Week 19.10.2026 – 25.10.2026

We will discuss how to bootstrap string theory in AdS, using N=4 SYM as a concrete example. The first lecture will review the basic ingredients of AdS/CFT: the definitions of AdS and CFT, the correspondence between Kaluza–Klein modes and single-trace operators, and the relation between string scattering in AdS and correlation functions of Kaluza–Klein modes in the CFT. The second lecture will focus on the stress-tensor multiplet four-point function, including its conformal block expansion, the constraints imposed by superconformal Ward identities, and its formulation in Mellin space. The third lecture will introduce the flat-space limit, explain how it acts on the conformal block expansion, and discuss the spectrum of single-trace operators that survives at strong coupling. In the final lecture, we will go beyond the flat-space limit and study the first AdS correction. We will introduce a worldsheet ansatz for this correction and show how matching it to the conformal block expansion fixes the correction completely.

Stochastic Exchange Models (SEMs) are continuous stochastic dynamics in which a certain conserved quantity is redistributed on a graph at the arrival times of a Poisson process on the edges. Examples in this class include the Averaging (AVG) process (a.k.a. repeated-averages dynamics), the Kipnis–Marchioro–Presutti (KMP) model, and their discrete counterparts. Recently, significant progress has been made in understanding the quantitative convergence of these models toward their equilibrium states, both in terms of their mixing times (e.g., the presence or absence of the cutoff phenomenon) and their spectral properties (e.g., the validity of an Aldous-like identity). In this talk, which will be based on a series of joint works with F. Sau, P. Caputo, and S. Kim, I will review some of these recent developments.

BPS decoupling limits of string and M-theory have led to profound insights into gravity and non-perturbative quantum theory, most notably with the AdS/CFT correspondence. A large class of such limits are related by U-duality to discrete lightcone quantization limits (DLCQs). However, there are natural BPS decoupling limits within the AdS/CFT correspondence that cannot be seen as DLCQ limits. Therefore, we introduce a new kind of BPS decoupling limit in which the relevant BPS states carry momentum along circle fibers geometrically twisted over a base space, unlike for a DLCQ limit. In the AdS/CFT correspondence these limits are associated with BPS states of the global symmetry algebra  PSU(2,2|4). In general, string fluctuations in a twisted BPS decoupling limit are described by non-relativistic sigma-models on emergent non-Lorentzian target spaces with a natural symplectic structure. This  further supports the broader connection between BPS limits and non-Lorentzian geometry. We introduce analogous limits of D-branes as well, probing the same symplectic structure as the string. For AdS/CFT, these results point to corners with tractable and explicit quantum-mechanical descriptions of non-perturbative regimes.

We develop a class of general diffusion-based stochastic particle optimisation methods for loss functions with intractable gradients. Specifically, we consider problems in which the loss gradient is an integral with respect to a parameter-dependent distribution, a structure that includes training generative models, fine-tuning, and learning latent-variable models. We introduce mean-field dynamics and its interacting-particle approximations, which contain several existing algorithms as special cases and provide a route for constructing new methods. Under well-posedness and joint contractivity assumptions, we prove exponential convergence and show that the continuous-time particle system admits a non-asymptotic error bound. We illustrate our framework by developing higher-order Langevin and momentum optimisers and evaluating them on maximum marginal-likelihood estimation and energy-based-model training.

Abstract: A Bianchi group is a group of the form PSL2(O_k), where O_k is the ring of integers of an imaginary quadratic number field k. In this talk, after revisiting the classical case of PSL2(Z) acting on the upper half-space, I will give an introduction to Bianchi groups and their action on the hyperbolic 3-space. I will then present a computational approach to defining a deformation retract of the hyperbolic 3-space which is invariant under this action. We will also see some 3D pictures of how this deformation retract looks like. Finally, I will outline how one can obtain a presentation of the Bianchi group from the deformation retract.