Séminaire d’Analyse
Invariant manifold theory is a fundamental tool in the study of local dynamics near invariant structures in smooth evolution systems. It ensures the existence of nonlinearly invariant structures from linear ones. The theory has been well developed for diffeomorphisms, ODEs, semilinear PDEs, and some quasilinear parabolic PDEs. However, it becomes subtle for quasilinear or more nonlinear PDEs due to regularity issues when there is no smoothing effect. In this talk, we consider a class of nonlinear PDEs whose linearizations satisfy certain energy estimates. We prove that the linear exponential dichotomy implies the existence of local stable/unstable manifolds of the equilibria. In particular the result applies to a class of nonlinear Hamiltonian PDEs including the capillary gravity water waves of one or two fluids, quasilinear wave and Schrödinger equations, KdV type equations, etc., for which the linear analysis is also discussed. Basically, for such systems under certain conditions, spectral instability implies the existence of stable and unstable manifolds, which in particular yields the nonlinear instability in rough Sobolev norms and/or the existence of solutions decaying in high Sobolev norms. This is a joint work with Jalal Shatah. 
 
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Schoenberg’s theorem on positive definite functions on spheres manifests in three distinct “avatars”: in the study of matrix preservers, the harmonic analysis of homogeneous spaces, and the representation theory of infinite-dimensional groups. I will discuss these connections, starting from the classical entrywise calculus of the Schur product theorem and extending to the asymptotic behavior of spherical functions on Olshanski spherical pairs.
As a geometric application, we will see how this framework recovers the linear programming bounds of Delsarte–Goethals–Seidel and Kabatiansky–Levenshtein for spherical codes, which in turn yield upper bounds for sphere packing densities. I will then extend Schoenberg’s classical framework to partially defined positivity preservers on discrete domains, and apply this machinery to the problem of soft thresholding for correlation matrices in high-dimensional statistics.
This is joint work with James Pascoe.
 
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Séminaire de géométrie arithmétique
A longstanding question in the theory of Shimura varieties concerns their perfectoidness at infinite level — a property that would reveal deep connections between étale and coherent cohomology. In this talk, we establish a criterion for perfectoidness via Sen theory, building on a new development of p-adic Hodge theory for general valuation fields that extends Tate’s foundational work on local fields. We further provide a conceptual explanation, based on the p-adic Simpson correspondence after Abbes-Gros, Liu-Zhu and Tsuji, for why Shimura varieties satisfy this criterion, at least in the case of modular curves. For general Shimura varieties, it follows through additional technical arguments due to Pan and Rodríguez Camargo. This yields the “pointwise perfectoidness” of Shimura varieties at infinite level, which suffices to establish the desired connection between different cohomologies. As an application, we show that integral completed cohomology groups vanish in higher degrees, thereby confirming a conjecture of Calegari and Emerton for arbitrary Shimura varieties.
 
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Séminaire Amplitudes et Gravitation sur l’Yvette (IHES/IPhT)
Black holes with sufficiently large initial charge and mass will Hawking-evaporate towards the extremal limit. The emission slows as the temperature approaches zero, but still reaches the point where a single Hawking quantum would make the object superextremal, removing the horizon. We take this semiclassical prediction at face value and ask: When the emission occurs, what is revealed?  Using a simple thin-shell model for the matter originally forming the black hole, we find that this matter *re-emerges* after the horizon is removed and subsequently expands back to large radius.  This expanding remnant has been bathed in the ingoing Hawking quanta during evaporation and presumably carries correlations with the outgoing quanta, offering the attractive possibility of studying information paradox issues in a setup where spacetime curvatures are globally small, so that quantum gravity is not required.
 
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Mathematics Inspired by Physics    A Conference in Honor of Yan Soibelman’s Contributions    June 1 & 2 2026    at IHES – Marilyn and James Simons Conference Center ; Léon Motchane Amphitheater   How to get to IHES
Registration is free but compulsory. 

Yan Soibelman, professor of mathematics at Kansas State University, has made influential contributions at the interface of geometry, algebra, and theoretical physics. His early work on quantum groups and braided categories helped shape modern interactions between representation theory and mathematical physics. His work is strongly inspired by ideas from quantum field theory and string theory, which he helps translate into rigorous mathematical frameworks.
In collaboration with Maxim Kontsevich, he has played a central role in the development of mirror symmetry, including various fundamental results in the theory of operads and A  ͚ categories. He also introduced key concepts such as motivic Donaldson–Thomas invariants, the Kontsevich–Soibelman wall-crossing formula, and cohomological Hall algebras. His work has had a lasting impact on areas where physics drives new mathematical insights.
 
The conference will celebrate Yan Soibelman’s contributions by bringing together experts to present recent advances & foster interaction across fields as well as engaging early-career researchers in these exciting developments.
Speakers:

Veronica Fantini, Laboratoire de Mathématique d’Orsay

Mikhail Kapranov, Kavli Institute

Bernhard Keller, Institut de mathématiques Jussieu-Paris Rive Gauche

Maxim Kontsevich, IHES

Pierre Schapira, Institut de mathématiques Jussieu-Paris Rive Gauche

Olivier Schiffmann, CNRS, Laboratoire de Mathématiques d’Orsay

Alexander Soibelman, IHES

Yan Soibelman, Kansas State University

Bruno Vallette, Université Sorbonne Paris Nord 

 
Organizing committee:     Mikhail Kapranov, Kavli Institute · Maxim Kontsevich, IHES
 

The parallel transport construction can be used to produce an equivalence of categories between the category of representations of the fundamental group of a smooth connected manifold and the category of flat bundles over this manifold. I will discuss an analogue of this construction when the field of real numbers is replaced by the field of p-adic numbers. Given a smooth rigid space X over ℚp, consider the ring D of differential operators on the base change of X to Fontaine’s period ring BdR+. Let Dt be the subalgebra spanned by functions and vector fields multiplied by a uniformizer t in BdR+. Thus, Dt is an algebra over BdR+, whose mod t reduction is the commutative algebra of functions on the cotangent space, and which is isomorphic to D after inverting t. The category of modules over Dt can be twisted by any μp∞ gerbe over the cotangent space. I will construct a functor from the category of étale BdR+-local systems on XCp to the category of modules over Dt twisted by the Simpson gerbe. The composition of this functor with the mod t reduction recovers the p-adic Simpson functor of Bhatt and Zhang.
 
This is a joint work in progress with Bhargav Bhatt, Ben Heuer and Alexander Petrov.
 
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Séminaire d’Analyse
A Jordan curve on the Riemann sphere can be encoded by its conformal welding homeomorphism, which is a circle homeomorphism. I will explain that this correspondence should be viewed as a canonical correspondence between a Jordan curve in the boundary of hyperbolic 3-space H3 and a positive curve on the boundary of AdS3 space.
The Loewner energy measures how far a Jordan curve is away from being a circle or, equivalently, how far its welding homeomorphism is away from being Möbius. It arises as the action of random curves SLE, Kähler potential of Weil-Petersson universal Teichmüller space, Fredholm determinant of Grunsky operator, free energy of Coulomb gas on a Jordan curve, and a renormalized volume of H3, etc. All these links refer to either the curve description or the welding description of the Loewner energy.
I will discuss two optimizing problems for the Loewner energy, one under the constraint for the curve to pass through n given points on the Riemann sphere and the other under the constraint for the welding curve to pass through n given points in the boundary of AdS3. These two problems exhibit many symmetries that are poorly understood, but do suggest that the Loewner energy sits right in the middle of two perspectives (curve/welding).
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A framed polytope is the convex closure of a finite set of points in ℝⁿ together with an ordered linear basis. An n-category is a category that is enriched in the category of (n-1)-categories. Although these concepts may initially appear to be distant peaks in the mathematical landscape, there exists a trail connecting them, blazed in the 90’s by Kapranov and Voevodsky. We will traverse this path, widening and improving it as we address some of their conjectures along the way.
 
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By commuting three vector fields on the line ℝ ∋ x with monomial coefficients 1, −2x, and −x2, we realise the Lie algebra sl(2) in its Chevalley basis; the bracket acts on the coefficients as the Wronskian determinant. Let us extend this model to a class of polynomial homotopy Lie algebras in which the N-ary brackets are given by the Wronskian determinants over multidimension; the generalised Vandermonde determinants then express the structure constants.
The alternated composition of N = 2p differential operators wj(x) ∂px of strict order p on the line ℝ ∋ x is again a differential operator of strict order p; its coefficient is the constant c(p) times the Wronskian determinant of the coefficients w1, …, wN. At p = 1, the sl(2) case fixes c(1) = 1; easy is c(2) = 2, then c(3) = 90. In a recent joint work with K. C. Shah, we reach the exact values c(p = 4) = 586 656, c(p = 5) ≈ 1.9 · 1012, and c(p = 6) ≈ 7.9 · 1021. The positive integer sequence c(p) seems to be new; to know c(p ⩾ 7) is an open problem.
Deform the binary Lie bracket to a formal sum of Wronskians with purely even (N = 2p) or arbitrary (N ∈ ℕ⩾2) arities, see arXiv:2510.02145 [math.RA]. Not only does the full bracket Δ satisfy the Jacobi identity Δ[Δ] = 0 for homotopy Lie algebra, but for every pair of arities ℓ, m ⩾ 2 the respective (ℓ, m)-term in the identity vanishes separately. Over base dimension d = 1, we spot an infinite sequence of finite-dimensional polynomial homotopy Lie algebras starting at sl(2) and with the Wronskians as the brackets; all the structure constants, unless zero due to repetitions, equal ±1 in a suitable basis.
Let the base dimension d ⩾ 1 be arbitrary: ℝd ∋ (x1,…,xd). We proved in arXiv:math.RA/04110185 that the complete generalised Wronskians – involving all the derivatives up to a given differential order k ⩾ 1 – still satisfy the table of Jacobi identities for strong homotopy Lie algebras. The arity N = (d+k  d) of such brackets grows with dimension d and order k but the steps, as k ↦ k + 1, grow as well: over d > 1 the gaps get larger and larger. In a recent work arXiv:2511.03848 [math.RA] we prove that by allowing the multivariate Wronskians be incomplete in their top differential order k > 1, we do preserve all the SH-Lie Jacobi identities.
For complete Wronskians of orders k ⩾ 1 over (multi)dimension d ⩾ 1 as the brackets, in a work in progress (joint with M. G. Ķēniņš) we exhaustively describe all the finite-dimensional polynomial N-ary SH-Lie algebra generalisations of sl(2); we express their structure constants in terms of the multivariate Vandermonde determinants. Relaxing the finite-dimensionality assumption and taking the (Laurent-)monomials in d variables for the generators, we obtain multivariate analogues of the Witt algebra from CFT.
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Séminaire Amplitudes et Gravitation sur l’Yvette (IHES/IPhT)
The upcoming LISA mission will be sensitive to asymmetric binary black hole mergers. The self-force formalism generates waveform models by expanding in the small mass ratio, where accuracy requirements necessitate results up to second order. This problem is predominantly tackled with numerical methods, which are notoriously challenging. Complementing these methods with analytical approximations (e.g., around the Newtonian limit) can significantly reduce computational costs and yield independent benchmarks. Motivated by this, I will introduce analytical self-force theory and present the most recent advancements in second-order calculations. I will also present the latest version of the Teukolsky package within the Black Hole Perturbation Toolkit. This package has been greatly expanded to facilitate analytical calculations in self-force and black hole perturbation theory, and has recently found applications in scattering.
 
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We show that two chain models of the gravity and the hypercommutative operads in genus zero are Koszul dual to each other. Precisely the model for the gravity operad is based on cacti without base points, and its bar construction arises from a cellular decomposition of the moduli space of stable curves. This is joint work with Tommaso Rossi.
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Séminaire Amplitudes et Gravitation sur l’Yvette (IHES/IPhT)
Radiation-reaction force encodes dissipative effects in a binary system emitting gravitational waves. Within the post-Newtonian framework, radiation-reaction terms enter the equations of motion starting at 2.5PN order and affect the system’s dynamics accordingly. These effects can be incorporated as corrections to the quasi-Keplerian orbital parameters in the center-of-mass frame. In this presentation, I will discuss the Lagrange method of variation of constants that we used to determine the radiation-reaction corrections for the quasi-hyperbolic orbit at 3.5PN order, and I will also report on our ongoing work at 4.5PN order.
 
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