Recently, classification problems of gapped ground state phases attract a lot of attention in quantum statistical mechanics. We explain about operator algebraic approach to these problems.

 

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Surprisingly, there exist connected components of character varieties of fundamental groups of surfaces in semisimple Lie groups consisting only of injective representations with discrete image. Guichard and Wienhard introduced the notion of Θ-positive representations as a conjectural framework to explain this phenomenon. I will discuss joint work with Jonas Beyrer in which we establish several geometric properties of Θ-positive representations in PO(p,q). As an application, we deduce that they indeed form connected components of character varieties.

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IHES Covid-19 regulations:

– all the participants who will attend the event in person will have to keep their mask on in indoor spaces
and where the social distancing is not possible;
– speakers will be free to wear their mask or not at the moment of their talk if they feel more comfortable
without it;
– Up to 25 persons in the conference room, every participant will be asked to be able to provide a health pass
– Over 25 persons in the conference room, every participant will be asked to provide a health pass which will
be checked at the entrance of the conference room.

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In this talk, we will present some recent results on the twisted dynamical zeta functions of Ruelle and Selberg on compact hyperbolic manifolds, and the Fried conjecture. This conjecture concerns the relation between the value at zero of the twisted Ruelle zeta function and a spectral invariant: the complex-valued analytic torsion.

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In this talk we will focus on discrete subgroups Γ of higher rank Lie groups G whose limit set is a Lipschitz manifold, i.e. locally the graph of a Lipschitz map. This is a not-uncommon feature, verified by Zariski-dense groups, usually caused by stable geometric properties of the embedding Γ → G. We will recall several examples of such groups, specially those coming from Higher rank Teichmüller Theory. The main purpose of the lecture is to explain a recent result, in collaboration with B. Pozzetti and A. Wienhard, where we prove that the critical exponent of a specific combination of eigenvalues (only depending on the dimension of the limit set) is independent of the representation. We will also explore some of its consequences.

I will discuss some questions related to Novikov’s problem for foliations on surfaces.

The problem is the following. Let M be a 3-periodic surface and H a plane intersecting M. Which kind of curves are realized as the intersection of M and H?

This problem was formulated by Sergey Petrovitch Novikov in the 80’s. He conjectured that the « trivial » situations (periodic and integrable) are generic. I will discuss the simplest situation when M is very symmetric. In this case, with Dynnikov and Skripchenko, we prove that the first return of the foliation induced by H is an Arnoux-Rauzy interval exchange transformation. I will give some properties of these maps (results with Arnoux, Cassaigne and Ferenczi). In the most interesting situation, I will mention a work in progress with Dynnikov, Mercat, Paris-Romaskevich and Skripchenko which is supposed to solve Novikov’s conjecture.

The talk will be accessible to a broad audience.

We study finite N aspects of the O(m) × O(N-m) vector model with quartic interactions in general 2 ≤ d ≤ 6 spacetime dimensions. This model has recently been shown to display the phenomenon of persistent symmetry breaking at a perturbative Wilson-Fisher-like fixed point in d=4-ε dimensions. The large rank limit of the bi-conical model displays a conformal manifold and a moduli space of vacua. We find a set of three double trace scalar operators that are respectively irrelevant, relevant and marginal deformations of the conformal manifold in general d. We calculate the anomalous dimensions of the single and multi-trace scalar operators to the first sub-leading order in the large rank expansion. The anomalous dimension of the marginal operator does not vanish in general, indicating that the conformal manifold is lifted at finite N. In the case of equal ranks we are able to derive explicitly the scaling dimensions of various operators as functions of only d.

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IHES Covid-19 regulations:

– all the participants who will attend the event in person will have to keep their mask on in indoor spaces
and where the social distancing is not possible;
– speakers will be free to wear their mask or not at the moment of their talk if they feel more comfortable
without it;
– Up to 25 persons in the conference room, every participant will be asked to be able to provide a health pass
– Over 25 persons in the conference room, every participant will be asked to provide a health pass which will
be checked at the entrance of the conference room.

==================================================================

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A many-body quantum system that is continually monitored by an external observer can be in two distinct dynamical phases, distinguished by whether or not repeated local measurements (throughout the bulk of the system) prevent the build-up of long-range quantum entanglement. I will describe the key features of such “measurement phase transitions” and sketch theoretical approaches to their critical properties that make connections with topics in classical statistical mechanics, such as percolation and disordered magnetism. Finally I will discuss random tensor networks with a tree geometry. These arise in a simple limit of the measurement problem, and they show an entanglement transition that can be solved exactly by a mapping to a problem of traveling waves.

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IHES Covid-19 regulations:

– all the participants who will attend the event in person will have to keep their mask on in indoor spaces
and where the social distancing is not possible;
– speakers will be free to wear their mask or not at the moment of their talk if they feel more comfortable
without it;
– Up to 25 persons in the conference room, every participant will be asked to be able to provide a health pass
– Over 25 persons in the conference room, every participant will be asked to provide a health pass which will
be checked at the entrance of the conference room.

==================================================================

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Scattering amplitudes in quantum field theories have intricate analytic properties as functions of the energies and momenta of the scattered particles. In perturbation theory, their singularities are governed by a set of nonlinear polynomial equations, known as Landau equations, for each individual Feynman diagram. The singularity locus of the associated Feynman integral is made precise with the notion of the Landau discriminant, which characterizes when the Landau equations admit a solution. In order to compute this discriminant, we present approaches from classical elimination theory, as well as a numerical algorithm based on homotopy continuation. These methods allow us to compute Landau discriminants of various Feynman diagrams up to 3 loops, which were previously out of reach. For instance, the Landau discriminant of the envelope diagram is a reducible surface of degree 45 in the three-dimensional space of kinematic invariants. We investigate geometric properties of the Landau discriminant, such as irreducibility, dimension and degree.

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In this talk I will discuss the notion of a GREG — a Group Random Element Generator — which is a generalization of a random walk on a group. Roughly, a Greg is a random sequence of group elements.

Associated with a Greg one obtains a pair of Furstenberg-Poisson boundaries, the spaces of ideal futures and ideal pasts. An important property that a Greg might have is the Asymptotic Past And Future Independence. Gregs satisfying this property, namely Apafic Gregs, are very well behaved. Geodesic Flows in a negatively curved environment, as well as classical random walks on groups, give rise to Apafic Gregs. After surveying the subject, I will focus on linear representations of Gregs and the associated invariant called the Lyapunov spectrum. As it turns out, under mild assumptions the Lyapunov spectrum would be simple and continuously varying.

The talk is based on joint work with Alex Furman.

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IHES Covid-19 regulations:

– all the participants who will attend the event in person will have to keep their mask on in indoor spaces
and where the social distancing is not possible;
– speakers will be free to wear their mask or not at the moment of their talk if they feel more comfortable
without it;
– Up to 25 persons in the conference room, every participant will be asked to be able to provide a health pass
– Over 25 persons in the conference room, every participant will be asked to provide a health pass which will
be checked at the entrance of the conference room.

==================================================================

Recently methods of quantum statistical mechanics have been fruitfully applied to the study of phases of quantum lattice systems at zero temperature. For example, a rigorous definition of a Short-Range Entangled phase of matter has been given and a classification of such phases in one spatial dimension has been achieved. I will discuss some of these developments, focusing on the topology and geometry of the space of Short-Range Entangled states. According to a conjecture of A. Kitaev, these spaces form a loop spectrum in the sense of homotopy theory. This conjecture implies that to any family of Short-Range entangled states in one dimension one can associate a gerbe on the parameter space. I will show how to construct such a gerbe. Thе curvature of this gerbe is a closed 3-form with quantized periods and can be regarded as a higher-dimensional generalization of the curvature of the Berry connection.

https://us02web.zoom.us/j/81778962715?pwd=QnpNS2ErSnBCTWRYUHphd1VMMysyZz09

ID de réunion : 817 7896 2715
Code secret : 800452

Mini-Cours

We show that the naive counts of rational curves in an affine log Calabi-Yau variety U, containing an open algebraic torus, determine in a surprisingly simple way, a family of log Calabi-Yau varieties, as the spectrum of a commutative associative algebra equipped with a multilinear form. This is directly inspired by a very similar conjecture of Gross-Hacking-Keel in mirror symmetry, known as the Frobenius structure conjecture. Although the statement involves only elementary algebraic geometry, our proof employs Berkovich non-archimedean analytic methods. We construct the structure constants of the algebra via counting non-archimedean analytic disks in the analytification of U. We establish various properties of the counting, notably deformation invariance, symmetry, gluing formula and convexity. In the special case when U is a Fock-Goncharov skew-symmetric X-cluster variety, we prove that our algebra generalizes, and in particular gives a direct geometric construction of, the mirror algebra of Gross-Hacking-Keel-Kontsevich. The comparison is proved via a canonical scattering diagram defined by counting infinitesimal non-archimedean analytic cylinders, without using the Kontsevich-Soibelman algorithm. Several combinatorial conjectures of GHKK follow readily from the geometric description. This is joint work with S. Keel; the reference is arXiv:1908.09861. If time permits, I will mention another application of our theory to the study of the moduli space of polarized Calabi-Yau pairs, in a work in progress with P. Hacking and S. Keel. Here is a plan for each session of the mini-course:

1) Motivation and ideas from mirror symmetry, main results.
2) Skeletal curves: a key notion in the theory.
3) Naive counts, tail conditions and deformation invariance.
4) Scattering diagram, comparison with Gross-Hacking-Keel-Kontsevich, applications to cluster algebras, applications to moduli spaces of Calabi-Yau pairs.

Registration is compulsory. Please click on the link below to receive the zoom link and password to join the mini-course online:

https://us02web.zoom.us/meeting/register/tZIvcOCorD8iG9ES5hqURXELfgJhQbXND8N1

Mini-Cours

We show that the naive counts of rational curves in an affine log Calabi-Yau variety U, containing an open algebraic torus, determine in a surprisingly simple way, a family of log Calabi-Yau varieties, as the spectrum of a commutative associative algebra equipped with a multilinear form. This is directly inspired by a very similar conjecture of Gross-Hacking-Keel in mirror symmetry, known as the Frobenius structure conjecture. Although the statement involves only elementary algebraic geometry, our proof employs Berkovich non-archimedean analytic methods. We construct the structure constants of the algebra via counting non-archimedean analytic disks in the analytification of U. We establish various properties of the counting, notably deformation invariance, symmetry, gluing formula and convexity. In the special case when U is a Fock-Goncharov skew-symmetric X-cluster variety, we prove that our algebra generalizes, and in particular gives a direct geometric construction of, the mirror algebra of Gross-Hacking-Keel-Kontsevich. The comparison is proved via a canonical scattering diagram defined by counting infinitesimal non-archimedean analytic cylinders, without using the Kontsevich-Soibelman algorithm. Several combinatorial conjectures of GHKK follow readily from the geometric description. This is joint work with S. Keel; the reference is arXiv:1908.09861. If time permits, I will mention another application of our theory to the study of the moduli space of polarized Calabi-Yau pairs, in a work in progress with P. Hacking and S. Keel. Here is a plan for each session of the mini-course:

1) Motivation and ideas from mirror symmetry, main results.
2) Skeletal curves: a key notion in the theory.
3) Naive counts, tail conditions and deformation invariance.
4) Scattering diagram, comparison with Gross-Hacking-Keel-Kontsevich, applications to cluster algebras, applications to moduli spaces of Calabi-Yau pairs.

Registration is compulsory. Please click on the link below to receive the zoom link and password to join the mini-course online:

https://us02web.zoom.us/meeting/register/tZIvcOCorD8iG9ES5hqURXELfgJhQbXND8N1