ICTS

What happens when smoothness—the cornerstone of classical differential geometry and backbone of the formulation of the Einstein equation—breaks down? This question drives major challenges in general relativity, where singularities, horizons, and low-regularity phenomena expose the limits of traditional approaches. In this talk, I present an overview of recent synthetic frameworks for Riemannian and Lorentzian manifolds that push geometry beyond the smooth category, integrating ideas from metric geometry, causality theory, and generalized curvature, and discuss their advantages and limitations in the context of general relativity.

I will overview some places where combinatorics can help us understand quantum field theory that I've been involved with

Quantum Hall effect is a two-dimensional experimental phenomenon whose understanding has revealed deep connections to mathematics, for example, topology and noncommutative geometry. I will present its extension to arbitrary higher dimensions and the construction of the topological effective actions underlying its dynamics. I will also discuss the calculation of the entanglement entropy for such systems.

According to a famous remark by Schrödinger, entanglement is the defining feature of quantum theory. While entanglement is a valuable and fragile resource in quantum systems with finitely many degrees of freedom, it is instead ubiquitous in quantum field theory, appearing already in the vacuum state of the simplest free field theories. In this talk, I will present a simple and operationally motivated framework for characterizing the spatial distribution of entanglement in quantum field theory, inspired by the limitations of realistic measurements. The approach combines Gaussian quantum information, finite sets of smeared observables, and the notion of partner systems into a geometric framework. I will then apply these methods to two physically interesting situations. First, I will study the production and distribution of entanglement during cosmic inflation. I will show that, although inflation entangles localized modes with the rest of the field, this entanglement is highly delocalized and effectively acts as noise for any pair of observables localized within the observable universe. This observation mitigates the need to invoke ad hoc quantum-to-classical mechanisms to explain the apparent classicality of cosmological perturbations. Finally, I will discuss how the same framework can be used to design experimentally accessible probes of vacuum entanglement in analogue gravity systems. I will argue that, thanks to low effective temperatures and recent advances in quantum state tomography, detecting entanglement between suitably constructed observables is within the capability of current experiments, even in the presence of realistic experimental errors.

The field equations governing the near-horizon limits of extremal black holes reduce to a quasi-Einstein condition on the horizon cross-section, relating the Ricci tensor, a one-form, and a parameter m. This geometric viewpoint leads to a unified picture including Einstein metrics and Ricci solitons. I will discuss recent mathematical results on the rigidity and structure of such quasi-Einstein manifolds, including classification theorems for static and divergence-free cases, and their implications for the uniqueness of near-horizon geometries.

We will report on work in progress studying the existence of Einstein metrics on families of (total spaces of) homogeneous torus bundles. This is based on joint work with Masoumeh Zarei.

In this talk, we discuss an graphical algorithm to find the largest positroid contained in a rank 2 matroid. Positroids appear in calculations for SYM N=4 theory, and identifying and calculating the boundaries between them have implications about amplitudes of that theory.

With O. Kravchenko and L. Ryvkin, we recently showed that multilocal observables on covariant fields with values in a vector bundle E are efficiently represented by distributional sections of a Poisson algebra bundle constructed on the dual jet bundle of E, over the space of unordered configurations of points in spacetime [https://arxiv.org/abs/2407.15287]. This model requires the usual fiberwise tensor product and also a symmetrized version of the external tensor product of bundles. It gives a very rich algebraic structure supporting a quadratic double Poisson bracket and a related Yang Baxter equation [Van den Berg 2004, Odesski, Roubtsov, Sokolov 2012]. It also provides a natural extension of multisymplectic geometry and a finite-dimensional framework for a quantization leading to QFT, suitable to describe the higher structures coming from symmetries and from on-shell quotients. In this talk I present Poisson algebra bundles and the main steps of the deformation quantization by Laplace pairing [Brouder, Fauser, Frabetti, Oekl 2004, Borcherds 2011, Herscovich 2017].

Kinematical Lie algebras encode the symmetry structures that govern spacetime kinematics and the evolution of physical systems. Assuming spatial isotropy, translational and rotational invariance, and invariance under inertial transformations, one can systematically classify all possible kinematical algebras relevant to physics. This classification includes, for example, the Poincaré algebra describing flat relativistic spacetimes. These algebras can be promoted to Lie groups and associated with homogeneous spacetimes, including a wide class of so-called non-Lorentzian geometries. Motivated by recent developments in non-Lorentzian physics, this talk revisits the light-cone formulation of quantum field theory and its connections to kinematical Lie algebras of the Bargmann, Galilei, and Carroll types. We further explore applications of these algebras to different areas of physics, such as the asymptotic structure of spacetime, collider physics, and certain condensed-matter systems.

We discuss various axiomatic approaches to quantum field theories. We introduce the definition of Functorial Field Theories and discuss how various mathematical disciplines including (but not limited to) category theory, topology, (higher) algebra and functional analysis come together in this area of study. As an example, we will focus on 2d chiral conformal field theories.