I will discuss the circumstances in which residual finiteness properties of an amalgamated free product $A\ast_c B$ may be deduced from the properties of $A$ and $B$, with particular regard to the pro-p residual properties.

In this lecture we describe the effective Chebotarev Theorem for global function fields and show how this can be used to describe the statistics of a polynomial map f in terms of its monodromy groups. With this tool in hand, we will provide a strategy to remove the remaining heuristic in the quasi-polynomial time algorithm for discrete
logarithm problems over finite fields of small characteristic.

The CHY formulae are a set of remarkable formulae describing the scattering amplitudes of a variety of massless theories, as  certain worldsheet integrals, localized on the solutions to certain polynomial equations (scattering equations). These formulae arise from a new class of holomorphic strings called Ambitwistor strings that encode exactly the dynamics of the supergravity (Yang-Mills) modes of string theory.



Despite some recent progress by W. Siegel and collaborators, it remains as an open question as to what extent this theory was connected to the full string theory. The most mysterious point being certainly that the localization equations of the ambitwistor string also appear in the zero tension limit of string theory (alpha’ to infinity), which is the opposite limit than the supergravity one (alpha’ to zero).



In this talk, I’ll report on some work in progress with E. Casali (Math. Inst. Oxford) and argue that the ambitwistor string is actually a tensionless string. Using some forgotten results on the quantization of these objects, we explain that the quantization of tensionless strings is ambiguous, and can lead either to a higher spin theory, or to the ambitwistor string, hence solving the previously mentioned paradox. In passing, we see that the degenerations of the tensile worldsheet that lead to tensionless strings make connection with Galilean Conformal Algebras and the (3d) BMS algebra.

I will describe a sketch of the proof of Grothendieck conjecture on fundamental groups.
 

Heterotic vacua, defined with a holomorphic bundle and connection satisfying hermitian Yang-Mills, realise four-dimensional chiral gauge theories. We exploit the rich interplay between four-dimensional physics, supersymmetry and  geometry to construct a natural Kaehler metric for the moduli space, with a shockingly simple Kaehler potential. Along the way, we discover a natural geometric structure for the heterotic moduli.