Tight security is increasingly gaining importance in real-world
cryptography, as it allows to choose cryptographic parameters in a way
that is supported by a security proof, without the need to sacrifice
efficiency by compensating the security loss of a reduction with larger
parameters. However, for many important cryptographic primitives,
including digital signatures and authenticated key exchange (AKE), we
are still lacking constructions that are suitable for real-world deployment.

This talk will present the first first practical AKE protocol with tight
security. It allows the establishment of a key within 1 RTT in a
practical client-server setting, provides forward security, is simple
and easy to implement, and thus very suitable for practical deployment.
It is essentially the "signed Diffie-Hellman" protocol, but with an
additional message, which is crucial to achieve tight security. This
message is used to overcome a technical difficulty in constructing
tightly-secure AKE protocols.

The second important building block is a practical signature scheme with
tight security in a real-world multi-user setting with adaptive
corruptions. The scheme is based on a new way of applying the
Fiat-Shamir approach to construct tightly-secure signatures from certain
identification schemes.

For a theoretically-sound choice of parameters and a moderate number of
users and sessions, our protocol has comparable computational efficiency
to the simple signed Diffie-Hellman protocol with EC-DSA, while for
large-scale settings our protocol has even better computational per-
formance, at moderately increased communication complexity.

 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.