# Upcoming Seminars & Events

## Primary tabs

##### CR invariant area functionals and the singular CR Yamabe solutions on the Heisenberg of H^{1}.

Joint seminar with Rutgers University.

##### Isoperimetric inequalities on surfaces

I will describe several recent results concerning extremal metrics and values of isoperimetric constants on different surfaces like the 2-sphere, the real projective plane or the Klein bottle. The idea is to find a metric realizing the supremum of a given eigenvalue over the whole set of Riemannian metrics on the surface. This problem (and its solutions) has a very strong link with the theory of harmonic maps.

##### Regularity and structure of scalar conservation laws

Scalar conservation law equations develop jump discontinuities even when the initial data is smooth. Ideally, we would expect these discontinuities to be confined to a collection of codimension-one surfaces, and the solution to be relatively smoother away from these jumps. The picture is less clear for rough initial data which is merely bounded. While a linear transport equation may have arbitrarily rough solutions, genuinly nonlinear conservation laws have a subtle regularization effect.

##### The mathematics of charged liquid drops

In this talk, I will present an overview of recent analytical developments in the studies of equilibrium configurations of liquid drops in the presence of repulsive Coulombic forces. Due to the fundamental nature of Coulombic interaction, these problems arise in systems of very different physical nature and on vastly different scales: from femtometer scale of a single atomic nucleus to micrometer scale of droplets in electrosprays to kilometer scale of neutron stars.

##### Uniqueness of blow-ups and asymptotic decay for Dirichlet energy minimizing multi-valued functions

***Please note special date and time. **In the early 1980's Almgren developed a theory of Dirichlet energy minimizing multi-valued functions, proving that the Hausdorff dimension of the singular set (including branch points) of such a function is at most (n-2) where n is the dimension of its domain. Almgren used this bound in an essential way to show that the same upper bound holds for the dimension of the singular set of an area minimizing n-dimensional rectifiable current of arbitrary codimension.

##### Noether-Lefschetz locus on singular threefolds

The classical Noether-Lefschetz theorem says that any curve in a very general surface X in P^3 of degree d \geq 4 is a restriction of a surface in the ambient space, in particular the Picard number of X is 1 (a property is very general if it holds in the complement of countably many proper closed subvarieties).

The Noether-Lefschetz locus S_d is the locus of the degree d \geq 4 surfaces in P^3 whose Picard number is greater than 1. I will discuss

generalizations and applications to singular ambient spaces, and also

##### Joint Equidistribution of CM Points

A celebrated theorem of Duke states that Picard/Galois orbits of CM points on a complex modular curve equidistribute in the limit when the absolute value of the discriminant goes to infinity.

The equidistribution of Picard and Galois orbits of special points in products of modular curves was conjectured by Michel and Venkatesh and as part of the equidistribution strengthening of the André-Oort conjecture. I will explain the proof of a recent theorem making progress towards this conjecture.

##### Stochastic homogenization: renormalization, regularity, and quantitative estimates

There has been a lot of work in recent years on the problem of understanding the behavior of solutions of PDEs with random coefficients, with most of the work focused on linear elliptic equations in divergence form. I will give an overview of recent joint works with Smart, Kuusi and Mourrat in which we introduce a ``renormalization group" method, which leads to a very precise, quantitative description of the solutions.

##### Two problems involving breakup of a liquid film

Understanding the breakup of a liquid film is complicated by the fact that there is no obvious instability driving breakup: surface tension favors a film of uniform thickness over a deformed one. Here, we identify two mechanisms driving a film toward (infinite time) pinch-off. In the first problem, we show how the rise of a bubble is arrested in a narrow tube, on account of the lubricating film pinching off. In the second problem, breakup of a free liquid film is driven by a strong temperature gradient across the pinch region.

##### TBA-Yu-Shen Lin

##### TBA-Xudong Zheng

##### TBA - Dr. Efi Efrati

##### TBA-Gal Mishne

##### TBA-Timo Seppalainen

##### Regularity of area-minimizing surfaces in higher codimension: old and new

The theory of integral currents, developed by Federer and Fleming in the 60s, gives a powerful framework to solve the Plateau's problem in every dimension and codimension. The interior and boundary regularity theory for the codimension

one case is rather well understood, thanks to the work of several mathematicians in the 60es, 70es and 80es.

##### TBA-Amina Abdurrahman

##### TBA-Joel Moreira

##### TBA-Suyoung Choi

##### TBA-Serguei Denissov

##### Irreducible SL(2,C)-representations of integer homology 3-spheres

We prove that the splicing of any two non-trivial knots in the 3-sphere admits an irreducible SU(2)-representation of its fundamental group. This uses instanton gauge theory, and in particular a non-vanishing result of Kronheimer-Mrowka and some new results that we establish for holonomy perturbations of the ASD equation.