# Portland State University

## Analysis Seminar 2018–2019

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.

### Spring Term Schedule 2019

Friday, April 19: in East Hall Room 236, 2PM -3PM

Peter Oberly will speak on: Recurrence in Dynamical Systems and Combinatorics

Abstract. In this talk, we will illustrate a technique which connects results in additive combinatorics with statements on recurrence in dynamical systems. In particular, we will show how Van der Waerden’s theorem on colorings of the integers is equivalent to a multiple recurrence theorem in topological dynamics, and how Szemerédi’s theorem on arithmetic progressions follows from a deep generalization of the Poincaré recurrence theorem.

Friday, April 12: in East Hall Room 236, 2PM -3PM

Pieter VandenBerge will speak on: Derivatives and their discontinuities

Abstract. The absolute value function is often cited when discussing functions that are not everywhere differentiable. Less often mentioned are functions that are everywhere differentiable, but for which the resulting derivative function fails to be continuous. The canonical example of such a function is $f(x) = x^2 \sin(1/x)$ (where $$f(0) = 0$$), for which $$f'(0)=0$$, but for which the lines tangent to the graph of f have slopes that oscillate between -1 and 1 ever faster as they approach the origin.

In this talk, we’ll discuss just how badly discontinuous such a derivative can become. We’ll begin with a few theorems that show derivatives cannot have so-called jump or removable discontinuities, then go on to explore functions with derivatives that are discontinuous on dense subsets of the real line. The majority of the talk will be accessible to students who have taken or are currently enrolled in Math 311.

Notes for this talk are here.

Friday, April 5: in Cramer Hall, Room 265 (!): 2 PM- 3 PM

Logan Fox will speak on: Fixed points and Fatou components

Abstract. In the study of complex dynamics it is often necessary to subdivide the plane into two complementary invariant sets: the Julia set and the Fatou set. In this talk, we will discuss how we can use attracting fixed points of analytic functions to determine components of the Fatou set. This will lead to a simple yet significant example of an entire function which has ‘wandering’ Fatou components.

Notes for this talk are here.

### Winter Term Schedule 2019

Friday, March 1: in East Hall Room 236: 2 PM–3 PM

Joel Shapiro will speak on: ‘Similarity’ of a function to its derivative … Part II

Abstract. Suppose $$\,f$$ is a continuously differentiable mapping of euclidean space with $$\,f(0)=0$$ and invertible derivative $$A=f'(0)$$. We’ll explore the possibility that $$f$$ is similar to $$A$$ in the sense that there is a homeomorphism $$\phi$$ for which $f\circ\phi = \phi\circ A.$ In a previous talk (not a prerequisite for this one) we saw how to view the map $$\phi$$ as a change of variable that linearizes $$f$$ near the origin. Today we’ll prove such a result for the real line, and explore the situation in higher dimensions.

Friday, February 22: in East Hall Room 236: 2 PM–3 PM

Prof. J. J. P. Veerman will speak on: Diffusion and consensus on weakly connected directed graphs

Abstract: We outline a complete and self-contained treatment of the asymptotics of (discrete and continuous) consensus and diffusion on directed graphs. Let G be a weakly connected directed graph with directed graph Laplacian L. In many or most applications involving digraphs, it is possible to identify a direction of flow of information. We fix that direction as the direction of the edges. With this convention, consensus (and its discrete-time analogue) and diffusion (and its discrete-time analogue) are dual to one another in the sense that dx/dt = -Lx for consensus, and dp/dt = -pL for diffusion. As a result, their asymptotic states can be described as duals.

We give a precise characterization of a basis of row vectors of the left null-space of L and of a basis of column vectors of the right null-space of L. This characterization is given in terms of the partition of G into strongly connected components and how these are connected to each other. In turn, this allows us to give a complete characterization of the asymptotic behavior of both diffusion and consensus in terms of these eigenvectors.

As an application of these ideas, we present a treatment of the pagerank algorithm that is dual to the usual one, and give a new result that shows that the teleporting feature usually included in the algorithm actually does not add information.

Slides for this talk are here.

Friday, February 15: No seminar today.

Instead, at 2 PM in Urban Center Room 204 the Department will welcome special guest:

Professor Krystal Taylor of The Ohio State University

who will speak on

The geometry of sets from the perspective of Fourier analysis and projection theory

Friday, February 8 in East Hall Room 236: 2PM–3 PM

Joel Shapiro will speak on: How similar is a function to its derivative?

Abstract. Suppose f is a continuously differentiable mapping of euclidean space with f(0)=0. The possible existence of a homeomorphism $$\phi$$ that makes f similar to f'(0) in the sense that $f\circ \phi =\phi\circ f'(0)$ provided a crucial step in last term’s proof of the Hartman-Grobman Theorem.

This kind of result is important in its own right, and interesting even in one dimension. We will explore.

Friday, January 25 in East Hall Room 236: 2PM–3 PM

Geoff Diestel will speak on: Problems in Convex Geometry

Abstract. Fundamental problems in convex geometry involve determining a convex body uniquely, or up to some equivalence, from measurements related its sections and projections. Such problems are easy to understand and difficult to solve. A survey of some solved and open problems will be given.

The talk will start with a brief discussion of the solved Busemann-Petty problem and its dual Shephard problem. Then focus will be on partially resolved problems related to volumes of central sections and projections of origin-symmetric bodies including the Nakajima problem which asks if a convex body with constant width and constant brightness(projection) must be a Euclidean ball. An affirmative answer is only known in three-dimensions and a general approach will be outlined.

Friday, January 18 in East Hall Room 236: 2PM–3 PM

Jim Rulla will speak on: Cubics, quartics, and the DFT

Abstract. The Discrete Fourier Transform (DFT), an important tool in science and engineering, turns out to be useful in algebra, too. In this talk, we’ll use the DFT to find the roots of polynomials of degrees 3 and 4 (cubics and quartics). Lagrange’s opinion was that the cubic requires particular artifices that do not present themselves naturally.

However no such “artifices” are required if we use the DFT. The technique is remarkably simple and easy to remember.

Notes for Jim’s talks on DFT and solutions by radicals are here.

Friday, January 11 in East Hall Room 236: 2PM–3 PM

Jim Rulla will speak on: Solution by Radicals and the Discrete Fourier Transform

Abstract. The Discrete Fourier Transform (DFT), an important tool in science and engineering, turns out to be useful in algebra, too. In this talk, we’ll use the DFT to find the roots of polynomials of degree 2, 3, and 4 (quadratics, cubics, and quartics). Lagrange’s opinion was that the cubic requires particular artifices that do not present themselves naturally, but no such “artifices” are required if we use the DFT. The technique is remarkably simple and easy to remember.

The first part of the talk will introduce the DFT, so no previous experience is required. This talk will concentrate on positive results — how to find the roots when we can. If there is interest, a second talk will concentrate on negative results — how Galois might have come up with his theory.

### Fall Term Schedule 2018

Friday, November 16 in East Hall Room 236: 2 PM – 3 PM

Joel Shapiro will speak on: A.E. Convergence and Maximal Functions (you can’t have one without the other)

Abstract. Suppose you have a sequence $$(T_n)$$ of linear transformations taking a Banach space $$B$$ into the space of functions measurable for some sigma-algebra. A famous theorem of Banach tells us that a certain type of uniform estimate on the maximal function $T^*f=\sup_nT_nf \qquad (f\in B)$ characterizes those operator sequences $$(T_n)$$ for which $$(T_nf)$$ converges a.e. for each $$f\in B$$.

In this talk I’ll explain Banach’s Principle, and show how it lurks behind the proofs of both the Lebegue Differentiation Theroem and the Birkhoff Ergodic Theorem.

Notes for this talk are here.

Friday, November 9 in East Hall Room 236: 2 PM – 3 PM

Peter Oberly will speak on: The Pointwise Ergodic Theorem and its Applications

Abstract. At the intersection of dynamical systems and measure theory, the pointwise ergodic theorem is a powerful tool with applications ranging from statistical mechanics to number theory. In this talk we will outline a proof of the theorem, and then illustrate its utility by proving Borel’s theorem on normal numbers. If time permits, we will use the ergodic theorem to derive the law of large numbers.

Notes to accompany this talk are here.

Friday, November 2 in East Hall Room 236: 2 PM – 3 PM

Prof. Mau Nam Nguyen will speak on: Smoothing Techniques and Applications to Optimization

Abstract. I will present smoothing techniques based on the Fenchel conjugate of convex functions with applications to a number of optimization problems in facility location, machine learning and image reconstructions.

Friday, October 26 in East Hall Room 236: 2 PM – 3 PM

Pieter VandenBerge will speak on: The Hartman-Grobman Theorem

Abstract. Numerous important phenomena in the real world—such as predator-prey relationships, atmospheric dynamics and electrical signals—can be effectively modeled with systems of non-linear ordinary differential equations. However unlike their linear counterparts, such systems may be impossible to solve analytically. They may nevertheless be “locally equivalent” to linear versions of themselves. The Hartman-Grobman Theorem tells us when such “linearization” is possible.

In this talk, we’ll explore this theorem in depth, quantify what it means for two systems to be “locally equivalent,” and look at counter-examples that show linearization is not always possible.

Friday, October 12 & 19 in East Hall Room 236: 2 PM – 3 PM

Logan Fox will speak on: An Introduction to the Hausdorff Measure

Abstract. This talk will cover the definition and basic properties of the Hausdorff measure; show the uniqueness of the Hausdorff dimension of a set; and prove that the sigma-algebra generated by the Hausdorff measure includes the Borel sets. This is followed by the result that, given the appropriate correction factor, the n-dimensional Hausdorff and Lebesgue measures agree on Borel sets. And, given time, the Hausdorff dimension and measure of the Cantor set will be explored as an example.

Notes to accompany these talks are here.

Friday, September 28 in East Hall Room 236: 2 PM – 3 PM

Joel Shapiro will speak on: The Basel Problem Made Easy Complex

Abstract. Complex analysis offers powerful tools that can be applied to problems in real analysis. To illustrate, I’ll show how complex methods can be used to evaluate $\sum_{n=1}^\infty \frac{1}{n^{2k}}$ for each positive integer k. For k=2 this is the Basel Problem popularized by Jacob Bernoulli 1689, and solved in 1735 by Leonhard Euler, who went on to discover the summation formula for general even powers.

Prerequisite: The arithmetic of complex numbers. All else will be explained.

Slides for this talk are here.