L4

We investigate Néron models of Jacobians of singular curves over strictly Henselian discretely valued fields, and their behaviour under tame base change. For a semiabelian variety, this behaviour is governed by a finite sequence of (a priori) real numbers between 0 and 1, called "jumps". The jumps are conjectured to be rational, which is known in some cases. The purpose of this paper is to prove this conjecture in the case where the semiabelian variety is the Jacobian of a geometrically integral curve with a push-out singularity. Along the way, we prove the conjecture for algebraic tori which are induced along finite separable extensions, and generalize Raynaud's description of the identity component of the Néron model of the Jacobian of a smooth curve (in terms of the Picard functor of a proper, flat, and regular model) to our situation. The main technical result of this paper is that the exact sequence which decomposes the Jacobian of one of our singular curves into its toric and Abelian parts extends to an exact sequence of Néron models. Previously, only split semiabelian varieties were known to have this property.

Compactifications of 6D Superconformal Field Theories (SCFTs) on four-manidolfds lead to novel interacting 2D SCFTs. I will describe the various Lagrangian and non-Lagrangian sectors of the resulting 2D theories, as well as their interactions. In general this construction can be embedded in compactifications of the physical superstring, providing a general template for realizing 2D conformal field theories coupled to worldsheet gravity, i.e. a UV completion for non-critical string theories.  

I will introduce a McKean—Vlasov problem arising from a simple mean-field model of interacting neurons. The equation is nonlinear and captures the positive feedback effect of neurons spiking. This leads to a phase transition in the regularity of the solution: if the interaction is too strong, then the system exhibits blow-up. We will cover the mathematical challenges in defining, constructing and proving uniqueness of solutions, as well as explaining the connection to PDEs, integral equations and mathematical finance.

Superradiance in black hole spacetimes is a phenomenon by which a field of spin 0 or 1 can extract energy from the background. Typically, one can imagine sending a wave packet with a given energy towards a black hole and receiving in return a superposition of wave packets carrying a total amount of energy that is larger than the energy sent in. It can be caused by rotation or by interaction between the charges of the black hole and the field. In the first case, the region where superradiance takes place (the ergoregion) has a clear geometrical localization depending only on the physical parameters of the black hole. For charge induced superradiance, this is not the case and we have a generalized ergoregion depending also on the physical properties of the field (mass, charge, angular momentum). In the most severe cases, the generalized ergoregion may cover the whole exterior of the black hole. We focus on charge-induced superradiance for spin 0 fields in spherically symmetric situations. Alain Bachelot wrote a thorough theoretical study of this question in 2004, which, to my knowledge, is the only work of its kind. When I was in Bordeaux, he and I discussed the possibility of investigating superradiance numerically. Over the years it became an actual research project, involving Laurent Di Menza and more recently Mathieu Pellen, of which this talk is an account. The idea was to observe numerically some superradiant behaviours and gain a more precise understanding of the phenomenon. We shall show an exact analogue of the Penrose process with the superradiance of wave packets and a slightly different behaviour for fields "emerging" inside the ergoregion. We shall also explore the related question of black hole bombs and present some recent observations. 

In this talk I will present recent results obtained within the
framework of perturbative algebraic quantum field theory. This novel
approach to mathematical foundations of quantum field theory allows to
combine the axiomatic framework of algebraic QFT by Haag and Kastler with
perturbative methods. Recently also non-perturbative results have been
obtained within this approach. I will report on these results and present
new perspectives that they open for better understanding of foundations of
QFT.

Using tropical geometry and new methods in the theory of Fukaya categories, we explain a mirror symmetry equivalence relating the Fukaya category of a hypersurface and the category of coherent sheaves on the boundary of a toric variety.

We construct maximal surfaces with Neumann boundary conditions in Minkowski space using mean curvature flow. In particular we find curvature conditions on a boundary manifold so that mean curvature flow may be shown to exist for all time, and give conditions under which the maximal hypersurfaces are stable under the flow.

Hilbert's Nullstellensatz asserts the existence of a complex point satisfying lying on a given variety, provided there is no (ideal-theoretic) proof to the contrary.
I will describe an analogue for curves (of unbounded degree), with respect to conditions specifying that they lie on a given smooth variety, and have homology class
near a specified ray.   In particular, an analogue of the Lefschetz principle (relating large positive characteristic to characteristic zero) becomes available for such questions.
The proof is very close to a theorem of  Boucksom-Demailly-Pau-Peternell on moveable curves, but requires a certain sharpening.   This is part of a joint project with Itai Ben Yaacov, investigating the logic of the product formula; the algebro-geometric statement is needed for proving the existential closure of $\Cc(t)^{alg}$ in this language.  
 

Hilbert's Nullstellensatz asserts the existence of a complex point satisfying lying on a given variety, provided there is no (ideal-theoretic) proof to the contrary.
I will describe an analogue for curves (of unbounded degree), with respect to conditions specifying that they lie on a given smooth variety, and have homology class
near a specified ray.   In particular, an analogue of the Lefschetz principle (relating large positive characteristic to characteristic zero) becomes available for such questions.
The proof is very close to a theorem of  Boucksom-Demailly-Pau-Peternell on moveable curves, but requires a certain sharpening.   This is part of a joint project with Itai Ben Yaacov, investigating the logic of the product formula; the algebro-geometric statement is needed for proving the existential closure of $\Cc(t)^{alg}$ in this language. 

Shortly after Mason & Skinner introduced the so-called ambitwistor strings, Berkovits came up with a pure-spinor analogue of the theory, which was later shown to provide the supersymmetric version of the Cachazo-He-Yuan amplitudes. In the heterotic version, however, both models give somewhat unsatisfactory descriptions of the supergravity sector.

In this talk, I will show how the original pure-spinor version of the heterotic ambitwistor string can be modified in a consistent manner that renders the supergravity sector treatable. In addition to the massless states, the spectrum of the new model --- which we call sectorized heterotic string --- contains a single massive level. In the limit in which a dimensionful parameter is taken to infinity, these massive states become the unexpected massless states (e.g. a 3-form potential) first encountered by Mason & Skinner."