The * What is ...?* seminar will run again in Summer 2023. Professor Vitaly Bergelson, Linus Ge, and Mikey Reilly will serve as coordinators/mediators for the seminar. The meetings will take place in Cockins Hall 240.

### MEETINGS: Tuesdays and Thursdays during summer semester at 4:15 pm.

The seminar's main goal is to expose culturally ambitious participants to some mathematical notions not taught in standard courses. These topics form an important part of mathematical folklore, and may prove useful for doing research and enhancing teaching.

Lectures will be given mostly by graduate and undergraduate student participants. See the archived topics below.

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2023 Talks

**Abstract: **Cantor spaces are nonempty, compact, totally disconnected metric spaces without isolated points. The most popular example is the Cantor ternary set, but it is far from the only one: other Cantor spaces include the rings of p-adic integers, some Julia sets and the space {0,1}^N with the product topology. The starting point of this talk is a theorem by Brouwer claiming that all Cantor spaces are homeomorphic to {0,1}^N, which we call "the Cantor space". After explaining several surprising topological properties of the Cantor space, we will use them to answer some questions, like: is the space Q of rational numbers homeomorphic to Q∩[0,1]? And to Q^2? Which topological spaces admit space filling curves?

**Abstract: **In 1936, Erdős and Turán conjectured that every set of integers A which was "large enough" contained arithmetic progessions of any length. This was resolved by Szemerédi in 1975 in his celebrated theorem. In 1977, Furstenberg gave a separate proof which reduced the problem of finding additive structure in sets of integers to multiple recurrence in a probability space. In this talk, we will present and prove the key correspondence principle used in his proof, as well as discuss corollaries in arithmetic combinatorics.

**Abstract**: The theory of Fourier series gives a way of expressing any periodic function as the limit of a sequence of sums of trig functions. In these sums, each trig function has a period which is an integer multiple of π, but what happens if we consider sums of this form in which the trig functions have arbitrary periods? This is what is known as a (Bohr) Almost Periodic function. In this talk, we will investigate the properties and some interesting applications of almost periodic functions.

**Abstract**: This is an attempt to give a motivated introduction to the étale fundamental group, at least in the affine case, for an audience in general position. If time is on my side, I will describe some of the ways it has been applied in algebra, geometry, and number theory.

** Abstract:** In 1900, Alfred Young introduced the combinatorial objects that we now call Young Tableaux while studying representations of the symmetric group. In this talk, we will define Young Tableaux and discuss the operations of “bumping” and “sliding” on Young Tableaux. We will use these concepts to state and discuss the Robinson-Schensted-Knuth correspondence (commonly called “RSK correspondence”). We will discuss a few applications of the RSK correspondence before moving on to an introductory discussion about group representations and how Young Tableaux appear in Representation Theory.

**Abstract:** A linear transformation on a finite dimensional complex vector space always has an eigenvalue; the same is not always true for the infinite-dimensional vector spaces. In the 1950s, functional analysts became interested in knowing whether one can still say something interesting about broad classes of linear operators. This talk aims to present the background leading up to the invariant subspace problem, Lomonosov’s proof for compact operators, and the current progress.

**Abstract:** The tire tracks formed by the rear wheel of a bicycle, once the front wheel follows a prescribed path, are known as tractrices (or curves of pursuit). Such curves were discovered by Newton in 1676 and studied further by Leibniz, Huygens and Euler. A version of bicycle was originally invented by Karl von Drais in 1817. Around the same time, several engineering devices to measure area of a small irregular shape, known as planimeters, came to be (Amsler 1854, Prytz 1886). The Prytz planimeter was based on a relation between the area enclosed by the front wheel track and the angular displacement of the bicycle frame (i.e., bicycle monodromy), which was a remarkably simple mechanical implementation of Green's theorem. These objects have reappeared on mathematical stage recently by the works of Foote (1998), Foote-Levi-Tabachnikov (2012), Bor-Levi-Perline-Tabachnikov (2020) and many others.

In this talk, I will present the differential equation determining tractrices, define bicycle monodromy, and explain its relation with the area enclosed by the front wheel track.

**Abstract: **In the 1960s, PhD student Robert Berger discovered the first aperiodic tiling of R^2 using a set of 20,426 distinct tile shapes. Around a decade later, Roger Penrose discovered a set of only two tiles displaying instances of five-fold symmetry which produce only aperiodic tilings of R^2. These tilings are examples of quasicrystals – structures that are aperiodic yet exhibit ordering in a certain sense. In this talk, we will provide motivation and formally define quasicrystals. In addition, we will explore interesting properties and applications of quasicrystals.

**Abstract: **As mathematicians (aspiring or otherwise), we deal with a variety of structures such as sets, lattices, graphs, vector spaces, etc. A central goal of Ramsey theory is to question their rigidity in the following sense: if one partitions the structure finitely, what properties must be inherited by one of the parts? We will take a look at some of the most classical results in this area and see how the Graham-Rothschild theorem provides a unified framework to rule them all. This theorem also enabled Graham and Rothschild to prove nine new results and make progress towards a (then) unsolved problem. If time permits, we will formulate some of these and discuss some trivia relating to Graham’s number

**Abstract:** In 1916, Lusin, who was then a young professor in Moscow, asked his student Souslin to read a paper that Lebesgue had published in 1905. Soon, Souslin told Lusin that he had found a serious mistake in Lebesgue's paper. Spurred by this, Souslin and Lusin founded the theory of analytic sets and thereby succeeded in salvaging an important part of Lebesgue's paper. Decades later, their ideas were extended by Choquet to solve an important problem in potential theory. Choquet's work, in turn, found important applications in the theory of continuous time stochastic processes. I will sketch some highlights of this story, with particular attention to Choquet's proof that analytic sets are capacitable.

Previous Years' What is ...? Talks

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