MEETINGS: Tuesdays and Thursdays during summer semester at 4:00 pm.
Here is the Zoom link.
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.
Title: What is Arrow's Impossibility Theorem?
Abstract: Consider an election where voters are asked to rank each of three candidates. Is there a “good” way to choose a winner? A famous result in social choice theory, Arrow’s impossibility theorem, tells us that the answer is no for a particular definition of “good”: A system satisfying certain conditions will always produce a dictator. Curiously, the generalization of this question to an infinite electorate quickly leads us into the world of ultrafilters. In this talk we will formulate Arrow’s theorem and give an elementary proof before introducing the ultrafilter perspective and investigating if an infinite electorate really eliminates our dictator.
Title: What is the limit of a sequence of graphs?
Abstract: Graph theory finds applications in many areas of mathematics, as well as physics, computer science, and many other disciplines. But what happens when the graphs under consideration get very large? Within the past 20 years, a few limit theories have been developed, with the aim of approximating features of very large graphs. We will discuss two of these theories, going into more detail about one of them: the bounded-degree random weak limit theory of Benjamini and Schramm.
Title: What is the Lovasz conjecture on Hamiltonian cycles?
Abstract: One version of the Lovasz conjecture on Hamiltonian cycles is that any connected Cayley graph of a finite group has a Hamiltonian cycle. We will review some known results pertaining to the Lovasz conjecture as well as state 2 of its variants. If time permits, we will also discuss the 1-2-3 conjecture, which is about inducing a proper vertex coloring of a graph from a coloring of its edges. After reviewing the results that support the 1-2-3 conjecture, we will discuss its analogue for directed graphs.
Title: What is Bayesian Decision Theory?
Abstract: Bayesian Decision Theory is a framework for taking advantage of both prior knowledge about the relative likelihood of certain states of nature as well as information about the cost (or "loss") associated with different kinds of errors. Using this info, you can quantifiably make mathematically optimal decisions based on the results of experiments you perform. In this talk, we will cover the basic flow of how to use decision theory and show applications of this perspective to estimation problems and hypothesis testing.
Title: What is Minkowski's Question Mark function?
Abstract: Minkowski's Question Mark Function, denoted ?, is a map which continuously extends a bijection between two Cantor subsets of the real numbers. As it turns out, ? connects to continued fractions and fractals in very unexpected ways. We will discuss these connections as well as other interesting properties of this function.
Title: What is the Alexander Horned Sphere?
Abstract: We will be discussing a famous counterexample to the so-called Jordan Curve Theorem in three dimensions, the Alexander Horned Sphere. We will also be discussing the phenomenon of mathematical pathology more widely, with other examples of I’ll behaved or strange objects.
Title: What is a Besicovitch set?
Abstract: A Besicovitch set is a set containing at least one unit line segment in every direction. It was conjectured that there was a minimum area to Besicovitch sets in the plane. However, there are several constructions of Besicovitch sets which are compact and of Lebesgue measure zero. In addition, Besicovitch sets is related to a number of areas of mathematics including harmonic analysis.
Title: What are the Euler—Lagrange equations?
Abstract: Many natural optimization problems in geometry and physics require one to maximize or minimize a functional. Building on the idea of stationarity underlying Fermat’s theorem (local extrema of differentiable functions occur where the derivative is zero), Euler and Lagrange discovered a system of differential equations whose solutions are local extrema of functionals. In this talk, we will introduce the calculus of variations and derive the Euler—Lagrange equations. Then we will solve the equations in several different contexts, producing a wide range of applications, including minimal surfaces and an alternative formulation of Newton's second law of motion.
Title: What is an Apollonian gasket?
Title: What are surreal numbers?
Abstract: In a game like chess or go, how do we know who is winning and which move is the best? John Conway studied how to evaluate and compare positions in a game, and defined surreal numbers to be the values of certain game positions. Surreal numbers are an extension of the real numbers, but are constructed in a simpler way.
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