L4

The 2d (massless) Sinh-Gordon model is amongst the simplest 2d quantum field theories that are expected to be integrable (= infinitely many symmetries), but without conformal symmetry. In this talk I will explain a rigorous construction of this model and its vertex correlations (= Laplace transforms) on the infinite cylinder using probability theory. A fundamental role is played by the Sinh-Gordon Hamiltonian and I will explain how the theory of Gaussian multiplicative chaos can be used to analyze this linear map. This talk will be based on joint work with Colin Guillarmou and Vincent Vargas.

We derive a lower bound, independent of the initial condition, for the solution of the KPZ equation on the torus, using its representation as the value function of a stochastic control problem.

With the same techniques we also prove a bound for its oscillation, again independent of initial conditions, which is related to Harnack's inequality for the (rough) heat equation.

In this talk, we will explore the emerging role of generative AI in mathematical research. Building on insights from the “Malliavin–Stein experiment”, carried out in collaboration with Charles-Philippe Diez and Luis Da Maia, we will discuss our experience and reflect on how AI might influence the way mathematics is conceived, proven, and created.

Given a finite poset $P$ we ask how small a family of subsets of $[n]$ can be such that it does not contain an induced copy of the poset, but adding any other subset creates such a copy. This number is called the saturation number of $P$, denoted by $\operatorname{sat}^*(n,P)$. Despite the apparent similarity to the saturation for graphs, this notion is vastly different. For example, it has been shown that the saturation numbers exhibit a dichotomy: for any poset, the saturation number is either bounded, or at least $2 n^{1/2}$. In fact, it is believed that the saturation number is always bounded or exactly linear. In this talk we will be discussing the most recent advances in this field, with the focus on the diamond poset, whose saturation number was unknown until recently.

Joint with Sean Jaffe.

We adapt Joyce's theory of wall-crossing for enumerative invariants of $\mathbb C$-linear additive categories to Pandharipande-Thomas stable pairs on smooth projective Fano 3-folds of "type C or D", and investigate implications for Pandharipande-Thomas generating functions with descendent insertions.

By analyzing the wall-crossing behavior from a stability condition where pairs are unstable to the standard stability condition for PT stable pairs, we derive an explicit formula expressing the PT stable pair invariants $[P_n(X,\beta)]^{virt}$ in terms of sheaf-theoretic invariants $[\mathcal M^{ss}_{(0,0,\beta_i, n_i - \beta_i.c_1/2)}(\tau_-)]_{\rm inv}$ for the moduli space of Gieseker semistable coherent sheaves on $X$ with Chern character $(0,0,\beta_i, n_i - \beta_i.c_1/2)$.

These enumerative invariants are defined as elements in the Lie algebra on the rational Betti homology of the piecewise-linear rigidified higher moduli stack of objects in the bounded derived category of X. Under tensoring by a line bundle, we exhibit a control over the periodicity of sheaf-theoretic invariants with respect to the Euler characteristic $n_i$, which we use to show that the sheaf-theoretic invariants form a quasi-polynomial in $n_i$ of degree $2$ with period given by the divisibility of $\beta_i$ in the lattice $H_2(X,\mathbb Z)/\text{torsion}$.

We use this periodicity in the sheaf-theoretic invariants to show that the descendent generating series for Pandharipande-Thomas stable pairs is the Laurent expansion of a rational function over $\mathbb Q$ in this setting, thus confirming a conjecture due to Pandharipande-Thomas from 2007. Furthermore, we construct a counterexample to a conjecture due to Pandharipande from 2017 on the location of the poles of the descendent generating series, and give a direct proof of a slightly modified conjecture on the location of these poles using wall-crossing techniques.

 

The ’t Hooft expansion is a powerful organizational framework for understanding QFTs as perturbations away from the large N limit and has deep connections to string theory and holography. In this talk, I will discuss categorical aspects of the ’t Hooft expansion, i.e. what one learns about topological defects from the ’t Hooft expansion and, correspondingly, topological strings and twisted holography. This talk is based off the paper arXiv:2411.00760 from last year as well as the more recent review paper arXiv:2511.19776.

I will discuss some of my work on thermal correlators in AdS/CFT. In particular, given a thermal correlator, how are the characteristic properties of bulk black holes encoded in such correlators? This includes exploring the spectrum of QNMs, the so-called thermal product formula, the photon ring, and geodesics bouncing off the black hole singularity. I will discuss how the latter might change when finite string effects are considered.