The What is ...? seminar will run again in Summer 2022. Professor Vitaly Bergelson and Ethan Ackelsberg 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.
Abstract: Given a set of n points in the plane, what is the minimum number of distinct distances they span? Originally posed by Erdős in 1946, the distinct distances problem motivated a great deal of development in combinatorial geometry. It has generated a family of questions, many of which are still active areas of research. In this talk, we will present the historical advancements, from Erdős's original result to the essential resolution by Guth-Katz in 2015, as well as generalizations and applications.
Abstract: In 1907, Oskar Perron discovered a remarkable property of square matrices with positive real entries, while studying convergence criteria for Jacobi's generalized continued fractions. Namely, they possess a unique positive eigenvalue which is strictly greater than the modulus of any other eigenvalue. This fact was generalized, and Perron's proof was vastly simplified by Georg Frobenius in 1912, to what is now known as Perron-Frobenius theorem. Since then this theorem has found applications in dynamics, physics (statistical mechanics), probability (Markov processes), economics etc. In this talk, I will state and prove this theorem. Time permitting, I will also present its motivation and applications.
Abstract: A classical result states that a simple (one dimensional) continued fraction associated to a real number is periodic if and only if the real number is a quadratic irrational. Multidimensional Continued Fractions are an attempt to generalize this result to cubic, and subsequently higher degree, irrationals. After briefly restating some results on simple (one-dimensional) continued fractions in terms of matrices, we will discuss multidimensional continued fractions. Then, we will prove a result relating Perron—Frobenius matrices and Multidimensional Continued Fractions.
Abstract: Ptolemy's inequality, named after the Greek astronomer and mathematician Claudius Ptolemy, is a theorem that relates the six distances of a quadrilateral in Euclidean space. The case of equality, known as Ptolemy's theorem, allowed him to write a precise trigonometric table in the 2nd century AD. In my talk, I will go over the history and importance of the theorem and prove generalizations to non-Euclidean geometries. These proofs have connections to other classical determinants in metric geometry which can be used, for example, to argue why the Earth is spherical. If time allows, I will discuss the degenerate case of the inequality as curvature goes to negative infinity.
Abstract: A rearrangement of a function f : ℝn → [0, ∞) is a function with the same distribution of values as f, in the sense that its superlevel sets have all the same measures. The symmetric decreasing rearrangement f* of f is the particular rearrangement that is radially symmetric and maximally concentrates large values near the origin of its domain. We will explore a proof of the Riesz rearrangement inequality (Riesz, 1930), which describes how a certain integral involving three functions can be maximized among all rearrangements of these functions by choosing their symmetric decreasing rearrangements.
Abstract: A group is called amenable if it satisfies certain "nice" measurement properties. As we will see, many well-behaved classes of groups are amenable to being amenable (but the converse need not be true!) and several seemingly different "nice" measurement properties turn out to be equivalent. We will discuss properties of (non)amenability and time permitting, we will use the concept of amenability to explore the classical Banach-Tarski paradox and its generalizations.
Abstract: In 1932, Erdős conjectured that for any infinite sequence x1, x2, ... of 1 and -1 and any m > 0, there exist integers k and d such that |xd + x2d + ... + xkd| > m. In other words, when restricted to the arithmetic progression d, 2d, ..., kd, the number of 1's and the number of -1's differ by at least m, hence the word "discrepancy''. This conjecture was open for 80 years until resolved by Terry Tao in 2015. Tao's solution built upon the work of Polymath Project 5 and his own theorem on logarithmic Elliot's conjecture. In this seminar, I will talk about this fascinating problem, its history and the main ideas of Tao's proof (if time permits).
Abstract: After a review of the classical Bernoulli numbers, we explore how Dirichlet L-functions lead to the generalized Bernoulli numbers introduced by Leopoldt in 1958.
Abstract: Polynomials, if constructed correctly, can capture some crucial combinatorial information. A neat illustration of this principle is the Combinatorial Nullstellensatz, which was first formulated and proven by Noga Alon in 1999. In that same paper, he gave simple proofs of 14 previously known theorems from additive number theory and graph theory. We will look at some of these solutions and time permitting, solve (arguably) the most difficult International Mathematics Olympiad problem.
Abstract: The "spectral gap" is a quantitative notion describing the connectivity of a graph. If a graph does not have many edges (for example, if each vertex has only 3 neighbors), then it cannot be too well connected, as one might expect. However, there are some graphs which are exceptionally well connected, despite having few edges. These are called Ramanujan graphs, and it is difficult to explicitly describe them, even though they are (almost) plentiful.
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