Motivated by a family of examples constructed by R. Schwartz based on Pappus’s theorem, we explore constructions of representations of discrete groups arising from incidence theorems in projective spaces. Schwartz recently proved that his construction parametrizes the boundary of a connected component of Anosov representations in a certain moduli space of representations of PSL(2,Z) into Isom(SL(3,R)/SO(3)). We use a generalized version of Pappus’s theorem to define a higher dimensional analogue of this family, and we prove that in RP3 these representations are « extended geometrically finite », a notion of relative Anosov representation introduced by T. Weisman. This is joint work in progress with W. Dimoune.
 

Given an element of a finite group G, one may ask for its presentation as a minimal product of generators from some predetermined subset S. This problem also has a graph-theoretic formulation: given a vertex of the Cayley graph associated with G and S, find the shortest path back to the identity. We present a machine learning approach to answering this version of the question, develop it as a method for helping resolve further problems in combinatorial group theory, and present several conjectures posed with its aid.
 
 
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We present a new approach to one of the central problems in AI: the construction of embeddings (latent representations) inspired by ideas from holographic string dualities, in particular the AdS/CFT correspondence. Our central conjecture is that, for a broad class of graphs, there exists a dual polygon together with a map (« holography ») that sends graph vertices to lattice paths inside the polygon in such a way that graph distances are well approximated by the areas enclosed between the corresponding lattice paths. This can be viewed as a combinatorial analogue of the « Complexity = Volume » principle proposed by L. Susskind and collaborators in 2014 in the context of AdS/CFT, and later explored for Cayley graphs by H. Lin, D. Melnikov, and others. From the perspective of AI, this suggests that effective embeddings of graph vertices may be realized as their holographic images. Understanding such holographic representations for natural language and other AI systems could provide a new mathematical framework for constructing more powerful latent representations and, ultimately, more effective AI architectures. As mathematical applications of the conjecture, we explain how it naturally leads to variants of the famous Babai conjecture predicting an n2 upper bound on the diameters of Sn Cayley graphs and conjecture of quasi-polynomiality of the diameters and word metrics. We also discuss connections with limit shapes of Young diagrams, integrable systems, and the description of conformal dimensions in conformal field theories arising as large-size limits of graph families. The talk is based on the CayleyPy series of papers, in particular CayleyPy-4: AI-Holography. Towards analogs of holographic string dualities for AI tasks (https://arxiv.org/abs/2603.22195).
The talk will be accessible to mathematicians. No prior background in artificial intelligence or string theory will be assumed.
 
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I will present some new results for obtaining rigorous lower bounds on the spectral gap of frustration-free quantum spin chains. Such spin chains are characterized by the fact that their ground state minimizes each local Hamiltonian term individually, and include paradigmatic models such as the AKLT and the Majumdar-Ghosh model. In my talk, I will discuss two new results, which significantly improve upon existing methods: First, a new approach based on semidefinite programming to obtain rigorous lower bounds on the spectral gap of frustration-free Hamiltonians. Second, a method based on the so-called « martingale method » of Fannes, Nachtergaele, and Werner, which not only significantly improves upon their method, but also results in a much simplified proof of the fact that a large class of frustration-free Hamiltonians (namely, parent Hamiltonians of well-conditioned Matrix Product States) are always gapped.
 
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Tensor networks offer a sign-problem-free approach to study lattice gauge theories, but extracting precise universal information associated with the deconfinement transition remains challenging. In this work, we study the deconfinement transition of (2+1)-dimensional ${mathbb Z}_N$ lattice gauge theories at finite temperature using a thermal tensor network approach, where the partition functions at finite temperature are formulated as three-dimensional tensor networks. These tensor networks are first contracted in the temporal direction, and the subsequent coarse-graining in the spatial directions yields a renormalized transfer matrix, the spectrum of which directly encodes the universal conformal field theory data. In particular, by numerically extracting the central charge and scaling dimensions, we verify that the universality class of the thermal deconfinement transition matches the prediction of the Svetitsky-Yaffe conjecture for N = 2, 3, 5. Moreover, we show that the ${mathbb Z}_5$ theory at finite temperature exhibits an intermediate phase with an emergent U(1) symmetry. Critical couplings are determined via Gu-Wen ratios and agree with existing Monte Carlo simulations. Finally, extrapolating these critical couplings at finite temperature enables us to determine the deconfinement transition points for N = 2, 3 at zero temperature.
 
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Séminaire de géométrie arithmétique
The logarithm matrix appears in the arithmetic of elliptic curves, and encodes the growth of arithmetic invariants in towers of number fields. Its definition is a bit involved. One goal of the talk is to present a simpler definition, in terms of 2-adic digits. We also present an answer to a question posed by Bruno Kahn on June 22, 2026, which makes the definition of the logarithm matrix even simpler.
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Due to the weather conditions, the seminar will take place on Zoom :
https://us02web.zoom.us/j/88107752144?pwd=73SFjfjHkQXLNPWnbJCWPtrhJkQWia.1
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We use the Fuzzy Sphere technology to extract new 3d Ising CFT data and examine its properties in light of expectations from ergodicity.  In our analysis, a microscopic construction of the special conformal generators plays an important role.  We also study 3d Ising Field Theory, namely, the Ising CFT deformed by a relevant operator.   Here we use both the  Fuzzy Sphere directly, as well as Hamiltonian truncation employing the new Ising CFT data.
 
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We consider « fool’s crowns » — parts of Riemann surfaces with holes with n marked points/bordered cusps on a hole boundary decorated by horocycles. We define volumes of the corresponding moduli spaces by postulating a decoration-independent action, which in the limit of n tending to infinity transforms into a Schwarzian action on boundary « trumpets ». We derive the corresponding volumes for arbitrary n and coupling constant kappa and describe the statistical model obtained in terms of Brownian bridges. This is a joint work with Timothy Budd (Nijmegen Univ.)
 
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Séminaire Amplitudes et Gravitation sur l’Yvette (IHES/IPhT)
Infrared divergences obscure key analytic properties of scattering amplitudes, exposing gaps in our understanding of unitarity, causality, and crossing symmetry in theories with long-range forces. In this talk, I will use a simple model to illustrate novel analytic features of long-range theories, including modifications to the connectedness structure of amplitudes and to the general optical theorem. Since the LSZ reduction formula does not apply to theories with long-range forces, I will also present a modified version of LSZ reduction for this model, which accounts for long-range interactions and yields IR-finite amplitudes without ambiguous scales or ill-defined integrals.
 
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Séminaire d’Analyse
The bispectral problem concerns the construction and the classification of operators possessing a symmetry between the space and spectral variables. Different versions of this problem have interesting connections to several areas of mathematics such as integrable systems, algebraic geometry, representation theory, classical orthogonal polynomials, etc. I will review the problem and some of these connections, and then discuss recent results related to quantum integrable systems and hypergeometric functions associated with several classical multivariate distributions.
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Séminaire Amplitudes et Gravitation sur l’Yvette (IHES/IPhT)
I will present the detailed construction of an extension of the Bondi framework which includes a non-vanishing twist, i.e. the relaxation of the hyperspace-orthogonality condition. This extension has several advantages which have been noted in recent work. First, it allows to access a Carroll-covariant structure on Scri. Second, and most importantly, the presence of a non-vanishing twist allows to write all the algebraically special solutions in finite form. This includes for example Kerr or supertranslated Schwarzschild, which would otherwise require an infinite 1/r expansion when written in the standard non-twisting Bondi gauge. This opens the possibility of defining symplectic symmetries for these algebraically special solutions. The construction of the algebraically general solutions near null infinity requires the use of a combination of the Newman-Penrose and metric formalisms, which will be presented in detail.
 
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Séminaire de géométrie arithmétique
Let k be a perfect field of characteristic p > 0. The Hodge-Witt sheaf is a sheaf with transfer on the category of smooth k-varieties. However, since it does not satisfy the cohomological A1-invariance, the Hodge-Witt cohomology is not representable in Voevodsky’s category of motives. One way to overcome this drawback is to consider the category of motives with modulus defined by Kahn-Miyazaki-Saito-Yamazaki. We define the Hodge-Witt sheaf for modulus pairs over k satisfying the properties called the cohomological cube invariance and the cohomological blow-up invariance. These imply that the Hodge-Witt cohomology is representable in the category of motives with modulus under resolution of singularities. Also, we try to discuss properties of our Hodge-Witt sheaves for modulus pairs and possible relationship with another construction by Ren-Rulling. This is partly joint work in progress with Veronika Ertl. 
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