Une carte serrée est une carte avec des sommets marqués, telle que tous ses sommets de degré 1 sont marqués. Pour un jeu donné de faces étiquetées de 1 à n de degrés prescrits (et en interprétant les sommets marqués comme des faces de degré 0), le nombre de cartes serrées distinctes que l’on peut construire sur la sphère est, comme l’a montré Norbury, un quasi-polynôme de degré n-3 dans les carrés des degrés des faces. Je montrerai comment obtenir la formule explicite de ce quasi-polynôme de manière purement bijective par une décomposition des cartes serrées planaires en tranches (« slices »). Je montrerai enfin comment en déduire une extension de la « formule des slicings » de Tutte (1962) au cas de cartes planaires ayant un nombre arbitraire de faces de degrés impairs.

(avec V. Delecroix, E. Goujard et P. Zograf)

Nous commencerons par le comptage de graphes à rubans, fait par M. Kontsevich et P. Norbury, et par le comptage de multi-géodésiques fermées simples sur les surfaces hyperboliques, effectué par M. Mirzakhani. En utilisant ces résultats, nous allons décrire la structure d’une multi-géodésique typique sur une surface de grand genre. Cette description est basée en partie sur les formules asymptotiques pour les corrélateurs de Witten-Kontsevich et pour les volumes de Masur-Veech de l’espace de modules de différentielles quadratiques en grand genre (les deux notions seront expliquées). Ces dernières formules, conjecturées par Delecroix-Goujard-Zograf-Zorich, ont été récemment démontrées par Amol Aggarwal.

We will discuss topological and ergodic aspects of the horocyclic flow on families of examples. This is joint work with Fernando Alcalde Cuesta.

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The Selberg trace formula provides a link between the length spectrum and the Laplacian spectrum of a hyperbolic surface. I will speak about a joint project with Maxime Fortier Bourque in which we are using this formula to probe extremal problems in hyperbolic geometry. These are questions of the form: what is the hyperbolic surface of a given genus with the largest kissing number or the largest spectral gap? Concretely, I will explain the general principle of our method, which is inspired by ideas from the world of Euclidean sphere packings. Moreover, I will explain why the Klein quartic, the most symmetric Riemann surface in genus 3, solves one of our extremal problems.

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Séminaire « Equations différentielles »

Gaudin model is the simplest case of the quantum Hitchin system corresponding to the rational curve with marked points. Algebraic Bethe ansatz provides a general way to solve the eigenproblem of this quantum integrable system, it expresses the eigenvectors and the eigenvalues as some explicit functions on the auxiliary parameters which satisfy  Bethe ansatz equations. The solutions of Bethe ansatz equations depend on the parameters of the integrable system, and this dependence is highly multivalued. We describe the monodromy of solutions of the Bethe ansatz equations and relate it to the RSK combinatorics of Young tableaux.

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Séminaire « Equations différentielles »

The aim of this talk is to give an overview on Frobenius Manifolds and its different facets. Frobenius manifolds arose in the process of axiomatisation of Topological Field Theory (TFT) and play a central role in the mathematical vision of mirror symmetry. However, recent results allow to go beyond this panorama.  In particular, we will discuss the existence of a new source of Frobenius Manifolds, which shows an unexpected bridge between algebraic geometry—involving moduli spaces of curves, Gromov—Witten like invariants, Saito’s space—and statistical manifolds, coming from information geometry.

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Séminaire Laurent Schwartz — EDP et applications

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Séminaire Laurent Schwartz — EDP et applications

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Séminaire Laurent Schwartz — EDP et applications

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IHES Covid-19 regulations:

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Lattice gauge theories, when restricted to the pure gauge sector (i.e. no matter, only gauge fields), typically show a phase transition between a topologically-ordered deconfined phase at weak coupling and a trivial confined phase at strong coupling (cf. Wegner 1971, Wilson 1974, Fradkin and Shenker 1979). The diagnosis for such a topological phase transition is a non-local gauge-invariant order parameter known as a Wegner-Wilson loop (WWL) defined along a chosen contour. The WWL features a perimeter law exp(-#P) in the topological phase and an area law exp(-#A) in the confined phase, where P is the perimeter and A the area of the contour. The trivial phase is described as having confined charges (“quark confinement”) and condensed fluxes. Whereas, the topological phase has free (deconfined) charges and fluxes. Two-dimensional quantum lattice gauge theories are special in that the excitations in the deconfined phase are anyons (cf. toric code model, Kitaev 2003). In the toric code, WWL were studied in detail by Halasz and Hamma 2012.

Here we extend such a study of WWL from lattice gauge theories built on gauge groups to string-net models (Levin-Wen 2005) built on more general objects known as unitary modular tensor categories. We use these WWL to study the different kind of anyonic excitations that are believed to be described at low-energy by a topological quantum field theory of the doubled achiral type.

Ref: A. Ritz-Zwilling, J.-N. Fuchs and J. Vidal, arxiv:2011.12609,  Phys. Rev. B 103, 075128 (2021).

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The need to improve the analytical knowledge of the gravitational waveforms emitted by binary systems has recently sparked a fervent activity in the application of  (classical and/or quantum) post-Minkowskian perturbation methods (expansion in G) to the  two-body relativistic gravitational dynamics and radiation. We shall discuss how the use of a classical Effective-Field-Theory worldline approach to gravitational scattering, combined with modern Quantum-Field-Theory integration techniques, allows one to compute both the gravitational-wave amplitude and the associated radiated four-momentum.

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Almost 40 years ago, Stanley noticed that some of the deep theorems of algebraic geometry have powerful combinatorial applications. Among other things, he used the hard Lefschetz theorem to rederive Dynkin’s theorem, and to characterize face numbers of simplicial polytopes.
Since then, several more deep combinatorial and geometric problems were discovered to be related to theorems surrounding the Lefschetz theorem. One first constructs a ring metaphorically modelling the combinatorial problem at hand, often modelled on constructions for toric varieties, and then tries to derive the combinatorial result using deep results in algebraic geometry. For instance:
– a Lefschetz property for implies that a simplicial complex PL-
embedded in R4 cannot have more triangles than four times the number of it’s edges. (Kalai/A.)
– a Hodge-Riemann type property implies the log-concavity of the coefficients of the chromatic polynomial. (Huh)
– a decomposition type property implies the positivity of the
Kazhdan-Lusztig polynomial. (Elias-Williamson)
At this point one can then hope that indeed, algebraic geometry provides the answer, which is often only the case in very special cases, when there is a sufficiently nice variety behind the metaphor.
It is at this point that purely combinatorial techniques can be attempted to prove the desired. This is the modern approach to the problem, and I will discuss the two main approaches used in this area: Firstly, an idea of Peter McMullen, based on local modifications of the ring and control of the signature of the intersection form. Second, an approach based on a theorem of Hall, using the observation that spaces of low-rank linear maps are of special form.