Monday, 29 December 2025
In the spirit of the meeting, I will present some open problems that arise within the Causal Set approach to the problem of quantum gravity, with a view to stimulating discussion and collaboration. I will give a very brief overview of the foundations of Causal Set Theory that includes an emphasis on the fact that the physical geometry of spacetime — as we know it — is Lorentzian and not Riemannian. This has far reaching consequences for the mathematics and physics of discrete spacetime. I will describe a number of examples of concrete problems.
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.
The transport properties of an extended system driven by active reservoirs is an issue of paramount importance, which remains virtually unexplored. We address this issue in the context of energy transport between two active reservoirs connected by a chain of harmonic oscillators. An active reservoir is modeled by a harmonic chain of overdamped run-and-tumble particles, the tumbling time-scale being a measure of the activity of the reservoir. These reservoirs satisfy a modified fluctuation-dissipation relation, which we illustrate by exactly computing the effective noise and dissipation kernels. We study the energy transport through a chain of harmonic oscillators driven by two active reservoirs of different activities and show that the stationary energy current shows remarkable features like negative differential conductivity and a non-trivial direction reversal. We also find nontrivial spatiotemporal velocity correlations in the stationary state, which distinguish this activity-driven nonequilibrium state from the usual thermally driven systems.
In my talk, I shall discuss the phenomenon of diffusion which I will describe using the Langevin and diffusion equations. The latter one belongs to the class of Fokker-Planck type equations. Then I will smoothly pass to the anomalous diffusion which appears when we study phenomena taking place in memory-dependent and/or heterogeneous media. In the rest of my talk, I will focus my presentation on discussing the so-called time smeared diffusion equations, i.e., the diffusion equations with a generalised fractional derivatives. To preserve physical applicability of solutions to such equations the integral kernels defining fractional derivatives cannot be arbitrary; they must be Sonin functions. I will show how to solve the time smeared diffusion equation using the Efros theorem, i.e., I will construct the appropriate integral decomposition, which, after certain assumptions, leads to the subordination procedure. Finally, I shall conclude my talk presenting an operator interpretation of the subordination.
Tuesday, 30 December 2025
I will overview some places where combinatorics can help us understand quantum field theory that I've been involved with
In this talk, I will first review the construction of invariants of three-manifolds from Chern–Simons field theory. I will then explain the equivalence between the SU(2) Chern–Simons invariants of closed three-manifolds and Lickorish’s invariant, which is defined using the bracket polynomial of cable links.
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.
Wednesday, 31 December 2025
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.
For a number of critical lattice models in 2D statistical physics, it has been proven that scaling limits of interfaces (with suitable boundary conditions) are described by random conformally invariant curves, called Schramm-Loewner evolutions (SLE). So-called partition functions of these SLEs (which also encode macroscopic crossing probabilities) can be regarded as specific correlation functions in the conformal field theory (CFT) associated to the lattice model in question. Although it is not clear how to define the latter mathematically, one can still make sense of many of the properties predicted for these CFTs. I give an overview of how one can rigorously connect 2D statistical physics and CFT in this way.
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.
We define Berezin-type and Fedosov-type deformation quantizations on arbitrary smooth manifolds by embedding them in CP^n or C^n and inducing the quantizations from the ambient space. If time permits we will mention some applications.
Causal set theory is an approach to quantum gravity in which spacetime is fundamentally discrete at the Planck scale and takes the form of a irregular Lorentzian lattice, or "causal set", from which continuum spacetime emerges in a large-scale (low-energy) approximation. Within this setting, we develop a quantum field theory formalism and derive a manifestly causal diagrammatic expansion for in-in correlators in local scalar field theories with finite polynomial interactions. The resulting expansion terminates at finite order in the interaction coupling, providing insight into how the underlying discreteness scale plays the role of an effective cut-off. In particular, we illustrate how this discreteness length can regularize expressions that diverge in the continuum limit.
Thursday, 01 January 2026
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.
I will introduce random tensor models by first reviewing their motivation coming from random geometric approach to quantum gravity. Then, I will selectively present some of the interesting research results, by highlighting recent results on enumeration of graphs representing tensor invariants, and reporting our recent work on a new notion of characteristic polynomials for tensors via Grassmann integrals and distributions of roots of random tensors. The latter two are based on arXiv:2404.16404[hep-th] and arXiv:2510.04068[math-ph]
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.
Friday, 02 January 2026
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.
Several classical relativity theorems, including the famous singularity theorems, have in their assumptions pointwise energy conditions. Those conditions bound the energy density (or similar quantities) on every spacetime point and are easily violated by quantum fields. One way to examine the applicability of those theorems in semiclassical gravity is to replace them with an averaged version, where the energy density is bounded on a segment of a causal geodesic. The index form method, used instead of the Raychaudhuri equation, provides a direct way of using those weakened conditions. In this talk I will explain how this method applies to several classical relativity theorems, the progress that has been made and the challenges ahead.