Amplitudes on the Coulomb branch of N=4 Super-Yang Mills (SYM) provide a rich playground to understand scattering amplitudes involving massive particles. We will discuss a symplectic Grassmannian integral representation of these amplitudes with a particular focus on three and four point amplitudes. We will discuss on-shell functions and the BCFW bridge on the Coulomb branch and possible future directions to obtain an amplituhedron-like geometry for this theory. Based on arxiv:2311.17763, and ongoing work with Veronica Calvo Cortes, Yassine El Maazouz and Amit Suthar.
 
 
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Séminaire de géométrie arithmétique
I will describe an arithmetic analogue of the theory of differential equations in which derivation operators acting on  functions are replaced  by Fermat quotient operators acting on numbers. I will then review a series of arithmetic applications of the theory including: the effective Manin-Mumford conjecture, finiteness  results for Heegner points (joint work with B. Poonen), construction of quotients of moduli spaces of abelian varieties by Hecke actions (joint work with A.Vasiu), and the curvature of the spectrum of the ring of integers (joint work with L.Miller).
 
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Séminaire Amplitudes et Gravitation sur l’Yvette (IHES/IPhT)
In this talk I present a new worldline action in which the worldline spin degrees of freedom are carried by bosonic oscillators. First, we construct a Hamiltonian for the worldline accounting for all spin-multipole moments at linear order in curvature, investigating the constraints on the Hamiltonian from a specific choice for the spin supplementary condition (SSC). Converting to the Lagrangian, we find a simple kinetic term for the spin DoFs in terms of bosonic oscillators, and also break the quadratic-in-spin ceiling of previous WQFT approaches involving Grassmann spinors. Injecting insight from the previously constructed Hamiltonian reveals a new class of operator redundancies in the EFT not related to equations of motion or integration by parts. Finally, a simple on-shell condition neatly ties up considerations related to the SSC, and paves the way for future higher-PM calculations involving spin.
 
 
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I will explain work joint with Aaron Landesman where we prove that for a finite group G and conjugacy invariant subset c, Hurwitz spaces parameterizing connected G-covers of the complement of a configuration of points on a disk with monodromy in c satisfy homological stability. We moreover compute the dominant part of the stable homology after inverting finitely many primes. This has applications to Malle’s conjecture over function fields, the Cohen—Lenstra—Martinet heuristics over function fields, as well as to the Picard rank conjecture.
 

Running Seminar
General Discussion 
Traces, Torsions, and the Neumann-Zagier Data for Specific Knots as Examples
 
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Séminaire Amplitudes et Gravitation sur l’Yvette (IHES/IPhT)
In this seminar, I will present the formalism we developed to solve analytically the Teukolsky equation and compute observables at first order in Self Force for the scattering problem, by combining a low-velocity (Post-Newtonian) and a large impact parameter (Post-Minkowskian) expansion. In particular, we generalize the usual techniques employed for studying bounded orbits, which presents a distinctive discrete frequency spectrum, in order to include the continuous Fourier spectrum of the unbound motion. Interestingly, this extension produces a rich mathematical structure on the non-local, logarithmic-in-frequency, sector.
I will briefly recap the application of these techniques on the scattering of a scalar charge, then I will present some preliminary results on the gravitational problem.
 
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Séminaire de géométrie arithmétique
I will present a new geometric perspective on the p-adic monodromy theorem of André, Kedlaya, and Mebkhout, which is based on the study of vector bundles on the analytic de Rham stack of the Fargues–Fontaine curve. I will then outline some applications to the p-adic Hodge theory of rigid-analytic varieties.   
This is based on joint work in progress with Anschütz, Le Bras, and Rodriguez Camargo. 
 
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Running Seminar
Given a hyperbolic knot K, state integrals are convergent integrals of products of Faddeev’s quantum dilogarithm associated with certain triangulations of S3K. Their asymptotic expansions are divergent power series conjectured to be resurgent and Borel-Laplace summable by Garoufalidis, Gu and Mariño. In this talk, I will prove this conjecture for the knots 41 and 52. This is based on a joint project with C. Wheeler, arXiv:2410.20973
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Séminaire de géométrie arithmétique
In this talk, we will explain relation between u-power torsions in Breuil–Kisin prismatic cohomology and various pathologies in p-adic cohomology theories, as well as mention some new results. Part of the talk will be based on earlier joint works with Tong Liu, we shall also report some recent ongoing projects with Ofer Gabber and Alexander Petrov separately.
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Séminaire Amplitudes et Gravitation sur l’Yvette (IHES/IPhT)
We show that Feynman integrals occurring in standard quantum field theories or perturbative worldline approaches to the scattering of  black holes are related to periods of Calabi-Yau varieties of various dimensions. After defining what mathematical properties a Calabi-Yau period motive has, we explain how the applications of the latter lead to an efficient analytic evaluation of the Feynman integrals in dimensional regularisation.
 
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The mapping class group of a surface is the group of isotopy classes of homeomorphisms of the surface. It acts on the space of conjugacy classes of morphisms from the fundamental group of the surface to some fixed Lie group. Such spaces are known as character varieties. In this talk we will investigate the rare phenomenon of finite orbits for mapping class group dynamics on character varieties. We will see how to construct non-trivial examples of finite orbits and give some intuition on how to classify all finite orbits when the target Lie group is SL(2,C). Most of this work is a collaboration with Samuel Bronstein.
 

When does a sequence of hyperbolic 3-manifolds with volume going to infinity have exponentially growing torsion homology? For arithmetic towers, the work of Bergeron-Sengun-Venkatesh suggests a set of conditions that conjecturally imply exponential growth of torsion homology. For nice sequences of hyperbolic 3-manifolds we use a different approach to find a condition implying exponential torsion homology growth: we give a condition on the spectrum of the Laplacian. I will give several motivations for this condition and show how to construct concrete examples of sequences satisfying it. This is based on joint work with Anshul Adve, Vikram Giri, Ben Lowe and Jonathan Zung.