The Seed seminar of mathematics and physics is a seminar series that aims to foster interactions between mathematicians and theoretical physicists, with both online and in-person events. It is holding a special day on Matrix models for quantum systems at IHES, with contributions from Guillaume Aubrun, Philippe Biane, Bertrand Eynard and Vladimir Kazakov. Registration is free and open until May 31, 2024.Invited speakers:Guillaume Aubrun (Institut Camille Jordan, Lyon)Philippe Biane (Laboratoire d’Informatique Gaspard Monge, Marne-la-Vallée)Bertrand Eynard (Institut de Physique Théorique, CEA Saclay)Vladimir Kazakov (Laboratoire de Physique de l’École Normale Supérieure, Paris)Scientific Committee: Thierry Bodineau ( IHES)Slava Rychkov (IHES)Organizing Committee: Ariane Carrance (CMAP)Matteo D’Achille (LMO)Edoardo Lauria (LPENS)

Seminar on Quantum Modularity and ResurgenceThe partition function of topological string theory is an asymptotic series in the topological string coupling and provides in a certain limit a generating function of Gromov-Witten (GW) invariants of a Calabi-Yau threefold. I will discuss how the resurgence analysis of the partition function allows one to extract BPS invariants of the same underlying geometry. I will further discuss how the analytic functions in the topological string coupling obtained by Borel summation admit a dual expansion in the inverse of the topological string coupling leading to another asymptotic series at strong coupling and to the notion of topological string S-duality. This S-duality leads to a new modular structure in the topological string coupling. I will also discuss relations to difference equations and the exact WKB analysis of the mirror geometry. This is based on various joint works with Lotte Hollands, Arpan Saha, Iván Tulli and Jörg Teschner as well as on work in progress.========Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide.

The simplest topologically ordered phase in 2+1D is the deconfined phase of Z2 lattice gauge theory. There are two reasonably well-understood ways to exit the deconfined phase: the Higgs transition, where electric charge (the « e » anyon) condenses, and the confinement transition, where magnetic charge (the « m » anyon) condenses. However, we can also exit the deconfined phase via the self-dual line in the phase diagram, where there is a symmetry between « e » and « m ». What happens here is more mysterious. If this transition is continuous, it may be the simplest critical point with no useful continuum Lagrangian (as yet). After reviewing the formulation of the model as the statistical mechanics of membranes, I will describe clear Monte Carlo evidence for the continuity of the self-dual transition. I will sketch why it is not a conventional « Landau » critical point. Separately, I will use the membrane formulation to describe a very concrete and intuitive way of understanding the emergent higher-form symmetries which appear in part of the phase diagram (and which are the reason that the Higgs and confinement transitions can be understood using Landau theory, despite lacking local order parameters). Work with Andres Somoza and Pablo Serna (https://arxiv.org/abs/2012.15845 and https://arxiv.org/abs/2403.04025). Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_physique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide.

Probability and analysis informal seminarOn a finite connected graph, the product of the non-zero eigenvalues of the Laplacian counts the rooted spanning trees, according to a theorem often attributed to Kirchhoff (1847), or sometimes to Sylvester (1857). Among many generalisations of this classical result, those of Zaslavsky (1982), Forman (1993) and Kenyon (2011) state that when we twist the Laplacian by putting a sign or a phase, complex or quaternionic, on each edge, its determinant counts covering forests of unicycles, with appropriate weights. A common feature of all these results is that the random subgraphs naturally associated to each of these situations (uniform spanning trees and random covering forests of unicycles), seen as random subsets of the (finite) set of edges of the ambient graph, are determinantal point processes. I will present some results of an ongoing joint work with With Adrien Kassel (CNRS, ENS Lyon) in which we investigate further extensions of these results to the covariant Laplacian associated with an arbitrary unitary connection, that is, to the Laplacian twisted by a unitary matrix on each edge. In a first part, I will describe the classical results of Kirchhoff and Forman, then (from a perhaps slightly unorthodox point of view) determinantal point processes on finite sets, and explain what the ones have to do with the others. In a second part, I will describe the measures on Grassmannians that we introduced with Adrien Kassel, explain why they are relevant to the understanding of the twistes Laplacian, and finally describe, to the extent that we understand them, the new random objects that appear over a graph endowed with a unitary connection.========Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide.

Seminar on Quantum Modularity and ResurgenceThe talk will focus on quantum modularity relations satisfied by the $q$-Pochhammer symbol  $(q)_N = (1-q) … (1-q^N)$ at $q=exp(2 pi i x)$. These formulas can be interpreted as finite analogues of the usual modularity for the Dedekind eta-function. We’ll discuss certain aspects which come very handy upon summing over $N$. We’ll explain how these can be used, in the context of Kashaev’s invariant of hyperbolic knots, to prove, in a few cases, Zagier’s quantum modularity conjecture by means of what we currently know on the Volume Conjecture. This is based on joint work with Sandro Bettin.========Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide.

The goal of this conference is to advance the dialogue and interactions between the LLM community and the larger world of mathematics in order to further the mathematical understanding of LLMs and contribute to solving some of the outstanding problems in the new field of LLMs. In particular we intend to investigate mathematical structures that can be used to understand LLMs in terms of what they implicitly learn and how.  At the same time, in the opposite direction the use of LLMs in order to do mathematics will be investigated. Registration is free and open until May 16, 2024.Invited speakers:François Charton (Meta AI Research)Andrew Dudzik (Google DeepMind)Amaury Hayat (École des Ponts ParisTech & CERMICS)Julia Kempe (NYU Center for Data Science & CIMS)Gabriel Synnaeve (Meta AI Research)Yiannis Vlassopoulos (Athena Research Center & IHES)Organizers: François Charton (Meta AI Research), Michael Douglas (Harvard University & IHES) & Yiannis Vlassopoulos (Athena Research Center & IHES) 

In my previous talk, on 9/4/24, I set Stark’s Conjectures in the more general context of Hilbert’s 12th Problem, highlighting the special complex functions used by number-theorists to study various cases in recent decades. I also surveyed the remarkable way that the same special functions have cropped up recently in Quantum and Statistical Physics, as indeed have SICs themselves in the case of the first order Stark Conjecture over real quadratic fields.In this second, more number-theoretic, talk I will focus on the latter case. After recalling the necessary details, I will motivate and explain some ongoing work which sets SICs in the context of the Heisenberg group over ${mathbb Z}_p$ (the p-adic integers), `Theta-pairings’ of p-adic measures and Coleman’s power series. This in turn motivates the search for `special measures’ to replace the complex functions mentioned above, in a possible p-adic theory of real-multiplication.Although this will necessarily be a more technical talk than the previous one, I shall still aim to make it largely accessible to non-number-theorists.========Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide.

Albert Schwarz started as a topologist/geometer in the early ’50s, then in the ’70s he began an exploration of mathematical aspects of quantum field theory and made numerous seminal contributions to this subject. He has been a regular visitor to IHES since 1995 and was involved in many significant collaborations. This May he will give a short course on his recent research. The mini-conference is a tribute to the 70th anniversary of Schwarz’s remarkable scientific career by several of his friends and colleagues. Registration is free and open until June 7, 2024.Invited speakers:Alain Connes (Collège de France & IHES)Anton Kapustin (Caltech)Maxim Kontsevich (IHES)Boris Pioline (LPTHE – Sorbonne Université)Albert Schwarz (University of California at Davis & IHES)Organizers: Anton Kapustin (Caltech) & Maxim Kontsevich (IHES) 

We show that holographic CFTs with a global U(1) symmetry that contain particles satisfying a form of the weak gravity conjecture, contain states that thermalize much slower than the typical thermalization time set by the black hole temperature. In the eikonal limit, these states correspond to metastable bulk configurations containing charged particles hovering outside the horizon of a Reissner–Nordström black hole in AdS. More generally, we study slow thermalization by computing the quasinormal modes of black holes due to charged scalar perturbations. We find that these states persist a finite distance away from the critical point in the EFT parameters, contrary to the ‘critical slowing down’ behavior expected near a phase transition. Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_physique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide.

The course is based on a minibook that will be published by Springer. The text below is a shortened preface  to this book.In the conventional exposition of quantum mechanics, we work in Hilbert space and examine operators within this space. Self-adjoint operators are associated with physical quantities. Physicists predominantly use this methodology, however, it has its limitations. In this course we explore alternative viewpoints; our exposition does not depend on standard textbooks. We consider the algebraic approach, where the initial point is an algebra of observables, an associative algebra with involution, in which the self-adjoint elements are observables. This approach is nearly as old as quantum mechanics itself. In addition, we discuss the geometric approach, where the initial point is a set of states. This viewpoint was advocated in my recent papers; it is much more general. We demonstrate within the framework of this approach that quantum mechanics can be viewed as classical mechanics where our devices permit us to observe only a subset of physical quantities. Furthermore, we show that using this approach we can construct a wide class of physical theories that generalize quantum mechanics.We highlight that the emergence of probabilities in quantum theory can be derived from decoherence caused by adiabatic interaction with a random environment. We underscore that the concept of a particle is not primary in quantum theory. If the theory is translation-invariant we define particles as elementary excitations of the ground state. Quasiparticles are elementary excitations of any translation-invariant state. We analyze the concept of scattering but we do not utilize the concept of a field and do not assume locality and Poincare invariance. We  discuss not only the conventional scattering matrix (related to scattering cross-sections) but also the concept of an inclusive scattering matrix, which is closely related to the concept of inclusive scattering cross-sections. Scattering matrix can be expressed in terms of Green’s functions by the well-known formula belonging to Lehmann, Symanczyk, and Zimmermann, and the inclusive scattering matrix can be expressed in terms of generalized Green’s functions, which first appeared in nonequilibrium statistical physics in Keldysh formalism. As a concrete realization of the geometric approach, we describe the formalism of L-functionals where states are represented by non-linear functionals corresponding to positive functionals on Weyl and Clifford algebras (to states in the algebraic approach). L-functionals can be applied to solve the infrared problem in quantum electrodynamics.

The course is based on a minibook that will be published by Springer. The text below is a shortened preface  to this book.In the conventional exposition of quantum mechanics, we work in Hilbert space and examine operators within this space. Self-adjoint operators are associated with physical quantities. Physicists predominantly use this methodology, however, it has its limitations. In this course we explore alternative viewpoints; our exposition does not depend on standard textbooks. We consider the algebraic approach, where the initial point is an algebra of observables, an associative algebra with involution, in which the self-adjoint elements are observables. This approach is nearly as old as quantum mechanics itself. In addition, we discuss the geometric approach, where the initial point is a set of states. This viewpoint was advocated in my recent papers; it is much more general. We demonstrate within the framework of this approach that quantum mechanics can be viewed as classical mechanics where our devices permit us to observe only a subset of physical quantities. Furthermore, we show that using this approach we can construct a wide class of physical theories that generalize quantum mechanics.We highlight that the emergence of probabilities in quantum theory can be derived from decoherence caused by adiabatic interaction with a random environment. We underscore that the concept of a particle is not primary in quantum theory. If the theory is translation-invariant we define particles as elementary excitations of the ground state. Quasiparticles are elementary excitations of any translation-invariant state. We analyze the concept of scattering but we do not utilize the concept of a field and do not assume locality and Poincare invariance. We  discuss not only the conventional scattering matrix (related to scattering cross-sections) but also the concept of an inclusive scattering matrix, which is closely related to the concept of inclusive scattering cross-sections. Scattering matrix can be expressed in terms of Green’s functions by the well-known formula belonging to Lehmann, Symanczyk, and Zimmermann, and the inclusive scattering matrix can be expressed in terms of generalized Green’s functions, which first appeared in nonequilibrium statistical physics in Keldysh formalism. As a concrete realization of the geometric approach, we describe the formalism of L-functionals where states are represented by non-linear functionals corresponding to positive functionals on Weyl and Clifford algebras (to states in the algebraic approach). L-functionals can be applied to solve the infrared problem in quantum electrodynamics.

The course is based on a minibook that will be published by Springer. The text below is a shortened preface  to this book.In the conventional exposition of quantum mechanics, we work in Hilbert space and examine operators within this space. Self-adjoint operators are associated with physical quantities. Physicists predominantly use this methodology, however, it has its limitations. In this course we explore alternative viewpoints; our exposition does not depend on standard textbooks. We consider the algebraic approach, where the initial point is an algebra of observables, an associative algebra with involution, in which the self-adjoint elements are observables. This approach is nearly as old as quantum mechanics itself. In addition, we discuss the geometric approach, where the initial point is a set of states. This viewpoint was advocated in my recent papers; it is much more general. We demonstrate within the framework of this approach that quantum mechanics can be viewed as classical mechanics where our devices permit us to observe only a subset of physical quantities. Furthermore, we show that using this approach we can construct a wide class of physical theories that generalize quantum mechanics.We highlight that the emergence of probabilities in quantum theory can be derived from decoherence caused by adiabatic interaction with a random environment. We underscore that the concept of a particle is not primary in quantum theory. If the theory is translation-invariant we define particles as elementary excitations of the ground state. Quasiparticles are elementary excitations of any translation-invariant state. We analyze the concept of scattering but we do not utilize the concept of a field and do not assume locality and Poincare invariance. We  discuss not only the conventional scattering matrix (related to scattering cross-sections) but also the concept of an inclusive scattering matrix, which is closely related to the concept of inclusive scattering cross-sections. Scattering matrix can be expressed in terms of Green’s functions by the well-known formula belonging to Lehmann, Symanczyk, and Zimmermann, and the inclusive scattering matrix can be expressed in terms of generalized Green’s functions, which first appeared in nonequilibrium statistical physics in Keldysh formalism. As a concrete realization of the geometric approach, we describe the formalism of L-functionals where states are represented by non-linear functionals corresponding to positive functionals on Weyl and Clifford algebras (to states in the algebraic approach). L-functionals can be applied to solve the infrared problem in quantum electrodynamics.