Dimension Dependence of Critical Phenomena in Percolation (3/6)
Atttention : Les deux premières Leçons auront lieu à l’IMO, Amphithéâtre Yoccoz, le 16 mai à 10h et à 14h
Retrouvez toutes ces informations sur le site de la Fondation Mathématique Jacques Hadamard :
https://www.fondation-hadamard.fr/fr/evenements/cours-avances/
Abstract:
In Bernoulli bond percolation, we delete or retain each edge of a graph independently at random with some retention parameter p and study the geometry of the connected components (clusters) of the resulting subgraph. For lattices of dimension d>1, percolation has a phase transition, with a infinite cluster emerging at a critical probability pc(d). It is believed that critical percolation at and near the critical probability exhibits rich, fractal-like geometry that is expected to be approximately independent of the choice of lattice but highly dependent on the dimension d. In particular, various qualitative distinctions are expected between the low dimensional case d<6, the high-dimensional case d>6, and the critical case d=6, but this remains poorly understood particularly in dimensions d=3,4,5,6.In this course, I will give an overview of of what is known about critical percolation, focussing on recent advances in long-range and hierarchical models for which various aspects of intermediate-dimensional critical phenomena can now be understood rigorously.
Dimension Dependence of Critical Phenomena in Percolation (4/6)
Atttention : Les deux premières Leçons auront lieu à l’IMO, Amphithéâtre Yoccoz, le 16 mai à 10h et à 14h
Retrouvez toutes ces informations sur le site de la Fondation Mathématique Jacques Hadamard :
https://www.fondation-hadamard.fr/fr/evenements/cours-avances/
Abstract:
In Bernoulli bond percolation, we delete or retain each edge of a graph independently at random with some retention parameter p and study the geometry of the connected components (clusters) of the resulting subgraph. For lattices of dimension d>1, percolation has a phase transition, with a infinite cluster emerging at a critical probability pc(d). It is believed that critical percolation at and near the critical probability exhibits rich, fractal-like geometry that is expected to be approximately independent of the choice of lattice but highly dependent on the dimension d. In particular, various qualitative distinctions are expected between the low dimensional case d<6, the high-dimensional case d>6, and the critical case d=6, but this remains poorly understood particularly in dimensions d=3,4,5,6.In this course, I will give an overview of of what is known about critical percolation, focussing on recent advances in long-range and hierarchical models for which various aspects of intermediate-dimensional critical phenomena can now be understood rigorously.
Dimension Dependence of Critical Phenomena in Percolation (5/6)
Atttention : Les deux premières Leçons auront lieu à l’IMO, Amphithéâtre Yoccoz, le 16 mai à 10h et à 14h
Retrouvez toutes ces informations sur le site de la Fondation Mathématique Jacques Hadamard :
https://www.fondation-hadamard.fr/fr/evenements/cours-avances/
Abstract:
In Bernoulli bond percolation, we delete or retain each edge of a graph independently at random with some retention parameter p and study the geometry of the connected components (clusters) of the resulting subgraph. For lattices of dimension d>1, percolation has a phase transition, with a infinite cluster emerging at a critical probability pc(d). It is believed that critical percolation at and near the critical probability exhibits rich, fractal-like geometry that is expected to be approximately independent of the choice of lattice but highly dependent on the dimension d. In particular, various qualitative distinctions are expected between the low dimensional case d<6, the high-dimensional case d>6, and the critical case d=6, but this remains poorly understood particularly in dimensions d=3,4,5,6.In this course, I will give an overview of of what is known about critical percolation, focussing on recent advances in long-range and hierarchical models for which various aspects of intermediate-dimensional critical phenomena can now be understood rigorously.
Dimension Dependence of Critical Phenomena in Percolation (6/6)
Atttention : Les deux premières Leçons auront lieu à l’IMO, Amphithéâtre Yoccoz, le 16 mai à 10h et à 14h
Retrouvez toutes ces informations sur le site de la Fondation Mathématique Jacques Hadamard :
https://www.fondation-hadamard.fr/fr/evenements/cours-avances/
Abstract:
In Bernoulli bond percolation, we delete or retain each edge of a graph independently at random with some retention parameter p and study the geometry of the connected components (clusters) of the resulting subgraph. For lattices of dimension d>1, percolation has a phase transition, with a infinite cluster emerging at a critical probability pc(d). It is believed that critical percolation at and near the critical probability exhibits rich, fractal-like geometry that is expected to be approximately independent of the choice of lattice but highly dependent on the dimension d. In particular, various qualitative distinctions are expected between the low dimensional case d<6, the high-dimensional case d>6, and the critical case d=6, but this remains poorly understood particularly in dimensions d=3,4,5,6.In this course, I will give an overview of of what is known about critical percolation, focussing on recent advances in long-range and hierarchical models for which various aspects of intermediate-dimensional critical phenomena can now be understood rigorously.
Factorizing Defects from Generalized Pinning Fields
We introduce generalized pinning fields in conformal field theory that model a large class of critical impurities at large distance, enriching the familiar universality classes. We provide a rigorous definition of such defects as certain unbounded operators on the Hilbert space and prove that when inserted on codimension-one surfaces they factorize the spacetime into two halves. The possible factorization channels are further constrained by symmetries in the bulk. As a corollary, we solve such critical impurities in the 2d tricritical Ising model and establish the factorization phenomena previously observed for localized mass deformations in the 3d O(N) model. If time permits, I will also explore the characteristics of these defects, including their density of states and the asymptotic behavior of one-point functions.
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Kernels In-depth: Higher Heun Equations
Running Seminar
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Running Seminar
========
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On the Classicality of the Motivic Galois Group
Séminaire de géométrie arithmétique
The classicality of the motivic Galois group is equivalent to the vanishing of all the higher degree operations on the cohomology of pairs of algebraic varieties. I will present a proof of this in a significant special case, namely the vanishing of degree-1 operations acting on the cohomology of smooth and projective varieties.
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Bootstrapping Dual Model Amplitudes with Regge Theory
Séminaire Amplitudes et Gravitation sur l’Yvette (IHES/IPhT)
Dual model amplitudes are meromorphic amplitudes involving the exchange of infinitely-many higher spin states. They are the staple of tree-level string theory, with the Veneziano and Virasoro-Shapiro amplitudes being their most famous representatives and also describe gauge theories in the large N limit.
Amplitudes of this class must satisfy very constraining consistency conditions, which makes them good candidates for bootstrap approaches. Yet, only recently they are receiving the attention of the S-matrix bootstrap community, partly because many standard S-matrix bootstrap techniques are not well suited for their study.
In this talk I will show some recent progress in the analysis of dual model amplitudes from a bootstrap perspective using techniques from Regge theory of complex angular momentum. After a short motivation/introduction to dual model amplitudes and presenting the main ideas of Regge theory, I will explain how these techniques can be used to study them and showcase some recent results: the proof that dual model amplitudes require infinitely many trajectories to be consistent, and a numerical method to build consistent dual amplitudes with customizable spectrum and high energy behavior.
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Torsion in the Context of the Asymptotic Expansion of Knot Invariants
Running Seminar
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Number Theoretic Aspects of 3-dimensional Topology
Running Seminar
The main aim of the seminar is to construct a bridge between 3D Chern-Simons resurgence techniques on the quantum side and Galois representations, L-functions and Frobenius traces on the classical topology side.
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Nonlinear Static Tidal Deformation of Black Holes
Séminaire Amplitudes et Gravitation sur l’Yvette (IHES/IPhT)
In gravity theories, the conservative tidal deformability of a self-gravitating object is parametrized in terms of a set of coefficients, commonly referred to as Love numbers. At leading order, these coefficients are obtained by solving the linearized static Einstein equations for the perturbations. In the talk, I will discuss the role of field nonlinearities in the calculation of the tidal deformability of compact objects. I will show that an infinite subset of nonlinear Love numbers of black holes is identically zero, a result that extends the well-known vanishing of linear Love numbers to higher orders in perturbation theory. Remarkably, this result follows from a fully nonlinear hidden symmetry structure in general relativity.
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