We will discuss 2d Euler and Boussinesq (incompressible) flows related to a possible boundary blow-up scenario for the 3d axi-symmetric case suggested by G. Luo and T. Hou, together with some easier model problems relevant for that situation.

We work in the context of Markovian rough paths associated to a class of uniformly subelliptic Dirichlet forms and prove an almost-Gaussian tail-estimate for the accumulated local p-variation functional, which has been introduced and studied by Cass, Litterer and Lyons. We comment on the significance of these estimates to a range of currently-studied problems, including the recent results of Ni Hao, and Chevyrev and Lyons.

The main goal of the talk is to set up a framework for constructing the likelihood for discretely observed RDEs. The main idea is to contract a function mapping the discretely observed data to the corresponding increments of the driving noise. Once this is known, the likelihood of the observations can be written as the likelihood of the increments of the corresponding noise times the Jacobian correction.

Constructing a function mapping data to noise is equivalent to solving the inverse problem of looking for the input given the output of the Ito map corresponding to the RDE. First, I simplify the problem by assuming that the driving noise is linear between observations. Then, I will introduce an iterative process and show that it converges in p-variation to the piecewise linear path X corresponding to the observations. Finally, I will show that the total error in the likelihood construction is bounded in p-variation.

Multiplicative chaos theory originated from the study of turbulence by Kolmogorov in the 1940s and it was mathematically founded by Kahane in the 1980s. Recently the theory has drawn much of attention due to its connection to SLEs and statistical physics.  In this talk I shall present some recent development of multiplicative chaos theory, as well as its applications to Liouville quantum gravity.

We consider the problem of large-scale regression where both the number of predictors, p, and the number of observations, n, may be in the order of millions or more. Computing a simple OLS or ridge regression estimator for such data, though potentially sensible from a purely statistical perspective (if n is large enough), can be a real computational challenge. One recent approach to tackling this problem in the common situation where the matrix of predictors is sparse, is to first compress the data by mapping it to an n by L matrix with L << p, using a scheme called b-bit min-wise hashing (Li and König, 2011). We study this technique from a theoretical perspective and obtain finite-sample bounds on the prediction error of regression following such data compression, showing how it exploits the sparsity of the data matrix to achieve good statistical performance. Surprisingly, we also find that a main effects model in the compressed data is able to approximate an interaction model in the original data. Fitting interactions requires no modification of the compression scheme, but only a higher-dimensional mapping with a larger L.
This is joint work with Nicolai Meinshausen (ETH Zürich).

I will discuss some properties of Hermitian metrics on compact complex manifolds, having constant Chern scalar curvature, focusing on the existence problem in fixed Hermitian conformal classes (the "Chern-Yamabe problem"). This is joint work with Daniele Angella and Simone Calamai.

By classical results of Mumford and Donagi, Mori-Mukai, Verra, the moduli spaces A_g of principally polarized abelian varieties of dimension g are unirational for g≤5 and are of general type for g≥7. Answering a conjecture of Kanev, we provide a uniformization of A6 by a Hurwitz space parameterizing certain curve covers. Using this uniformization, we study the geometry of A6 and make advances towards determining its birational type. This is a joint work with Donagi-Farkas-Izadi-Ortega.

We describe the cohomology of moduli spaces of points on schemes over Abelian varieties and give explicit calculations for schemes in dimensions less that three. The construction of Gulbrandsen allows one to consider virtual motives in dimension three. In particular we see a new proof of his conjectures on the Euler numbers of generalized Kummer schemes recently proven by Shen. Joint work in progress with Junliang Shen.

Donaldson-Thomas theory for Calabi-Yau 3-folds is a complexification of Chern-Simons theory. In this talk, I will discuss joint work with Naichung Conan Leung on the complexification of Donaldson theory.