The * What is ...?* seminar will run again in Summer 2024. Professor Vitaly Bergelson, Linus Ge, and Mikey Reilly will serve as coordinators/mediators for the seminar. The meetings will take place in EA 160.

### MEETINGS: Tuesdays and Thursdays from 4:40-5:40 pm in EA 160.

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.

##
2024 Talks

**Abstract: **In the *L*^{1} and *L*^{∞} norms, the unit balls are not "round" in the way that the other *L ^{p}* norms are. As a consequence of this, the

*L*

^{1}and

*L*

^{∞}spaces do not have many of the nice properties that the other

*L*spaces have; for example, they are not reflexive. In this talk, we will make these feelings precise by defining

^{p}*uniform convexity*and discussing how it allows us to prove useful properties of functional spaces.

**Abstract: **In this talk we describe a non-Archimedean extension of the real numbers: the ordered field *R of hyperreal numbers, which contains infinitesimal numbers ε such that 0<ε<1/n for any natural number n. We construct *R as an example of an ultraproduct construction; the branch of analysis which explores these constructions is known as non-standard analysis.

Time permitting, we will explain how to define a “limit” of a sequence of probability measure preserving systems using Loeb measures, another example of an ultraproduct construction. These limits of measure preserving systems are useful in the study of measurable dynamics.

**Abstract**: Every introductory analysis textbook defines uniform continuity of a function on its entire domain. It is also worthwhile to define uniform continuity of a function on a subset of its domain. To my surprise, I have been unable to find a single textbook that defines this in the way that is most useful in applications. I will explain how this concept should be defined and, to illustrate the value of defining it that way, I will sketch the proof that the Fourier series of a function f of bounded variation on [0,1] converges to f uniformly on each compact subset of the set of points of continuity of f. This is a refinement of the Dirichlet-Jordan theorem. Along the way, we will learn about Fejér’s theorem and Hardy’s Tauberian theorem, and we will see an elementary proof for the rate of decay of the Fourier coefficients of a function of bounded variation on [0,1].

**Abstract**: This talk explores packing spheres in 3D for maximum density. We'll discuss the Kepler conjecture, now theorem, on the densest possible arrangement. We'll delve into its history, the various attempts at proofs, as well as techniques and proofs used in other related problems, including circle packing, square packing, and the kissing number problem.

** Abstract:** Given any continuous function on [0,1], one can completely determine its behavior through integration over smaller and smaller intervals. In fact, locally Riemann integrable functions are also completely characterized by integration over intervals (up to an equivalence relation). Generalizing the notion of integration and the concept of intervals, we define generalized functions, also known as distributions. Distributions also have well defined derivatives, known as weak derivatives, which are central to the study of partial differential equations.

An alternative suggested title was "What is the easiest way to win a Fields Medal?"

**Abstract:** In 1879 Hermann Schubert wrote a paper exhibiting a “calculus” designed to answer problems in enumerative geometry concerning intersections of linear subspaces. The rigor of this “calculus” was questionable, leading to David Hilbert’s 15th question, which asked to put this calculus on a rigorous foundation. Throughout the 20th century, this question was largely answered in many cases. In this talk, we will discuss a bit of the history of this subject, introduce two important spaces in the subject (Projective Space and the Grassmannian), and move on to do Schubert Calculus on the Grassmannian (i.e. answer some enumerative geometry questions using the Grassmannian).

**Abstract:** In the theory of dynamical systems, one is given a set *S *and a self-map *f *:*S *→* S*, and the goal is to study the iterates of *f*. One natural question we can ask is whether *f *has any fixed points, or more generally periodic points. The intermediate value theorem implies that every continuous map *f *: [0,1] → [0,1] has a fixed point. The map *f *:[0,1] → [0,1] given by *f*(*x*)=1-*x* is continuous and has periodic points only of orders 1 and 2. A surprising result by the Ukrainian mathematician Oleksandr Mykolayovych Sharkovsky from 1964, sharpened by Li and Yorke in 1975, implies that if *f *:[0,1] → [0,1] is *any *continuous map with a periodic point of order 3, then, in fact, *f *has a periodic point of order *n* for each positive integer *n*. In this talk, I will present a proof of this surprising and beautiful result, using nothing more than the intermediate value theorem.

**Abstract:** In 1973 Gustafson proved the following result: the probability that two randomly chosen elements of a finite non-abelian group commute is at most 5/8. In this talk, we will give a proof of this theorem, and discuss some other variants of such probabilistic questions in group theory.

**Abstract:** In a 1968 paper, number theorist Kurt Mahler asked if there are any real numbers *x* such that the fractional part of 3* ^{n}*x/2

*is less than 1/2 for all positive integers*

^{n}*n*. This question, which is still unanswered, is now commonly called

*Mahler's 3/2 problem*. In this talk we will discuss Mahler's problem and prove some related results. We will also look at a few things relating to Waring's problem which help motivate why Mahler might have asked his question in the first place.

**Abstract:** What is the maximal area bounded by a closed differentiable curve in * R^{2}*? This is an example of an Isoperimetric problem. These types of problems were originally asked by the Ancient Greeks who understood that an island having bigger perimeter does not mean it has bigger area. From this observation the Greeks asked is there an inequality that relates the perimeter and area of a closed curve. In this talk, we will use tools from Differential Geometry and Convex Geometry to prove two different Isoperimetric problems.

**Abstract:** Helly's theorem (1913) states that if any *d+1* (or fewer) of the convex sets *K*_{1},...,*K*_{n} in **R**^{d} have nonempty intersection, then the intersection ∩^{n}_{i = 1}*K _{i}* of the entire family is itself nonempty. Given the ubiquity of convexity throughout mathematics, the significance of this statement is not too surprising. Indeed, the elegance of its formulation is overshadowed only by the diversity of its applications, which include results from geometry, combinatorics, approximation theory, and (with enough imagination) the social sciences. We will look at many of these and, time permitting, explore related topics in convex geometry.

**Abstract:** To the model theorist, the notion of completeness is at the core of many beautiful theorems. We exhibit the eponymous Completeness Theorem (without proof) and briefly discuss its high-level meaning which should be of interest to any mathematician. From this and the notion of complete theories, we easily deduce nontrivial results in graph theory, algebra, and geometry, all of whose statements contain no obvious logic influence! These include the De Bruijn-Erdős theorem, the Noether-Ostrowski irreducibility criterion, and the Ax-Grothendieck theorem. This talk will be made accessible to mathematicians without any logic background by using minimal formalism and blackboxing as needed.

Previous Years' What is ...? Talks

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