It is well known that expected signatures can be used as the “moments” of the law of stochastic processes. Inspired by this fact, we introduced higher rank expected signatures to capture the essences of the weak topologies of adapted processes, and characterize the information evolution pattern associated with stochastic processes. This approach provides an alternative perspective on a recent important work by Backhoff–Veraguas, Bartl, Beiglbock and Eder regarding adapted topologies and causal Wasserstein metrics.

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The dimer model, a classical model of statistical mechanics, is the uniform distribution on perfect matchings of a graph. In two dimensions, one can define an associated height function which turns the model into a random surface (with specified boundary conditions). In the 1960s, Kasteleyn and Temperley/Fisher found an exact "solution" to the model, computing the correlations in terms of a matrix called the Kasteleyn matrix. This exact solvability was the starting point for the breakthrough work of Kenyon (2000) who proved that the centred height function converges to the Dirichlet (or zero boundary conditions) Gaussian free field. This was the first proof of conformal invariance in statistical mechanics.

In this talk, I will focus on a natural modification of the model where one allows the vertices on the boundary of the graph to remain unmatched: this is the so-called monomer-dimer model, or dimer model with free boundary conditions. The main result that we obtain is that the scaling limit of the height function of the monomer-dimer model in the upper half-plane is the Neumann (or free boundary conditions) Gaussian free field. Key to this result is a somewhat miraculous random walk representation for the inverse Kasteleyn matrix, which I hope to discuss.

Joint work with Marcin Lis (Vienna) and Wei Qian (Paris).

It is a basic fact of convexity that the volume of convex bodies is a polynomial, whose coefficients contain many familiar geometric parameters as special cases. A fundamental result of convex geometry, the Alexandrov-Fenchel inequality, states that these coefficients are log-concave. This proves to have striking connections with other areas of mathematics: for example, the appearance of log-concave sequences in many combinatorial problems may be understood as a consequence of the Alexandrov-Fenchel inequality and its algebraic analogues.

There is a long-standing problem surrounding the Alexandrov-Fenchel inequality that has remained open since the original works of Minkowski (1903) and Alexandrov (1937): in what cases is equality attained? In convexity, this question corresponds to the solution of certain unusual isoperimetric problems, whose extremal bodies turn out to be numerous and strikingly bizarre. In combinatorics, an answer to this question would provide nontrivial information on the type of log-concave sequences that can arise in combinatorial applications. In recent work with Y. Shenfeld, we succeeded to settle the equality cases completely in the setting of convex polytopes. I will aim to describe this result, and to illustrate its potential combinatorial implications through a question of Stanley on the combinatorics of partially ordered sets.

We investigate whether Swampland constraints on the low-energy dynamics of weakly coupled string vacua in AdS can be related to inconsistencies of their putative holographic duals or, more generally, recast in terms of CFT data. In the main part of the talk, we shall illustrate how various swampland consistency constraints are equivalent to a negativity condition on the sign of certain mixed anomalous dimensions. This condition is similar to established CFT positivity bounds arising from causality and unitarity, but not known to hold in general. Our analysis will include LVS, KKLT, perturbative and racetrack stabilisation, and we shall also point out an intriguing connection to the Distance Conjecture. In the final part we will take a complementary approach, and show how a recent, more rigorous CFT inequality maps to non-trivial constraints on AdS, mentioning possible applications along the way.

We study several matrix diffusion processes constructed from a unitary Brownian motion. In particular, we use the Stiefel fibration to lift the Brownian motion of the complex Grass- mannian to the complex Stiefel manifold and deduce a skew-product decomposition of the Stiefel Brownian motion. As an application, we prove asymptotic laws for the determinants of the block entries of the unitary Brownian motion.