I will discuss the quantum-field-theory origins of a classic result of Goresky-Kottwitz-MacPherson concerning the Koszul duality between the homology of G and the G-equivariant cohomology of a point. The physical narrative starts from an analysis of supersymmetric quantum mechanics with G symmetry, and leads naturally to a definition of the category of boundary conditions in two-dimensional topological gauge theory, which might be called the "G-equivariant Fukaya category of a point." This simple example illustrates a more general phenomenon (also appearing in C. Teleman's work in recent years) that pure gauge theory in d dimensions seems to control the structure of G-actions in (d-1)-dimensional QFT. This is part of joint work with C. Beem, D. Ben Zvi, M. Bullimore, and A. Neitzke.

Topological field theories give rise to a wealth of algebraic structures, extending
the E_n algebra expressing the "topological OPE of local operators". We may interpret these algebraic structures as defining (slightly noncommutative) algebraic varieties and stacks, called moduli stacks of vacua, and relations among them. I will discuss some examples of these structures coming from the geometric Langlands program and their applications. Based on joint work with Andy Neitzke and Sam Gunningham. 

We will discuss the percolation model on the hexagonal grid. In 2001 S. Smirnov proved conformal invariance of its scaling limit through the use of a tricky auxiliary combinatorial construction.

We present a more conceptual approach, implying that the construction in question can be thought of as geometrically natural one.

The main goal of the talk is to make it believable that not all nice and useful objects in the field have been already found.

No background is required.

We present sharp gradient estimates for the solution of the filtering equation and report on its applications in a high order cubature method for the nonlinear filtering problem.

I will present a mathematical model for the genetic evolution of a population which is divided in discrete colonies along a linear habitat, and for which the population size of each colony is random and constant in time. I will show that, under reasonable assumptions on the distribution of the population sizes, over large spatial and temporal scales, this population can be described by the solution to a stochastic partial differential equation with constant coefficients. These coefficients describe the effective diffusion rate of genes within the population and its effective population density, which are both different from the mean population density and the mean diffusion rate of genes at the microscopic scale. To do this, I will present a duality technique and a new convergence result for coalescing random walks in a random environment.

In this paper, we investigate the limit properties of frequency of empirical averages when random variables are described by a set of probability measures and obtain a law of large numbers for upper-lower probabilities. Our result is an extension of the classical Kinchin's law of large numbers, but the proof is totally different.

keywords: Law of large numbers,capacity, non-additive probability, sub-linear expectation, indepence

paper by: Zengjing Chen School of Mathematics, Shandong University and Qingyang Liu Center for Economic Research, Shandong University

Based on classical invariant theory, I describe a complete set of elements of the signature that is invariant to the general linear group, rotations or permutations.

A geometric interpretation of some of these invariants will be given.

Joint work with Jeremy Reizenstein (Warwick).

We present an algebraic formulation for the flow of a differential equation driven by a path in a Lie group. The formulation is motivated by formal differential equations considered by Chen.

Recently, Deya, Gubinelli, Hofmanova and Tindel ('16) (also Bailleul-Gubinelli '15) have provided a general approach in order to obtain a priori estimates for rough partial differential equations of the form
(*)    du = Au dt + Bu dX
where X is a two-step rough path, A is a second order operator (elliptic), while B is first order. We will pursue the line of this work by presenting an L^p theory "à la Krylov" for generalized versions of (*). We will give an application of this theory by proving boundedness of solutions for a certain class

The celebrated Black-Scholes theory shows that one can get a risk-neutral option price through hedging. The Cameron-Martin-Girsanov theorem for diffusion processes plays a key role in this theory. We show that one can get some statistical arbitrage from a sequence of well-designed repeated trading at their prices according to the ergodic theorem for stationary process. Our result is based on both theoretical model and the real market data.