Spring Term Schedule 2022

Friday, June 3 (FMH Room 462, and remotely via Zoom)

Scott Lindstrom (Curtin University, Australia) will speak on:

Using dynamical geometry tools to study algorithms

Abstract: I will describe how I used the free geometry tool Cinderella to discover and describe a class of algorithms that may be useful for nonlinear optimization. This presentation will be conducted entirely within the dynamical interface, so that all of the concepts are very visible. It should be accessible to, and the tools-based approach may be useful for, a wide audience.

The presentation is principally based on this recent article.

Friday, May 27

Gary Sandine will speak on:

Finding a needle in a very large haystack

Abstract: Linear programming, a straightforward and elegant method for framing and solving optimization problems, arose during WWII from decision problems involving optimal resource allocation. Since then it has had profound effects throughout numerous industries.

Large linear programs can be solved efficiently on modern computers using a method such as the simplex method. However the requirement of integer solutions makes these problems much harder to solve. This talk will be about using linear programming methods to frame and solve integer programs, and will include an example of a binary integer programming problem that recently arose in my work in PSU OIT and rekindled my interest in this subject.

Friday, May 13 & 20

Jim Rulla will speak on:

Entropy, Planck, and Zeta

Abstract: The Riemann zeta function shows up in pyrometry, the measurement of temperature using the color of radiated light. Pyrometry uses Planck’s black body equation, the discovery of which marks the beginning of quantum mechanics. Planck derived his equation by thinking about entropy.

These talks use entropy to prove the fundamental results leading to (and including) Planck’s derivation of the black body equation. You don’t have to believe in entropy, just be willing to solve differential equations in which it appears. No mathematics beyond lower-division calculus is required.

Friday, April 22, 29 & May 6

Robert Lyons (Digimarc and PSU) will speak on:

Metrics and Connections

Abstract: Surfaces embedded in euclidean 3-space have been studied for centuries. In this first presentation we’ll talk about the classic theory of metrics and connections. Then we’ll discuss Cartan’s connection on frame bundles and how this led to the theory of connections on Principal bundles.

Joel Shapiro will speak on:

How to Prove an Almost-Everywhere Convergence Theorem

Abstract. We’ll discuss a “two-step program,” introduced in 1926 by Stephan Banach, for proving almost-everywhere convergence theorems. Although published almost a century ago, Banach’s program lies at the heart of all modern-day proofs of almost-everywhere convergence.

Friday, April 1 & April 8

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

The Birkhoff Ergodic Theorem

Abstract: The ergodic theorem (there are various versions) is arguably one of the most important theorems in mathematics. In essence, it offers a means to replace the study of long-term behavior of complex systems by much simpler statistical reasoning. Considerations of this nature gave rise to an important branch of theoretical physics (statistical physics). It is also widely applied in number theory, probability theory, functional analysis, and other fields of study.

Winter Term Schedule 2022

Friday, March 4

Joel Shapiro will speak on:

Isomorphisms(?) between \(\ell^p\) spaces

Abstract. For different \(p\) and \(q\) the spaces \(\ell^p\) and \(\ell^q\) are not the same. But might they be be isomorphic to each other? Could each (infinite dimensional) closed subspace of one of them be isomorphic to a closed subpace of the other? We will investigate.

Friday, February 25

Joel Shapiro will speak on:

Schur’s Lemma and Beyond: a Journey on the Gliding Hump

Abstract. After reviewing the notion of weak convergence we’ll prove Schur’s Lemma: Every weakly convergent sequence in \(\ell^1\) is norm convergent. Key to the argument will be the Gliding Hump Lemma, which has far-reaching applications. We’ll use it to examine essential difference between the different \(\ell^p\) spaces.

Friday, February 11

Joel Shapiro will speak on

Schur’s Lemma and the weak topology of a Banach space

Abstract. We’ll use the notion of {\em weak topology} on a Banach space, to better understand the importance of Schur’s Lemma, a striking result from 1920 that asserts for the classical sequence space \(\ell^1\): Every weakly convergent sequence is norm-convergent.

For example,each point of the unit ball of a Banach space is a weak-limit point of the unit sphere. Schur’s Lemma says that, at least in \(\ell^1\), this cannot happen sequentially. If not sequentially, then how? We’ll investigate.

Logan Fox will speak on

The Krein-Milman Theorem for Conical Bicombings

Abstract. In general, metric spaces lack a linear structure, so convexity is not well defined. However, if we equip a metric space with a notion of ‘betweenness,’ which we call a bicombing, then we can generalize the definitions of convexity by replacing linear segments with the segments defined by the bicombing. Assuming the bicombing satisfies a weak convexity condition, called a conical bicombing, we will show that every compact convex set is the closed convex hull of its extreme points.

Familiarity with the Krein–Milman theorem (and its proof) for locally convex spaces is beneficial, but not required. Given time, we will discuss other questions concerning convexity in metric spaces.

Notes for this talk are here.

Friday, January 21

Joel Shapiro will speak on

Schur’s Lemma and weak convergence in Banach spaces

Abstract. A sequence in a Banach space \(X\) is said to converge weakly if it converges pointwise on the Banach space’s dual (the space of continuous linear functionals on \(X\)). Norm convergence always implies weak convergence, but the converse does not usually hold. However Schur’s Lemma, a surprising result from 1920, asserts that for the sequence space \(\ell^1\) the converse does hold: every weakly convergent sequence in \(\ell^1\) turns out to be norm convergent.

In this talk we’ll spend some time discussing the notion of weak convergence, and the topology that gives rise to it. Then (time permitting) we’ll prove Schur’s Lemma.

Friday, January 14

Christopher Aagaard will speak on

Minimal Separating Sets for Compact Surfaces

Abstract. The minimal separating sets of Riemann surfaces are isomorphic to embedded topological graphs. In this talk we will discuss the classification of minimal separating sets in Riemann surfaces, the connection between graph embeddings and the symmetric groups, and a method for their enumeration.

Fall Term Schedule 2021

Friday, November 5, 2–3 PM

Joel Shapiro will speak on

Banach Basis Basics II

After reviewing the basics of Schauder bases for Banach spaces, we’ll consider an important question that arose during last week’s talk: that of continuity for linear functionals and projections associated with a Schauder basis. We’ll give examples based on Fourier analysis that show how this continuity figures in the study of subspaces of \(L^p\) spaces.

Friday, October 29, 2–3 PM

Joel Shapiro will speak on

Banach Basis Basics

Abstract: Thanks to the Axiom of Choice, every vector space has a basis: a linearly independent set of vectors whose linear combinations exhaust the space. For infinite dimensional Banach spaces, the Polish mathematician Juliusz Schauder proposed a more useful concept of basis, with norm-convergent infinite linear combinations replacing the usual (finite) ones.

In this talk we will carefully define these Schauder bases, discuss their fundamental properties, consider the problem of their existence, and give Schauder’s original example of such a basis for C([0,1]).

Friday, October 22, 2–3 PM

Mau Nam Nguyen will speak on

Convex Separation and the Krein-Milman Theorem

Abstract: In this talk we discuss convex separation theorems in locally convex topological vector spaces. As an application, we present a proof of the Krein-Milman theorem on extreme points of convex sets.

Friday, October 15, 2–3 PM

Logan Fox will speak on

Geodesic Bicombings and Convex Sets

Abstract: We introduce the notion of a geodesic bicombing on a metric space and explore some of the various qualities of bicombings. Assuming our bicombings satisfy certain convexity conditions, we can think of these spaces as nonlinear generalizations of normed space. This leads us to a discussion of convex sets in these generalized spaces. After revisiting Hormander’s embedding theorem, we show how the space of closed bounded convex sets admits a geodesic bicombing.

Friday, October 8, 2–3 PM

Gary Sandine (PSU) will speak on

A Radon-Nikodym type theorem for lattice homomorphisms

Abstract: I will present about ordered vector spaces and lattice homomorphisms - linear functions between ordered vector spaces that preserve the partial order structure. The talk will conclude with a proof of a Radon-Nikodym type theorem for lattice homomorphisms which hinges on a parallel to the Lebesgue decomposition of a measure with respect to another.