Seminars and short courses

Extremal black holes are special types of black holes which have exactly zero temperature. I will present a proof that extremal black holes form in finite time in gravitational collapse of charged matter. In particular, this construction provides a definitive disproof of the “third law” of black hole thermodynamics. We also show that extremal black holes take on a central role in gravitational collapse, giving rise to a new conjectural picture of “extremal critical collapse.” This is joint work with Ryan Unger (Princeton).

I will give an introduction to factorization homology using configuration spaces, and discuss the nonabelian Poincaré duality theorem of Segal, Salvatore, Lurie, and Ayala–Francis​, which relates factorization homology to mapping spaces. Time permitting, I will also talk about the Ayala–Francis axiomatic approach to factorization homology, which positions factorization homology as a "homology theory for manifolds."

Superconformal ‘type B’ quantum mechanical sigma models arise in a variety of interesting contexts, such as the description of D-brane bound states in an $AdS_2$ decoupling limit. Focusing on $N= 2B$ models, we study superconformal indices which count short multiplets and provide an alternative to the standard Witten index, as the latter suffers from infrared issues. We show that the basic index receives contributions from lowest Landau level states in an effective magnetic field and that, due to the noncompactness of the target space, it is typically divergent. Fortunately, the models of interest possess an additional target space isometry which allows for the definition of a well-behaved refined index. We compute this index using localization of the functional integral and find that the result agrees with a naive application of the Atiyah-Bott fixed point formula outside of it’s starting assumptions. In the simplest examples, this formula can also be directly verified by explicitly computing the short multiplet spectrum.

In this talk I will explain a general mechanism, based on derived symplectic geometry, to glue the local invariants of singularities that appear naturally in Donaldson-Thomas theory. Our mechanism recovers the categorified vanishing cycles sheaves constructed by Brav-Bussi-Dupont-Joyce, and provides a new more evolved gluing of Orlov’s categories of matrix factorisations, answering a conjecture of Kontsevich-Soibelman and Y.Toda. This is a joint work with B. Hennion (Orsay) and J. Holstein (Hamburg). The talk will be accessible to a general audience.

This will be a survey talk about recent progress on norm and pointwise convergence problems for classical and multiple ergodic averages along polynomial orbits. A celebrated theorem of Szemeredi asserts that every subset of integers with nonvanishing upper Banach density contains arbitrarily long arithmetic progressions. We will discuss the significance of using ergodic theory and Fourier analysis in solving this problem. We will also explain how this problem led to the conjecture of Furstenberg-Bergelson-Leibman, which is a major open problem in pointwise ergodic theory. Relations with number theory and additive combinatorics will be also discussed.

I will describe a map that associates to every deformation of an object in a higher category a collection of generalized symmetries of the object. Building on work by Lurie, we will see that the failure of this map to be an equivalence can be quantified. Under favorable circumstances, the map is an equivalence, and this leads to an explicit description of the space of deformations in terms of solutions to certain equations. I will discuss applications of these results to topological field theory and holomorphic symplectic geometry. This talk is based on joint work with Bhanu Kiran.

Mirror symmetry is a somewhat mysterious phenomenon that relates the geometry of two distinct Calabi-Yau manifolds. In the realm of trying to understand this relationship an equation for a connection on a line bundle in a Kahler manifold appeared. This is commonly called the deformed Hermitian-Yang-Mills equation and I will explain what it is and some current joint work with Benoit Charbonneau and Rosa Sena-Dias which explicitly solves this equation on a specific setting. This helps in understanding the problem of the existence of solutions and explore (or rule out) possible stability conditions.

The topological complexity and its higher versions are homotopy invariants which were introduced by Farber and Rudyak in order to give a topological measure of the complexity of the motion planning problem. We will discuss some properties of these invariants for closed manifolds with abelian fundamental group. In particular, we will give sufficient conditions for the (higher) topological complexity of such a manifold to be non-maximal. This is based on joint works with N. Cadavid, D. Cohen, J. González and S. Hughes.

We introduce the classical results of de Rham and Berger on the holonomy of a Riemannian manifold. We compare these to the situation of parallel skew-torsion, where we obtain Riemannian submersions from reducible holonomy. If time permits I will give an introduction to 3-(α, δ)-Sasaki manifolds and their submersion onto quaternionic Kahler manifolds.

It has been recently suggested that the strong Emergence Proposal is realized in equi-dimensional M-theory limits by integrating out all light towers of states with a typical mass scale not larger than the species scale, i.e the eleventh dimensional Planck mass. Within the BPS sector, these are transverse M2- and M5-branes, that can be wrapped and particle-like, carrying Kaluza-Klein momentum along the compact directions. We provide additional evidence for this picture by revisiting and investigating further the computation of $R^4$-interactions in M-theory à la Green-Gutperle-Vanhove. A central aspect is a novel UV-regularization of Schwinger-like integrals, whose actual meaning and power we clarify by first applying it to string perturbation theory. We consider then toroidal compactifications of M-theory and provide evidence that integrating out all light towers of states via Schwinger-like integrals thus regularized yields the complete result for $R^4$-interactions. In particular, this includes terms that are tree-level, one-loop and space-time instanton corrections from the weakly coupled point of view. Finally, we comment on the conceptual difference of our approach to earlier closely related work by Kiritsis-Pioline and Obers-Pioline, and conjecture a correspondence between two types of constrained Eisenstein series.

The classical Kantorovich-Rubinstein duality theorem established a significant connection between Monge optimal transport and the maximization of a linear form on 1-Lipschitz functions. This result has been widely used in various research areas, particularly to demonstrate a bridge between Monge transport theory and some class of optimal design problems in mechanics.

The aim of this talk is to present a similar theory when the linear form is maximized over all real $C^{1,1}$ functions with a Hessian matrix spectral norm not exceeding one. It turns out that this new maximization problem can be viewed as the dual of a specific optimal transport problem. The task is to find a minimal three-point plan with given first two marginals, where the third is assigned to be larger than both in the sense of convex order. The existence of optimal plans allows to express solutions of the underlying Beckman problem as a combination of rank-one tensor measures supported by a graph. In the context of two-dimensional mechanics, this graph encodes the optimal location of a grillage to support a given bending load.