I will report on an ongoing project in collaboration with Yannick Guedes Bonthonneau and Tobias Weich. The goal of this work is to define a natural spectrum associated with Anosov representations, consisting of complex hypersurfaces in the complexified dual Cartan subalgebra. The leading hypersurface corresponds to a well-known object in the literature — the so-called critical hypersurface of the representation. To some extent, this spectrum generalizes a similar notion in the rank-one case, known as the set of Pollicott-Ruelle resonances (and the leading resonance), which is known to encode the exponential decay of correlations, among other properties. I will describe the main consequences of this spectral approach, namely the meromorphic extension (to the full complexified dual Cartan subalgebra) of dynamical zeta functions and Poincaré series associated with the representation. If time permits, I will discuss specific values of these functions, the sharp quantitative decay of correlations for the refraction flow, and the perspectives for future work.
 

I will present several results on the topology of closed manifolds of dimension at least 4 that admit a (real) properly convex projective structure, all related to a vanishing theorem of Kobayashi from 1984 for the rational Pontryagin classes. In arbitrary dimensions, I will outline a classification of locally symmetric manifolds admitting properly convex projective structures. Focusing on dimension 4, I will then present a result on the geometric decomposition which is the analogue of a theorem of Benoist in dimension 3, and the construction of examples that realize all possible positive values of the Euler characteristic. Based on joint work with Stefano Riolo and Leone Slavich.
 

Running Seminar
In abstract Hodge theory, Deligne’s delta splitting measures how far a mixed Hodge structure is from being split as a real mixed Hodge structure. An allied notion, developed by S. Bloch, R.Hain et al., is that of a height for a special class of mixed Hodge structures called Biextensions.
The idea of a Biextension is closely related to algebraic cycles homologous to zero. Given two such cycles in complementary codimensions in an ambient smooth and projective variety, a certain cohomology group associated to the pair provides an example of a Biextension-type mixed Hodge structure.  The height associated with such a Biextension has been well studied and has been an active area of research for the past few decades.
 In an ongoing project, the speaker, along with J. I. Burgos Gil and G. Pearlstein, has developed a theory of mixed Hodge structures and heights associated with Bloch’s higher cycles, that generalizes the above study of Biextensions (doi.org/10.1112/plms.12443 and arXiv:2410.17167v2 [math.AG]).
 In the talk, I will explain the current state of the art of this project after reviewing the established theory.
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In joint work in progress with Hevesi, Thorne and Whitmore, we prove that automorphic Galois representations over CM fields are de Rham and have the expected Hodge–Tate weights. The novelty of our result is that we do not impose any assumptions on the residual mod p Galois representation. To achieve this, we prove a theorem on the cohomology of certain unitary Shimura varieties, which gives a bound on the maximal order of a torsion class in the lower half of degrees of cohomology. After placing this theorem in the right context and discussing its proof, I will sketch how it relates to the de Rham property of automorphic Galois representations.
 
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Running Seminar
A survey of results on regulators in Calabi-Yau families will be given.
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Séminaire Amplitudes et Gravitation sur l’Yvette (IHES/IPhT)
Observations of gravitational waves (GW) have the potential to unravel the mysteries of cosmic origins, constrain the equations of state of compact objects, and serve as a test of general relativity. To achieve these scientific goals, it is crucial to develop highly precise and accurate waveform templates to minimize systematic errors. My talk will focus on computing state-of-the-art effective two-body Hamiltonians and fluxes using techniques from quantum field theory and scattering amplitudes. These Hamiltonians and fluxes serve as fundamental building blocks for waveform templates, and dictate the accuracy of the waveform model. In particular, I will describe the effects of spinning and tidally deformed compact objects on evolution of the binary system. I will present computations of the linear-in-spin Hamiltonian at 4.5 post-Newtonian (PN) order, as well as the quadratic-in-spin Hamiltonian at 5PN order. Furthermore, I will describe oscillation modes of neutron stars (NS) induced by tidal interactions, particularly the fundamental-mode dynamical tides, which are expected to play a crucial role in constraining the NS equation of state in upcoming GW observing runs. I will discuss the computation of fluxes up to 2PN for adiabatic tidal interactions and effective Hamiltonians up to 3PN for both dynamic and adiabatic tidal interactions. The latter is particularly interesting as it requires introducing counterterms to remove divergences, leading to a renormalization group flow of the post-adiabatic Love number.
 
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Séminaire Amplitudes et Gravitation sur l’Yvette (IHES/IPhT)
The S-matrix bootstrap is a non-perturbative framework that exploits general physical principles such as unitarity, analyticity, and crossing symmetry to constrain scattering amplitudes and gain insights into strongly interacting dynamics where perturbation theory fails. I will briefly review the state of the art of the modern bootstrap and motivate the need for a new framework that incorporates multi-particle unitarity. I will then describe the Atkinson-Mandelstam method and explain how deep-learning techniques enable its practical realization. Finally, I will present some of the results obtained using this approach.
 
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Séminaire Amplitudes et Gravitation sur l’Yvette (IHES/IPhT)
We study scattering on the Coulomb branch of planar N=4 SYM.  At weak ‘t Hooft coupling scattering amplitudes can be computed using Feynman diagrams and the relevant degrees of freedom are quarks and gluons. At strong ‘t Hooft coupling the dynamics is captured by string scattering in flat space.  Dual conformal symmetry makes it possible to constrain the unitarity properties and the spectrum of the amplitude: bound states are encoded in the cusp anomalous dimension of a Maldacena-Wilson line with operators inserted at the cusp, which is fixed by integrability.  In this talk, we show how to combine dual conformal invariance, integrability, worldsheet bootstrap and the S-matrix bootstrap to explore scattering at finite ‘t Hooft coupling.  [Based on 2510.19909]
 
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Running Seminar
I will explain how to find Hodge points on the bases of Calabi-Yau families together with the respective Hodge cycles in their fibers by looking at their mod p reductions.
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Seed Seminar of Mathematics and Physics
Anti-de Sitter space acts as an infra-red cutoff for asymptotically free theories, allowing interpolation between a weakly-coupled small-sized regime and a strongly-coupled flat-space regime. I will discuss this interpolation in the context of Yang-Mills theories in AdS from the perspective of boundary conformal theories and its implications for the confinement/deconfinement transition. We find indications that at the transition a singlet scalar operator becomes marginal, destabilizing the deconfined phase existing at a small size and leading to a confined phase that smoothly connects to flat space.
 
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Seed Seminar of Mathematics and Physics
I will present a concrete construction of 2d Ising partition functions on lattice, with non-abelian (more generally non-invertible) global symmetry. The construction realizes the Ising model as the boundary theory of a 3d symmetry topological field theory (SymTFT) with specific boundary conditions. Suitable choices of boundary conditions correspond to going to different topological sectors of the global symmetry or gauging arbitrary non-anomalous subsymmetry. Electric-magnetic type duality of the SymTFT results in dual descriptions of the Ising model that generalizes the Kramers-Wannier duality of Z/2Z symmetric Ising models.
 
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Seed Seminar of Mathematics and Physics
The conformal block expansion of correlation functions is a fundamental tool in the bootstrap program. Block computation can be related to a specific class of harmonic functions on the conformal group. We generalise this concept to the case of thermal conformal field theories and compute one-point blocks for spinning representations in three dimensions. Specifically, we derive a universal Casimir equation and solve it using recursion relations. Finally, we use the blocks in several examples to explore some features of OPE coefficients.
 
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