## Yale math graduate student seminar

Fridays 12:30pm in LOM 201

Current organizers: Elijah and Fernando

### Spring 2019

May 2 **(Thursday 12 am) Junior colloquium by Yifeng Liu, pre-talk by Elad.**

Apr 26 **No seminar -- **GATSBY will feature lunch and general interest talks.

Apr 19 **Guangyu**

**Title:** What is a Period

**Abstract:** Periods are number arising as integrals of algebraic functions over domains defined by algebraic equations or inequalities. We will introduce this mysterious but important class of numbers and its connection with many other topics, including (Picard-Fuchs) differential equations, hypergeometric functions, modular forms and L functions.

Apr 12 **Michael**

**Title: **The mother of all 3-manifolds

**Abstract: **We will define an operation on 3-manifolds called Dehn surgery. Then we will show that any closed, connected, orientable 3-manifold can be obtained from Dehn surgery on the 3-sphere.

**Note:** 12:00 at LOM 200.

Apr 5 **Kolya**

**Title:** How to approximate functions (without real math)

**Abstract: **We here are not the most practical people. Many of us would think it exciting that one can fit a polynomial of degree n to any n+1 given real values – and stop at that. But how difficult would it be to write down a function of 100×100 binary variables – 0 or 1, black or white – expressing whether a 100×100 black-and-white picture shows a kitten, a skunk, or a telephone booth?

For better or for worse, in the real world, people like to (approximately) interpolate these kinds of functions through a huge amount of very noisy observed data in order to make decisions and predictions. This is important in the natural sciences, economics and finance, etc. In one particular application, we see this a wide range of modern technology, in the form of “intelligent” machines, yielding many pretty pictures.

I will give an introduction to the practical methods used for this purpose and their mathematical foundations, and show how little is really known about the mathematical properties. The pretty pictures are included.

Mar 29 **Pratyush**

**Title: **Mixing in hyperbolic manifolds

**Abstract: **We will start with introducing definitions and concepts in dynamical systems and hyperbolic geometry and illustrate them with examples. In addition to being of interest in itself, mixing properties have numerous applications to prove theorems in homogeneous dynamics and we will see an example of this. We will conclude with a tour of some current research in this topic.

Mar 8 **James**

**Title:** A tale of 2-groups and other stories from infinite group theory

**Abstract**: Is every finitely generated group in which every element has finite order itself finite? What if there is an N such that every element has order dividing N? We will learn about the history of such problems and resolve a few.

Mar 1 **Aaron**

**Title: **Illumination and Blocking in Rational Polygons

**Abstract:** Suppose that you are in a room with mirrored walls. The classical illumination problem asks: if you put a light source at a given point, will it light up the whole room? Are there rooms which can never be fully lit up? Now suppose that there are other people standing in the room: how does this affect the points which are illuminated? Starting from first concepts, I will explain Lelièvre, Monteil, and Weiss’s solution in the case when the room is a polygon whose angles are rational multiples of π. Along the way, we will get a glimpse at the rich interaction between the dynamics of billiards and the geometry of translation surfaces, as well as the breakthrough results of Eskin, Mirzakhani, and Mohammadi.

Feb 22 **Shiyue**

**Title: **Moduli Space of Linear Series with Imposed Ramification

**Abstract:** The moduli space of linear series on a curve $X$ of genus $g$ is well understood in Brill-Noether theory. We will introduce the main theorems of classical Brill-Noether variety, and then discuss the singularities of the moduli space with imposed ramification conditions studied by Chan-Osserman-Pflueger using degeneration techniques.

Feb 15 **Tal**

**Title:** Nuclear Reactor or How I Learned to stop worrying and love the reactor

**Abstract:** This talk will be a brief introduction to the world of nuclear reactors: From the basic building blocks of the physical processes, through what makes a nuclear reactor a nuclear reactor, passing through the safety and philosophy of safety, to the current global situation of nuclear energy and a glimpse to the future of it.

Feb 8 **Byungmin**

**Title:** Spectrum of random symmetric matrices

**Abstract:** Suppose we have a machine that generates random symmetric Bernoulli matrices: n×n symmetric matrices whose entries are ±1 with probability half and half. Using this machine, generate many matrices and draw the histrogram of their eigenvalues. What will be the shape of this histogram? In this talk, we prove that its shape converges to the semicircle as n→∞ (after suitable normalization).

Feb 1 **Elad**

**Title:** Introduction to Harmonic analysis over finite fields

**Abstract:** In this talk we will define the Fourier transform for functions defined over a finite abelian group. We will prove the analogs of Plancherel's theorem and of Fourier inversion formula. Then we'll focus on finite fields and introduce the Mellin transform. We'll introduce and prove the analog of Tate's local functional equation, which will lead us to the introduction of Gauss sums. If time permits, we will use Gauss sums and some ring theory/finite field theory to prove the law of quadratic reciprocity.