# Seminars & Events for 2016-2017

##### Kernel Fusion for Manifold Learning and Signal Processing

Kernel-based methods are useful for various machine learning tasks. A kernel is a symmetrical positive definite function constructed on the graph of the data. Spectral analysis of the kernel can lead to an efficient representation.

##### The sphere packing problem in dimensions 8 and 24.

The sphere packing problem is to find an arrangement of non-overlapping unit spheres in the $d$-dimensional Euclidean space in which the spheres fill as large a proportion of the space as possible. In this talk we will present a solution of the sphere packing problem in dimensions 8 and 24. In 2003 N. Elkies and H.

##### 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.

##### Maximising the number of induced cycles

How many holes can a graph on n vertices contain? How many induced cycles? For sufficiently large n, we determine the maximum number of induced cycles, the maximum number of holes, and the maximum number of

##### Modular forms and optimization in Euclidean space

In this talk we will show how modular forms can be applied to energy minimization problems in Euclidean space. Namely, we will explain Cohn-Elkies linear programming method and present explicit constructions of

corresponding certificate functions. In particular, we will discuss optimization in dimensions 8 and 24.

##### Nonlinear echoes and Landau damping with insufficient regularity

In this talk, we will discuss recent advances towards understanding the regularity hypotheses in the theorem of Mouhot and Villani on Landau damping near equilibrium for the Vlasov-Poisson equations. We show that, in general, their theorem cannot be extended to any Sobolev space on the 1D torus.

##### Lagrangian cell complexes and Markov numbers

Joint work with Ivan Smith. Let p be a positive integer. Take the quotient of a 2-disc by the equivalence relation which identifies two boundary points if the boundary arc connecting them subtends an angle which is an integer multiple of (2 pi / p). We call the resulting cell complex a 'p-pinwheel'. We will discuss constraints on Lagrangian embeddings of pinwheels.

##### Cornered Asymptotically Hyperbolic Metrics

This talk will concern cornered asymptotically hyperbolic spaces, which have a finite boundary in addition to the usual infinite boundary. After introducing the setting, we will present a normal form near the corner for these spaces.

##### Loop erased random walk, uniform spanning tree, and bi-Laplacian Gaussian field in the critical dimension

Critical lattice models are believed to converge to a free field in the scaling limit, at or above their critical dimension. This has been (partially) established for Ising and Phi^4 models for d \ge 4. We describe a simple spin model from uniform spanning forests in Z^d whose critical dimension is 4 and prove that the scaling limit is the bi-Laplacian Gaussian field for d\ge 4.

##### 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.

##### Recent advances in explicit constructions of Ramsey graphs

In his 1947 paper that inaugurated the probabilistic method, Erdős proved the existence of (2 log n)-Ramsey graphs on n vertices. Matching Erdős' result with a constructive proof is an intriguing problem in combinatorics that has gained a significant attention in the literature. In this talk we will present recent works towards this goal.

##### Recent progress on Serre weight conjectures

I will discuss some recent results on Serre weight conjectures in dimension >2, based on the study of certain tame type deformation rings. This is joint work with (various subset of) D. Le, B. Levin and S. Morra.

##### Homology cobordism and surgery on alternating knots

We discuss a few computations in Pin(2)-monopole Floer homology, and their applications. Our main protagonists are homology spheres obtained by surgery on alternating knots.

##### On the steady-states of the Navier-Stokes equations in the plane

Despite the seminal work of Jean Leray, the stationary Navier-Stokes equations in two-dimensional unbounded domains are still not completely understood mathematically. More precisely, the behavior at infinity of the weak solutions is an open question. The Stokes paradox states that the linearization of the Navier-Stokes equations have no bounded solutions in general.

##### Area-Minimizing Surfaces in Asymptotically Flat Three-Manifolds

We will discuss joint work with M. Eichmair in which we show that asymptotically flat three-manifolds with non-negative scalar curvature do not admit unbounded area-minimizing boundaries unless the ambient manifold is flat.

##### Efficient Numerical Methods for Thermodynamic Averaging and Statistical Inference

Molecular models and data analytics problems give rise to gargantuan systems of stochastic differential equations (SDEs) whose paths ergodically sample multimodal probability distributions. An important challenge for the numerical analyst (or the chemist, or the physicist, or the engineer, or the data scientist) is the design of efficient numerical methods to generate these paths. For SDEs, the

##### Cost of splitting Lagrangians

Assume that the derived Fukaya category of a symplectic manifold admits a collection of triangular generators. By definition, this means that any other Lagrangian submanifold which is an object of this category can be decomposed in terms of exact triangles involving the generators.

##### Critical points and the Gibbs measure of a spherical spin glass model

For integers N let H_N(x) be an isotropic Gaussian field on the N-dimensional unit sphere, meaning that Cov(H_N(x),H_N(y)) is a function, f_N, of the inner product of <x,y>. The spherical spin glass models of statistical mechanics are such random fields, with f_N = N f with the function independent of the dimension N.

##### The degenerate special Lagrangian equation

I will discuss a degenerate form of the special Lagrangian equation that arises as the geodesic equation for the space of positive Lagrangians. Considering graph Lagrangians in Euclidean space, the equation reduces to a second order fully non-linear PDE for a single real function.

##### Extremal Cuts of Sparse Random Graphs

The Max-Cut problem seeks to determine the maximal cut size in a given graph. With no polynomial-time efficient approximation for Max-Cut (unless P=NP), its asymptotic for a typical large sparse graph is of considerable interest.