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.

Let X be a smooth projective curve (of genus greater than or equal to 2) over C and M the moduli space of vector bundles over X, of rank 2 and with fixed determinant of degree 1.Then the Fourier-Mukai functor from the bounded derived category of coherent sheaves on X to that of M, given by the normalised Poincare bundle, is fully faithful, except (possibly) for hyperelliptic curves of genus 3,4,and 5

 This result is proved by establishing precise vanishing theorems for a family of vector bundles on the moduli space M.

 Results on the deformation  and inversion of Picard bundles (already known) follow from the full faithfulness of the F-M functor