# Seminars & Events for Graduate Student Seminar

##### A toy model in fluid dynamics

Distribution of energy among various Fourier modes and how they fluctuate in a turbulent flow are central questions in fluid dynamics. I will speak about a toy model (introduced by Friedlander Katz Pavlovic) which aims to understand energy transport.

##### The Gauss Sphere Problem

The Gauss Sphere Problem is a long-open problem in analytic number theory concerning the number of lattice points in a sphere. In this talk I will discuss a variant of this problem: that of summing a polynomial over the lattice points in a sphere. We will use tools from Fourier analysis, exponential sums and modular forms, to sharpen our estimates step by step.

##### On complex geometry of some non-Kahler manifolds

In the talk we discuss a construction of a large family of complex manifolds. This family includes Hopf and Calabi-Eckmann manifolds. Although the question whether a given differentiable manifold admits a complex structure is extremely hard, our construction completely answers it in the case of manifolds with "large" group of U(1)^m-symmetries. We discuss several very basic geometric characteristics of these manifolds including Dolbeault cohomology and field of meromorphic functions.

##### Uniform strong primeness in matrix rings

A ring $R$ is uniformly strongly prime if some finite $S \subseteq R$ is such that for $a,b \in R$, $aSb = \{0\}$ implies $a$ or $b$ is zero, in which case the bound of uniform strong primeness of $R$ is the smallest possible size of such an $S$. The case of matrix rings $R$ is considered. Via vector multiplication and bilinear equations, we obtain alternative definitions of uniform strong primeness, together with new theorems restricting the bound of uniform strong primeness of these rings.

##### Counting n particles in the plane

How do algebraic geometers "count" the number of ways of putting n points in the plane? I'll explain what Euler characteristic is, what a Hilbert scheme is, and how to compute the Euler characteristic of the Hilbert scheme of n points in the plane. The answer may surprise you, especially if you don't know what those words mean yet. Everything will be defined from scratch.

##### Symplectic surfaces vs pseudoholomorphic curves

Although the first appearence of symplectic manifolds was in physics, they have played a fundamental role in pure math for the last 30 years. A big breakthrough came after the work of Mikhail Gromov, who started the study of pseudoholomorphic curves in symplectic manifolds. This make a connection with ideas coming from algebraic geometry. The same kind of ideas allowed Andreas Floer to introduce Floer's homology which has revolutionized low dimensional topology in the last 20 years. I plan to talk about topological properties of those manifolds and the basic facts about the moduli space of pseudoholomorphic curves in a symplectic manifold and how, once we have constructed it, this gives amazing applications to geometry. I'm also giving a proof of the nonsqueezing theorem of Gromov.

##### Braid Groups and Categorification

Braid groups are fundamental objects in mathematics and categorification is the process of replacing set-theoretic theorems by category-theoretic analogues. We will connect the two by discussing a categorification of the Temperley-Lieb Algebra which results in a braid group action. We will conclude by discussing how a braid group action leads to a homology theory of links - triply graded link homology recently discovered by Mikhail Khovanov.

##### The p-adic uniformization of some curves

The classical uniformization theorem classifies Riemann surfaces according to their universal covers, and thus provides an important tool for the study of Riemann surfaces (curves over complex numbers). In the p-adic world, similar results hold for some classes of curves. In this talk, I want to show how Tate did this for some elliptic curves and how Mumford generalized it to curves of higher genus.

##### Nonabelian Hodge theory and uniformization

Classical Hodge theory provides a link between the topology and the analytic geometry of a compact Kaehler manifold X via harmonic forms. Similarly, in nonabelian Hodge theory (developed by Simpson based on works of Hitchin, Corlette, Donaldson, Uhlenbeck-Yau, and others), harmonic metrics on vector bundles are used to study the fundamental group of X, a nonabelian topological invariant. In this talk, we give an introduction to these topics. As an application, we sketch a proof of the old uniformization theorem for Riemann surfaces from a Hodge-theoretic point of view.

##### Playing with the waves outside a black hole

In the recent years, a lot of research has been devoted to the study of the behaviour of scalar waves in the exterior region of black hole spacetimes. This is intimately connected to the problem of non linear stability of such black hole spacetimes as solutions to the Einstein equations. In this talk, we will explore some interesting theorems and results in this area. Of course no physics background is required!

##### Rationality of Algebraic Varieties

Finding a rational parametrization for a system of polynomial equations has been studied for a long time, and it leads to the concept of rational varieties. Bézout’s theorem implies that degree 1 and 2 hypersurfaces are rational. It is a hard problem to determine whether a general variety is rational. For a smooth projective variety to be rational, it is necessary that any (tensor power of) differential form must vanish. This condition is also sufficient for curves and surfaces due to Riemann and Castelnuovo respectively. However, this is not true for higher dimensional varieties by the works of Clemens-Griffiths, Iskovskikh-Manin, Artin-Mumford, Kollár, etc.. In this talk, I will give an introduction to the rationality problem and prove the unirationality of cubic threefolds.

##### Finding Rational Curves

Rational curves are one of the most basic objects to look at on an algebraic variety, but they have a large impact on the geometric structure of the variety in which they sit. In this talk we will talk about what are rational curves, why they are useful, and how they are found. In particular we will discuss their role in birational geometry and look at Mori's proof of the bend and break theorem, used to prove certain rational curves exist on an algebraic variety.

##### Mobius randomness and homogeneous dynamics

Qualitative approaches to understanding the randomness of the primes offer a first step toward the extremely difficult quantitative challenge of sharply bounding sums involving the Mobius function. Recently, Sarnak has conjectured such a qualitative description that subsumes many previously known examples: any observable sequence of complex numbers from a zero-entropy topological dynamical system must fail to correlate with the Mobius function. I will outline this conjecture, go over some interesting known cases, and describe a bit of my thesis work, in which I aim to prove that the conjecture is true for all zero-entropy dynamics on homogeneous spaces of semisimple Lie groups.

##### Visualizing low-dimensional manifolds

Knots, 3-manifolds, and 4-manifolds are fundamental objects in topology. I will describe how to represent these objects using knot diagrams, Heegaard diagrams, and Kirby diagrams. We'll see how these pictures are useful for defining and computing invariants.

##### An introduction to the 3-D Euler Equations and the breakdown of smooth solutions

In this talk, I will introduce 3-D Euler equations, basic properties of them including derivations, formulations and related concepts like vorticity. Then I will introduce a classic result about the breakdown of smooth solutions, by Beale-Kato-Majda.

##### Geometric Analysis of Wave Equations

In this talk I will give a basic introduction to wave equations, with particular emphasis on geometric techniques which have proven useful in the study of nonlinear wave equations. Time permitting, I will also discuss some recent results on shock formation and on low-regularity local well-posedness.

##### Quantum knot invariants and the volume conjecture

I'll sketch some of the basics of quantum topology, including how to get knot invariants from a ribbon category and how to construct non-cocommutative Hopf algebras. Next we'll turn to hyperbolic geometry and discuss how hyperbolic structures on knot complements arise. Finally, I'll discuss how these two topics are linked via the volume conjecture, and give some directions of current work.

##### Defining varieties over number fields

Shimura varieties are complex algebraic varieties obtained as quotients of Hermitian symmetric spaces. However, it turns out that they admit canonical models defined over number fields. In this talk I will explain how to define the modular curve X_0(N) over the rationals and introduce the concepts of special points and canonical models for general Shimura varieties.

##### Riemann-Hurwitz

Using some simple ideas from topology, we'll prove Fermat's Last Theorem and the ABC conjecture, more or less. We'll also describe how to tile your favorite Riemann surfaces with attractive wallpaper.

##### Tropical Curves and Brill-Noether Theory

In the past few years, tropical geometry has established itself as an important new field bridging algebraic geometry and combinatorics whose techniques have been used successfully to attack problems in both fields. I'll talk about the basics of tropical geometry, with an emphasis on tropical curves. If time permits, I'll show how tropical geometry can be applied to prove results in Brill-Noether thoery of line bundles on algebraic curves.