In this talk, I will try to address the question of how to do algebraic topology with stratified spaces and schemes in a meaningful way. I will begin by recalling and motivating some of the main historical attempts to approach this question, both in topology and in algebraic geometry, passing through the work of Nash, Quinn, Siebenmann, Edwards–Hastings, Deligne, and, more recently, Lurie and Barwick–Glasman–Haine. I will then propose my own answer, based on a construction I call the stratified shape.Time permitting, I will discuss some applications of the stratified shape to proper homotopy theory (joint with Georg Lehner), homology manifolds (joint with Markus Land), and a generalization of Lurie’s exodromy equivalence (joint with Peter Haine and Guglielmo Nocera).
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Séminaire Amplitudes et Gravitation sur l’Yvette (IHES/IPhT)
In principle, rapidly rotating black holes could naturally act as particle colliders because geodesics particles that collide can exhibit center-of-mass (CM) energies that diverges in the limit that the black hole approaches extremality and the collision radius approaches the horizon, provided that one particle has a critically tuned angular momentum. In this talk, I present work to understand the naturalness and observability of these critical-particle collisions around near-extremal BHs. I will highlight 1) the landscape of conditions under which these critical-particle collisions occur, 2) the CM energy achievable for critical-particle collision where one particle falls from in from infinity and the other particle is part of an accretion disk, and 3) the observability as measure by the escape probability and expected energy of emission born of a critical-particle collision.
 
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Séminaire d’Analyse
I will discuss some recent works related to maximal operators and Nikodym sets in the plane. Nikodym constructed a measure-zero Borel set in the plane that contains a line segment passing through each point in the unit square. Fenton and Falconer asked whether a similar construction is possible when “lines through a point” are replaced by ”circles with a prescribed center.” This was answered in the negative by Bourgain and Marstrand. We study a generalization of this question, in which “lines through a point” or “circles with prescribed center” are replaced by general analytic families of plane curves. Using decouping, we prove new maximal function and local smoothing estimates for families of curves satisfying a multi-parameter analogue of Sogge’s cinematic curvature condition. Combining with a recent nonlinear generalization of Falconer’s prescribed projection theorem to construct Nikodym-type sets, we characterize when Nikodym-type sets exist for analytic families of plane curves.
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Séminaire Amplitudes et Gravitation sur l’Yvette (IHES/IPhT)
I study the evolution of the event horizon in a “two-timescale” perturbative expansion around a stationary black hole. This leads to strong constraints on the “coarse-grained” horizon dynamics. At leading order, the horizon exhibits an “adiabatic rigidity”: the leading corrections to its angular velocity and inaffinity remain uniform on each horizon cut, while evolving on the slow timescale. We then provide a systematic procedure for transforming a perturbative bulk solution, given for example in Lorenz gauge, to an ingoing Newman–Unti gauge adapted to the horizon, allowing the perturbed horizon geometry to be extracted directly from the metric. Finally, we formulate “horizon conservation laws” associated with horizon symmetries, including energy, dynamical entropy, and angular momentum. We expand the charges through second order and their fluxes through third order in perturbation theory, providing a framework for future applications to horizon absorption and backreaction in extreme mass-ratio inspirals.
 
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We provide a simple derivation of the conformal dispersion relation of Carmi and Caron-Huot which reconstructs a CFT four-point function from its double discontinuity. We exhibit a hidden conformal symmetry which explains the appearance of the « magic variable » x as a simple cross-ratio, built from four physical cross-ratios. We also introduce a regulator in the double discontinuity operation and then prove that one subtraction suffices for all integrals to converge.
 
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Séminaire Amplitudes et Gravitation sur l’Yvette (IHES/IPhT)
Sommerfeld effects describe long-distance corrections to radiation emitted by a short-distance source. They play an important role in atomic physics and dark matter phenomenology, but the same mechanism also arises in gravitational-wave radiation from compact binaries. In this talk, I will develop a systematic framework for gravitational Sommerfeld effects by combining worldline effective field theory with black hole perturbation theory. A key advantage of this framework is that it organizes high-order perturbative calculations without requiring any nontrivial loop integration, allowing us to compute the waveform to tenth order in the perturbative expansion and beyond. The formulation also reveals an exact connection between the radiation waveform and Compton scattering, as well as the underlying structures of the classical renormalization group of the multipoles and tidal Love numbers. I will conclude by discussing how these results can be incorporated into waveform resummation models for gravitational-wave signals. This talk is based on arXiv:2604.14112 with Chih-Hao Chang and Zihan Zhou, and work in progress.
 
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Séminaire de géométrie arithmétique
In 1952, Mordell erroneously claimed that the surface xyz = G(x,y) has infinitely many integral points for any G ∈ ℤ[x,y]. I will discuss a classification for finiteness and Zariski density of (positive) integral points on these log K3 surfaces when the coefficients of G are non-negative. This combines a dynamical argument proposed by Mordell, the theory of generalized cluster algebras, and the Killing–Cartan classification of semisimple Lie algebras. This talk is partially based on joint work with Antoine de Saint Germain.
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October 12-15, 2026    at IHES – Marilyn and James Simons Conference Center
How to get to IHES

 
This workshop explores the mathematics connected to the pentagram map and its generalizations.  
With 3 talks per day, spread out over 4 days, leaving time for individualized interactions, it will not be confined to the special topic of the pentagram map, but rather an exploration of the kind of mathematics that it relates to: integrable systems, algebraic dynamics, projective geometry and cluster algebras.
The pentagram map is a geometrically flavored dynamical system defined on polygons by a simple and projectively natural straight-line construction. 
Starting with a polygon, one intersects the shortest diagonals in cyclic order to produce a new polygon with the same number of vertices.   
Though quite elementary to define, the pentagram map turns out to have a deep structure connected to completely integrable systems,  symplectic geometry, cluster algebras, dimers,  algebraic dynamics,  and moduli spaces.  
The same goes for the many generalizations of the pentagram map, some of which are still quite mysterious.

Organizing Committee:
Richard E. Schwartz (Brown Univ. & Gretchen and Barry Mazur Chair at IHES) and Robert C. Penner (Schlumberger Chair for Mathematical Sciences at IHES)
Invited speakers:

Vladimir DRAGOVIĆ (UT Dallas)

Anna FELIKSON (Durham University)

Vladimir FOCK (IRMA, Strasbourg)

Misha GEKHTMAN (University of Notre Dame)

Max GLICK (Google)

Anton IZOSIMOV (University of Glasgow)

Rinat KEDEM (University of Illinois Urbana-Champaign)

Valentin OVSIENKO (CNRS – Université de Reims Champagne-Ardenne)

Jean-Baptiste STIEGLER (Université Paris-Saclay)

Sergei TABACHNIKOV (Penn State University)

Béatrice de TILIÈRE (Université Paris Dauphine – PSL)

Max WEINREICH (CUNY Baruch College)

Michał ZWIERZYŃSKI (Warsaw University of Technology)

The values of the Riemann zeta-function $zetaleft(alpha+beta nright)$ with $alpha>1$, $beta>0$ and $n$ a positive integer can be viewed as moments of a probability distribution. In my talk I will construct the corresponding random matrix model and determine its asymptotic spectral curve, which exhibits an unusual feature, the complete asymptotics of its partition function and uniform approximations to its correlation functions using a discrete version of Riemann-Hilbert problem with Airy and Bessel asymptotics. This relates a problem in number theory to ideas of Vershik and Okounkov in the representation theory of the symmetric group.
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Séminaire Amplitudes et Gravitation sur l’Yvette (IHES/IPhT)
In this talk I will describe an amplitude-based framework for extracting the frequency-dependent tidal response of neutron stars while preserving the tidal resonances associated with their normal modes. The response is obtained by matching the amplitude for gravitational-wave scattering off a neutron star, equivalently gravitational Compton scattering, computed using relativistic stellar perturbation theory, to the corresponding amplitude in worldline effective field theory. The tidal contribution is isolated through the difference between the neutron-star and black-hole scattering phases. Unlike conventional treatments, the construction allows for a post-Newtonian expansion without Taylor expanding the response in frequency or assuming a harmonic-oscillator model for the internal dynamics.
I will present the tidal response through relative third post-Newtonian (3PN) order and use currently available numerical results to illustrate its frequency dependence. The results reproduce the known static Love numbers in the low-frequency limit, exhibit resonances at the expected stellar mode frequencies, and capture their damping through gravitational-wave emission. I will also discuss the renormalisation of the tidal response required through 3PN order. Time permitting, I will outline how the resulting response can be incorporated into binary dynamics and ultimately into gravitational waveforms. 
 
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Séminaire de géométrie arithmétique
A Siegel domain is a finite-volume subset of a symmetric space whose translates under the action of the relevant arithmetic group cover the entire space. In this talk, we consider an analogue for the p-adic period spaces introduced by Rapoport and Zink. We give a partial criterion for the existence of a Siegel domain and discuss its expected applications.
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I will describe how one can use modern machine learning methods to attack open problems in group theory. I will show how ideas from AI, pure math, integrability, string theory and quantum information meet here in a highly productive way. I will focus on much-studied Cayley graphs of permutation groups, which are often very hard to explore due to their enormous size. I will describe how a modern ML approach overcomes this challenge and provides efficient pathfinding on these huge graphs. The outcome is a large set of new results and conjectures that in particular sharpen famous bounds on the diameters of these graphs. These results also hint at the existence of a novel, discrete string-like duality for this setting. It provides a realisation of the paradigm « complexity = volume » from AdS/CFT holography, and may potentially be applicable to a wider range of AI tasks. Lastly I will outline some new results for combinatorics of quantum periods and how this could help in improving these ML approaches. Based on arXiv:2509.19162, 2603.22195 and work to appear. Our code is publicly available as the CayleyPy library.
 
 
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