# Seminars & Events for Graduate Student Seminar

##### Geometry of Nonlinear Wave Equations

I will give an overview of a geometric approach to the issue of local well-posedness for quasilinear wave equations, and sketch its application to the proof of the Bounded L^2 Curvature Conjecture.

##### The Formation of Shocks in Nonlinear Wave Equations

I will discuss long time existence properties of certain quasilinear wave equations. The focus will be on a kind of singularity that occurs even when the initial data is very regular. Moreover, we shall see that the blow up is very geometric in nature.

##### Ergodic theory and negative curvature

Ergodic theory is a very robust method to understand/distinguish dynamical systems. To a Riemannian manifold (M,g) we can associate a canonical dynamical system, its geodesic flow. During this talk we will discuss the ergodic theory of geodesic flow on negatively curved manifolds. We will prove results about entropy of this flow and explain how starting from that we can conclude information about isometry groups and connection to open problems.

##### Lefschetz Pencils

We will discuss Lefschetz pencils and will prove Lefschetz's theorem on hyperplane sections (without using transcendental methods), which will give us a more or less full understanding of the (co)homology of nonsingular complex algebraic varieties up to the middle dimension. Time permitting we will also discuss the Picard Lefschetz monodromy formulas and sketch a proof that will follow roughly the same lines as our proof of the weak Lefschetz theorem.

##### Singular points of complex surfaces and Heegaard-Floer homology

I will describe some basic constructions of Singularity Theory and their relation with Low Dimensional Topology. In particular, in the second part of the talk, I want to discuss some relations between complex surface singularities and Heegaard Floer homology.

##### Quantum mechanics and low-dimensional topology

We'll discuss some applications of quantum mechanics (and, time permitting, quantum field theory) to Morse theory and low-dimensional topology, offering proof sketches of the Morse inequalities and new viewpoints on topological invariants.

##### Tunnell's Work on the Congruent Number Problem

The congruent number problem is a classical diophantine problem which asks to determine which integers are the areas of right triangles with rational sides. We explain Tunnell's theorem on congruent numbers using elliptic curves and modular forms.

##### Sharp Sobolev Inequalities and Applications

Interest in extremal cases of Sobolev embeddings is not just intrinsic but also arises from some useful geometric information they can provide. We'll first discuss tools used to obtain these inequalities before turning to applications on manifolds such as rigidity results.

##### A survey on the h-cobordism theorem and its applications

Smale's h-cobordism theorem has been one of the most influential results in topology, so much that the conventional line of separation between "low" and "high" dimensions is drawn between dimensions 4 and 5 because the h-cobordism holds true in dimension greater or equal to 5. During the talk I will try to give the key idea of the h-cobordism theorem (the so-called Whitney trick), and why it holds only in high dimensions. Time permitting, I will discuss some related problems, such as the generalised Poincaré conjecture.

##### The Modularity Theorem

The Shimura–Taniyama–Weil conjecture, now known as the modularity theorem, states that all elliptic curves over the field of rational numbers are modular. In this talk we will not attempt to discuss the proof; instead, we will have a more modest goal — to understand the statement, i.e., what it means for an elliptic curve to be modular. We will introduce elliptic curves, modular forms, Hecke algebras and eventually the Shimura construction. Time permitting, we will also see how the modularity theorem relates to other theorems and conjectures.

##### Capsets, Sunflower-free sets in {0,1}^n, and the slice rank method

In this talk we will look at the recent breakthrough of Ellenberg and Gijswijt and Croot, Lev and Pach, which used polynomial method to obtain exponential upper bounds for the Capset problem, that is upper bounds for the size of the largest set in F_3^n which contains no three term arithmetic progressions. In particular we will look at Tao's reformulation of this approach using the so called "Slice Rank Method," and how it can be used to give exponential upper bounds for the Capset and Sunflower-free problems.

##### Lagrangian analyticity in fluid mechanics

The incompressible Euler equations imply that the Lagrangian trajectories of individual water particles are automatically analytic in time, as long as the velocity field is slightly better than differentiable in the space variables. We'll discuss some of the mechanisms in different fluid models that allow for this property, and some that break it.

##### Remarkable identities in the counting functions for cubic and quartic rings

Let h(D) be the number of cubic rings having discriminant D, and let h'(D) be the number of cubic rings having discriminant -27D where the traces of all elements are divisible by 3. (In each case, weight rings by the reciprocal of their number of automorphisms.) At first glance, there is no relation between these two quantities, nor was there expected to be until in 1997, Y. Ohno computed them up to |D| = 200 and found a stunning coincidence that generalizes both cubic reciprocity and the Scholz reflection principle. The next year, J. Nakagawa verified Ohno's conjecture. I will explain the main ideas in my streamlined version of Nakagawa's proof, and an extension to quartic rings still in progress.

##### Maximum principle for tensors and Ricci flow invariant curvature conditions

Maximum principle is an important feature of parabolic/elliptic PDEs having numerous applications. In particular, it allows to bound solutions to these equations by the initial/boundary values. The Ricci flow is an evolution equation for a Riemannian metric resembling parabolic heat equation and it is highly desirable to have analogues of maximum principle for various geometric objects (primarily curvature tensor) associated to a metric. In 1986 Richard Hamilton proved a general maximum principle for tensors, which allowed to find many conditions on curvature tensor preserved under the Ricci flow. We discuss Hamilton's approach and formulate the most general approach to Ricci flow invariant curvature conditions formulated in 2013 by Burkhard Wilking.

##### Lagrangian Floer homology on a surface with boundary

We will describe an algebraic framework, which allows one to compute Lagrangian Floer homology HF(L0,L1) of two immersed curves on a surface. To a surface we assign a certain algebra. To each immersed curve L we assign an A_infinity module M(L) (over the surface algebra). Then computing Lagrangian Floer homology reduces to a computation of Hom(M(L0),M(L1)). We will describe this construction on a particular example of a 2-sphere with four discs deleted.

##### Ergodic Theory and Number Theory

The foundations of Ergodic Theory lie in Statistical Physics, but nowadays the field has many fruitful connections with Number Theory. We will explore some of these surprising links and discuss Furstenburg's remarkable proof of Szemerédi's Theorem, which says that any subset of the natural numbers of positive upper Banach density contains arbitrarily long arithmetic progressions.

##### Community Detection in the Stochastic Block Model

The Stochastic Block Model is one of the simplest models for random graphs in which there are different types of vertices. The key question that we consider is whether or not one can determine which vertices are of which type from the graph. This talk will cover some of the approaches to distinguishing between different types of vertices, such as acyclic belief propagation.

##### The Matching Game

In graphs, we often want to find matchings (sets of edges such that no two edges share a common vertex) with particular properties. We discuss algorithms and properties of matchings, with practical applications. No previous graph theory knowledge is expected.

##### What simple curves on surfaces know

Simple closed curves on Riemann surfaces carry an interesting structure. An example is a novel symmetry satisfied by generating series for collections of such curves. This symmetry reflects the global geometry of certain moduli spaces (containing Teichmuller spaces), whose Diophantine study reveals new insight on mapping class groups. In this talk, we give an elementary discussion of some of these ideas.

##### modular forms of weight 1

In this talk, we will discuss a classical result of Deligne and Serre on the existence of Galois representations associated to newforms of weight 1. When time permits, we'll discuss Langlands, Fontaine-Mazur conjecture in this context.