The security of pairings-based cryptography relies on the difficulty of two problems: computing discrete logarithms over elliptic curves and, respectively, finite fields GF(p^n) when n is a small integer larger than 1. The real-life difficulty of the latter problem was tested in 2006 by a record in a field GF(p^3) and in 2014 and 2015 by new records in GF(p^2), GF(p^3) and GF(p^4). We will present the new methods of polynomial selection which allowed to obtain these records. Then we discuss the difficulty of DLP in GF(p^6) and GF(p^12) when p has a special form (SNFS) for which two theoretical algorithms were presented recently.

We review some recent progress in computing massless spectra 

and moduli in heterotic string compactifications. In particular, it was   

recently shown that the heterotic Bianchi Identity can be accounted 

for by the construction of a holomorphic operator. Mathematically,

this corresponds to a holomorphic double extension. Moduli can 

then be computed in terms of cohomologies of this operator. We 

will see how the same structure can be derived form a 

Gukov-Vafa-Witten type superpotential. We note a relation between 

the lifted complex structure and bundle moduli, and cover some 

examples, and briefly consider obstructions and Yukawa 

couplings arising from these structures.

It has been recently pointed out that maximal gauged supergravities in four dimensions often come in one-parameter families. The parameter measures the combination of electric and magnetic vectors that participate in the gauging. I will discuss the higher-dimensional origin of these dyonic gaugings, when the gauge group is chosen to be ISO(7). This gauged supergravity arises from consistent truncation of massive type IIA on the six-sphere, with its dyonically-gauging parameter identified with the Romans mass. The (AdS) vacua of the 4D supergravity give rise to new explicit AdS4 backgrounds of massive type IIA. I will also show that the 3D field theories dual to these AdS4 solutions are Chern-Simons-matter theories with a simple gauge group and level k also given by the Romans mass.

The 3d/3d correspondence is about the correspondence between 3d N=2 supersymmetric gauge theories and the 3d complex Chern-Simons theory on a 3-manifold.

In this talk I will describe codimension 2 and 4 supersymmetric defects in this correspondence, by a combination of various existing techniques, such as state-integral models, cluster algebras, holographic dual, and 5d SYM.

We will briefly review the notions of Dirac cohomology and of $A_{\mathfrak{q}}(\lambda)$ modules of real reductive groups, and recall a formula for the Dirac cohomology of an $A_{\mathfrak{q}}(\lambda)$ module. Then we will discuss to what extent an $A_{\mathfrak{q}}(\lambda)$ module is determined by its Dirac cohomology. This is joint work with Jing-Song Huang and David Vogan.

Abstract: First order asymptotic of scalar renewal sequences with infinite mean characterized by regular variation has been classified in the 60's (Garsia and Lamperti). In the recent years, the question of higher order asymptotic for renewal sequences with infinite mean was motivated by obtaining 'mixing rates' for dynamical systems with infinite measure. In this talk I will present the recent results we have obtained on higher order asymptotic for renewal sequences with infinite mean and their consequences for error rates in certain limit theorems (such as arcsine law for null recurrent Markov processes).

TBC

Abstract: We consider different notions of rough paths on manifolds and study some of the relations between these definitions. Furthermore, we explore extensions to manifolds modelled along infinite dimensional Banach spaces.