What is ... ? Seminar

Abstract: In the L¹ 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 L¹ and L^∞ spaces do not have many of the nice properties that the other L^p spaces have; for example, they are not reflexive. In this talk, we will make these feelings precise by defining uniform convexity and discussing how it allows us to prove useful properties of functional spaces.

Talk notes

Mikey Reilly

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.

Talk notes

Saúl Rodríguez Martín

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].

Talk notes

Neil Falkner

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.

Matthew Stone

 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?"

Linus Ge

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).

Avery McNeill

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.

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Sachin Gautam

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Kabir Belgikar

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Hunter Handley