J.P. Veerman (PSU) will speak on:

Dynamics of Chemical Reaction Networks

Abstract: The study of the dynamics of chemical reactions, and in particular phenomena such as oscillating reactions, has led to the recognition that many dynamical properties of a chemical reaction can be predicted from graph theoretical properties of a certain directed graph, called a Chemical Reaction Network (CRN). In this graph, the edges represent the reactions and the vertices the reacting combinations of chemical substances.

In contrast with the classical treatment, in this work, we heavily rely on a recently developed theory of directed graph Laplacians to simplify the traditional treatment. We show that much of the dynamics of these polynomial systems of differential equations can be understood by analyzing the directed graph Laplacian associated with the system.

Our new theory allows a more concise mathematical treatment and leads to considerably stronger results. In particular, (i) we show that our Laplacian deficiency zero theorem is markedly stronger than the traditional one and (ii) we derive simple equations for the locus of the equilibria in all (Laplacian) deficiency zero cases.

Slides for this talk are here.

Friday, February 17, remotely via Zoom.

Joel Shapiro will speak on:

SVD for the Volterra operator

Abstract: Last week we showed that the Volterra operator \[Vf(x) = \int_0^x f(t)dt \qquad (0\le x\le 1)\] mapped \(L^2([0,1])\) into itself, with norm \(=\frac{2}{\pi}\), and that \(V\) had some interesting features that don’t show up in finite dimensional linear algebra (it’s 1-1 but not onto, and it has no eigenvalues).

This week we’ll link up with previous talks by Jim and Sheldon by deriving the Singular-Value Decomposition of the Volterra operator (familiarity with these previous SVD talks is not required to understand this one).

Slides for this talk are here

Friday, February 10, remotely via Zoom.

Joel Shapiro will speak on:

The Volterra Operator

Abstract: The Volterra operator is the mapping that takes a function \(f(x)\) to \(\int_0^x f\). In first-year Calculus we encountered this operator as it acted on continuous functions. In this talk we’ll show that it maps the Lebesgue-Hilbert space \(L^2([0,1])\) into itself, and we’ll compute its norm. Along the way we’ll answer the usual questions one asks about linear transformations: is it injective? Surjective? What are its eigenvalues? Surprises await!

In a future talk we’ll go on to find the Volterra operator’s singular-value decomposition.

Slides for this talk are here

Friday, January 27 and February 3, remotely via Zoom.

Sheldon Axler (San Francisco State) will speak on:

The Singular-Value Decomposition

Abstract: These talks will present the singular value decomposition from the viewpoint of linear maps. We will start at the beginning, so no prior knowledge of the singular value decomposition is needed. To understand this talk you only need to be familiar with the terminology in the following important theorem: If T is a self-adjoint operator on a finite-dimensional inner product space V, then there exists an orthonormal basis of V consisting of eigenvectors of T.

The second talk will discuss applications of the SVD to:

Friday, January 13 & January 20 at 1 PM, remotely via Zoom

Jim Rulla will speak on:

The Singular-Value Decomposition for Analysts

Abstract: Singular Value Decompositions (SVDs) represent linear operators by their “stretching” factors, called singular values. An operator’s largest singular value is well-known to analysts as the operator’s norm.

We’ll use elementary methods to derive SVDs by thinking about norms. Then we’ll apply SVDs to topics that might interest analysts, including (numerical) rank and instability, pseudo-inverses, data analysis, and sound.

Fall Term Schedule 2022

Friday, November 18 at 1 PM in FMH 462, and remotely via Zoom

Christopher Aagard will speak on:

A problem in Complex Analysis, Topology, and Representation Theory

Abstract: Holomorphic maps between compact Riemann surfaces take the form of ramified covering maps. A classical question in Riemann surface theory is: given a ramification profile of a covering of the Riemann sphere, how many maps are there satisfying that ramification profile? Hurwitz’s work on the problem showed its equivalence with problems in topology and representation theory of the symmetric groups.

Outside of occasional bursts of interest, the problem didn’t capture much attention until the 1990s when it was discovered that it’s also related to questions in mathematical physics and algebraic geometry. This talk will survey (mostly) classical results in the theory of Hurwitz numbers, introducing the equivalent problems and techniques for solving them in the analytic, topological, and representation theoretic settings, after which we’ll briefly discuss some modern techniques and open questions.

Friday, November 4, at 1:10 PM in FMH 462 and remotely via Zoom.

Serge Phanzu (PSU) will speak on:

Pure Quasinormal Operators Have Supercyclic Adjoints

Abstract: We prove that every pure quasinormal operator has a supercyclic adjoint. It follows that if an operator has a pure quasinormal extension then the operator has a supercyclic adjoint. Our result improves a theorem of Wogen who proved that every pure quasinormal operator has a cyclic adjoint.

Friday, October 28 at 1 PM in FMH 462, and remotely via Zoom (coordinates TBA).

Serge Phanzu (PSU) will speak on:

Some Fundamental Results in Hypercyclicity

Abstract: In 1982, Kitai proved that hyponormal operators cannot be hypercyclic. It turns out that normal, quasinormal, and subnormal operators cannot have hypercyclic vectors. A few years later, Chan and Sanders asked whether a weakly hypercyclic hyponormal operator exists on a Hilbert space. Sanders gave a negative answer to the question. Concerning the supercyclicity of the operators, Hilden and Wallen proved earlier that normal operators do not have supercyclic vectors. A stronger result was found in 1997 by Bourdon, who proved that hyponormal operators are not supercyclic.

Friday, October 21 at 1 PM, Remotely via Zoom:

Sheldon Axler (San Francisco State) will speak on:

The Marvelous Minimal Polynomial

Abstract: Suppose V is a finite-dimensional vector space over some field and T is a linear map from V to V. There is a unique monic polynomial p of smallest degree such that p(T) = 0 (here monic means that the coefficient of the highest order term equals 1). This polynomial p is called the minimal polynomial of T.

As will be discussed in this talk, the minimal polynomial of T is easily computable (usually very quickly, even when V has high dimension), and its zeros are exactly the eigenvalues of T. Furthermore, the minimal polynomial of T tells us whether or not T is diagonalizable. The minimal polynomial of T also tells us whether or not there exists a basis of V with respect to which T has an upper-triangular matrix. In addition, the minimal polynomial tells us whether or not there exists a basis of V consisting of generalized eigenvectors of T. Finally, in the case when V is a real or complex inner product space, the minimal polynomial leads to an easy proof of the finite-dimensional spectral theorem.

Friday, October 14: in FMH 462 @ 1 PM, and remotely via Zoom

Christopher Aagard (PSU) will speak on:

Minimally Separating Sets in Compact Surfaces and Branched Covers of the Riemann Sphere

Abstract: Abstract: A minimally separating set in a topological space is a subset whose omission disconnects the space, but for which the omission of any proper subset does not. Peter Veerman recently showed that for compact surfaces, minimally separating sets are isomorphic to embedded topological graphs and, for surfaces of low genus, classified minimally separating sets in terms of homology groups.

In genus greater than 1, the problem becomes more complicated. We’ll present: a general classification of minimal separating sets for compact surfaces, an algorithm for their enumeration, and will show that minimally separating sets correspond to branched covers of the Riemann sphere, ramified over three points.

Friday, October 7 (Remotely via Zoom)

Joel Shapiro will speak on:

Almost-everywhere convergence (for almost everyone).

Abstract. Almost a century ago Stefan Banach proved that for the most common classes of operator sequences, a.e. convergence is equivalent to a “maximal condition” on distribution functions.

Today, Banach’s paper forms the basis for every a.e.-convergence theorem. We’ll discuss—mostly without proofs—why this is true, and why Banach’s paper is not better known.

Seminar Schedule and lecture notes for: