# Open Neighborhood Seminar

### Harvard University Math Department

## Spring 2021

### February 3

**Speaker:**Melanie Matchett Wood (Harvard)**Title:**Universality in random algebraic structures**Abstract:**We will explore the idea of universality in probability theory, as seen in the central limit theorem, in which wide and varied input distributions can all give identical limiting outputs. We will then explore examples of universality in random algebraic structures, such as random vector spaces over finite fields. We will explain how this is connected to recent research on random integral matrices and distributions of class groups of number fields.

### February 17

**Speaker:**Alex Wright (UMichigan)**Title:**Universes of evenly curved surfaces**Abstract:**We will begin by discussing hyperbolic geometry, and how it can be used to build "evenly curved" metrics on a donut with one point removed. We will then discuss Maryam Mirzakhani's computation of the "size" of the universe of all such metrics (the Weil-Petersson volume of the moduli space of complete hyperbolic metrics on a punctured torus).

### March 3

**Speaker:**Jenny Wilson (UMichigan)**Title:**Save the Pilgrim!**Abstract:**An evil mathematician has kidnapped the Harvard Pilgrim! To win his freedom, a group of undergrads must each find their name in a row of boxes. The odds look dire—but we’ll use some probability theory and combinatorics to find a strategy that dramatically improves our chances. Can you help save our hapless mascot?

### March 17

**Speaker:**Matt Emerton (UChicago)**Title:**Periodicity**Abstract:**The periodicity of certain functions has been a topic of study throughout the long history of mathematics. It was a topic of study for you in high school trigonometry, and perhaps again in college Fourier analysis! In this talk I will revisit some of the ideas related to periodicity and Fourier analysis, and explain some of their relationships to other topics in mathematics, such as group theory and number theory. In particular, I hope to say something about the important role of *non-abelian* groups of periods in contemporary mathematics.

### April 7

**Speaker:**Jordan Ellenberg (UW Madison)**Title:**Outward-facing mathematics**Abstract:**I will talk, pretty casually, about random walks, which I first learned about as part of my Harvard undergrad thesis in finite group theory, and which turn out to be at the heart of the mathematical analysis of gerrymandering; along the way I will talk about the project of doing mathematics in a way that engages with the world outside the math department walls.

### April 21

**Speaker:**Akhil Matthew (UChicago)**Title:**Algebraic topology and sums of squares formulas**Abstract:**It is a classical fact that the product of a sum of two squares with a sum of two squares is naturally a sum of two squares. (One can also replace "two" by "four" or "eight.") But in general, it is not known exactly when a product of the sum of m squares with a sum of n squares can be represented as a sum of p squares. I will discuss how methods of algebraic topology have been used to study this question. In particular, the tools of algebraic topology produce tools to obstruct the existence of such formulas in general. Moreover, these tools can be adapted to study the analogous question in positive characteristic.

**Organizers:**Ana Balibanu (ana@math.harvard.edu) and Elden Elmanto (elmanto@math.harvard.edu). Please drop us an email if you are curious about the seminar!