The Loewner energy for Jordan curves first arises from the large deviations of Schramm-Loewner evolution (SLE), a family of random fractal curves modeling interfaces in 2D statistical mechanics. In a certain way, this energy measures the roundness of a Jordan curve, and we show that it is finite if and only if the curve is a Weil-Petersson quasicircle and connect it to determinants of Laplacians. Furthermore, the Loewner energy is a Kahler potential on the Weil-Petersson Teichmueller space identified with the space of Weil-Petersson quasicircles. Intriguingly, this class of finite energy curves has more than 20 equivalent definitions arising in very different contexts, including Teichmueller theory, geometric function theory, hyperbolic geometry, spectral theory, and has been studied since the eighties with motivations from string theory. The myriad of perspectives on this class of curves is both luxurious and mysterious. I will overview the links between Loewner energy and SLE, Weil-Petersson quasicircles, and other branches of mathematics it touches on. I will highlight how ideas from random conformal geometry inspire new results on Weil-Petersson quasicircles and discuss further directions.​

The Loewner energy for Jordan curves first arises from the large deviations of Schramm-Loewner evolution (SLE), a family of random fractal curves modeling interfaces in 2D statistical mechanics. In a certain way, this energy measures the roundness of a Jordan curve, and we show that it is finite if and only if the curve is a Weil-Petersson quasicircle and connect it to determinants of Laplacians. Furthermore, the Loewner energy is a Kahler potential on the Weil-Petersson Teichmueller space identified with the space of Weil-Petersson quasicircles. Intriguingly, this class of finite energy curves has more than 20 equivalent definitions arising in very different contexts, including Teichmueller theory, geometric function theory, hyperbolic geometry, spectral theory, and has been studied since the eighties with motivations from string theory. The myriad of perspectives on this class of curves is both luxurious and mysterious. I will overview the links between Loewner energy and SLE, Weil-Petersson quasicircles, and other branches of mathematics it touches on. I will highlight how ideas from random conformal geometry inspire new results on Weil-Petersson quasicircles and discuss further directions.​

The Loewner energy for Jordan curves first arises from the large deviations of Schramm-Loewner evolution (SLE), a family of random fractal curves modeling interfaces in 2D statistical mechanics. In a certain way, this energy measures the roundness of a Jordan curve, and we show that it is finite if and only if the curve is a Weil-Petersson quasicircle and connect it to determinants of Laplacians. Furthermore, the Loewner energy is a Kahler potential on the Weil-Petersson Teichmueller space identified with the space of Weil-Petersson quasicircles. Intriguingly, this class of finite energy curves has more than 20 equivalent definitions arising in very different contexts, including Teichmueller theory, geometric function theory, hyperbolic geometry, spectral theory, and has been studied since the eighties with motivations from string theory. The myriad of perspectives on this class of curves is both luxurious and mysterious. I will overview the links between Loewner energy and SLE, Weil-Petersson quasicircles, and other branches of mathematics it touches on. I will highlight how ideas from random conformal geometry inspire new results on Weil-Petersson quasicircles and discuss further directions.​

The Loewner energy for Jordan curves first arises from the large deviations of Schramm-Loewner evolution (SLE), a family of random fractal curves modeling interfaces in 2D statistical mechanics. In a certain way, this energy measures the roundness of a Jordan curve, and we show that it is finite if and only if the curve is a Weil-Petersson quasicircle and connect it to determinants of Laplacians. Furthermore, the Loewner energy is a Kahler potential on the Weil-Petersson Teichmueller space identified with the space of Weil-Petersson quasicircles. Intriguingly, this class of finite energy curves has more than 20 equivalent definitions arising in very different contexts, including Teichmueller theory, geometric function theory, hyperbolic geometry, spectral theory, and has been studied since the eighties with motivations from string theory. The myriad of perspectives on this class of curves is both luxurious and mysterious. I will overview the links between Loewner energy and SLE, Weil-Petersson quasicircles, and other branches of mathematics it touches on. I will highlight how ideas from random conformal geometry inspire new results on Weil-Petersson quasicircles and discuss further directions.​

Numerical evidence suggests that the Random Field Ising Model loses Parisi-Sourlas SUSY and the dimensional reduction property somewhere between 4 and 5 dimensions, while a related model of branched polymers retains these features in any d. I will present a recent theory, developed in 2019-2021 jointly with A. Kaviraj and E. Trevisani and published in [1-4], which aims to explain these facts.

Outline:
1. Random Field Ising Model: phase diagram, well-established facts and experiments.
2. Numerical results for the dimensional reduction of critical exponents: “no” for d=3,4, “yes” for d=5.
3. Parisi-Sourlas supersymmetry implies dimensional reduction
4. Generalities about RG fixed point disappearance
5. Loss of Parisi-Sourlas SUSY via dangerously irrelevant operators?
6. Replica field theory. Cardy field transform “derivation » of Parisi-Sourlas SUSY and its potential loopholes.
7. Replica symmetric interactions in the Cardy basis
8. Leader and follower interactions
9. Classification of leaders
10. Anomalous dimension computations and results. Evidence for the SUSY fixed point instability below ~4.5
11. Future directions and open problems.

Literature:
[1] A. Kaviraj, S. Rychkov, E. Trevisani, « Random Field Ising Model and Parisi-Sourlas Supersymmetry I. Supersymmetric CFT, » [arXiv:1912.01617] JHEP 2004 (2020) 090
[2] A. Kaviraj, S. Rychkov, E. Trevisani, « Random Field Ising Model and Parisi-Sourlas Supersymmetry II. Renormalization Group”, [arXiv:2009.10087] JHEP 03 (2021) 219
[3] A. Kaviraj, S. Rychkov, E. Trevisani, « The fate of Parisi-Sourlas supersymmetry in Random Field models”, [arXiv:2112.06942] Phys.Rev.Lett. 129 (2022) 045701
[4] A. Kaviraj, E. Trevisani, « Random Field φ^3 Model and Parisi-Sourlas Supersymmetry », [arXiv:2203.12629] JHEP 08 (2022) 290

Numerical evidence suggests that the Random Field Ising Model loses Parisi-Sourlas SUSY and the dimensional reduction property somewhere between 4 and 5 dimensions, while a related model of branched polymers retains these features in any d. I will present a recent theory, developed in 2019-2021 jointly with A. Kaviraj and E. Trevisani and published in [1-4], which aims to explain these facts.

Outline:
1. Random Field Ising Model: phase diagram, well-established facts and experiments.
2. Numerical results for the dimensional reduction of critical exponents: “no” for d=3,4, “yes” for d=5.
3. Parisi-Sourlas supersymmetry implies dimensional reduction
4. Generalities about RG fixed point disappearance
5. Loss of Parisi-Sourlas SUSY via dangerously irrelevant operators?
6. Replica field theory. Cardy field transform “derivation » of Parisi-Sourlas SUSY and its potential loopholes.
7. Replica symmetric interactions in the Cardy basis
8. Leader and follower interactions
9. Classification of leaders
10. Anomalous dimension computations and results. Evidence for the SUSY fixed point instability below ~4.5
11. Future directions and open problems.

Literature:
[1] A. Kaviraj, S. Rychkov, E. Trevisani, « Random Field Ising Model and Parisi-Sourlas Supersymmetry I. Supersymmetric CFT, » [arXiv:1912.01617] JHEP 2004 (2020) 090
[2] A. Kaviraj, S. Rychkov, E. Trevisani, « Random Field Ising Model and Parisi-Sourlas Supersymmetry II. Renormalization Group”, [arXiv:2009.10087] JHEP 03 (2021) 219
[3] A. Kaviraj, S. Rychkov, E. Trevisani, « The fate of Parisi-Sourlas supersymmetry in Random Field models”, [arXiv:2112.06942] Phys.Rev.Lett. 129 (2022) 045701
[4] A. Kaviraj, E. Trevisani, « Random Field φ^3 Model and Parisi-Sourlas Supersymmetry », [arXiv:2203.12629] JHEP 08 (2022) 290

Numerical evidence suggests that the Random Field Ising Model loses Parisi-Sourlas SUSY and the dimensional reduction property somewhere between 4 and 5 dimensions, while a related model of branched polymers retains these features in any d. I will present a recent theory, developed in 2019-2021 jointly with A. Kaviraj and E. Trevisani and published in [1-4], which aims to explain these facts.

Outline:
1. Random Field Ising Model: phase diagram, well-established facts and experiments.
2. Numerical results for the dimensional reduction of critical exponents: “no” for d=3,4, “yes” for d=5.
3. Parisi-Sourlas supersymmetry implies dimensional reduction
4. Generalities about RG fixed point disappearance
5. Loss of Parisi-Sourlas SUSY via dangerously irrelevant operators?
6. Replica field theory. Cardy field transform “derivation » of Parisi-Sourlas SUSY and its potential loopholes.
7. Replica symmetric interactions in the Cardy basis
8. Leader and follower interactions
9. Classification of leaders
10. Anomalous dimension computations and results. Evidence for the SUSY fixed point instability below ~4.5
11. Future directions and open problems.

Literature:
[1] A. Kaviraj, S. Rychkov, E. Trevisani, « Random Field Ising Model and Parisi-Sourlas Supersymmetry I. Supersymmetric CFT, » [arXiv:1912.01617] JHEP 2004 (2020) 090
[2] A. Kaviraj, S. Rychkov, E. Trevisani, « Random Field Ising Model and Parisi-Sourlas Supersymmetry II. Renormalization Group”, [arXiv:2009.10087] JHEP 03 (2021) 219
[3] A. Kaviraj, S. Rychkov, E. Trevisani, « The fate of Parisi-Sourlas supersymmetry in Random Field models”, [arXiv:2112.06942] Phys.Rev.Lett. 129 (2022) 045701
[4] A. Kaviraj, E. Trevisani, « Random Field φ^3 Model and Parisi-Sourlas Supersymmetry », [arXiv:2203.12629] JHEP 08 (2022) 290

Numerical evidence suggests that the Random Field Ising Model loses Parisi-Sourlas SUSY and the dimensional reduction property somewhere between 4 and 5 dimensions, while a related model of branched polymers retains these features in any d. I will present a recent theory, developed in 2019-2021 jointly with A. Kaviraj and E. Trevisani and published in [1-4], which aims to explain these facts.

Outline:
1. Random Field Ising Model: phase diagram, well-established facts and experiments.
2. Numerical results for the dimensional reduction of critical exponents: “no” for d=3,4, “yes” for d=5.
3. Parisi-Sourlas supersymmetry implies dimensional reduction
4. Generalities about RG fixed point disappearance
5. Loss of Parisi-Sourlas SUSY via dangerously irrelevant operators?
6. Replica field theory. Cardy field transform “derivation » of Parisi-Sourlas SUSY and its potential loopholes.
7. Replica symmetric interactions in the Cardy basis
8. Leader and follower interactions
9. Classification of leaders
10. Anomalous dimension computations and results. Evidence for the SUSY fixed point instability below ~4.5
11. Future directions and open problems.

Literature:
[1] A. Kaviraj, S. Rychkov, E. Trevisani, « Random Field Ising Model and Parisi-Sourlas Supersymmetry I. Supersymmetric CFT, » [arXiv:1912.01617] JHEP 2004 (2020) 090
[2] A. Kaviraj, S. Rychkov, E. Trevisani, « Random Field Ising Model and Parisi-Sourlas Supersymmetry II. Renormalization Group”, [arXiv:2009.10087] JHEP 03 (2021) 219
[3] A. Kaviraj, S. Rychkov, E. Trevisani, « The fate of Parisi-Sourlas supersymmetry in Random Field models”, [arXiv:2112.06942] Phys.Rev.Lett. 129 (2022) 045701
[4] A. Kaviraj, E. Trevisani, « Random Field φ^3 Model and Parisi-Sourlas Supersymmetry », [arXiv:2203.12629] JHEP 08 (2022) 290

Let G be a finitely generated group acting faithfully by linear transformations on a finite-dimensional complex vector space. The theorems of Malcev, Selberg, or Tits provide important properties satisfied by G. To what extent do these properties continue to hold when G is acting by polynomial (instead of linear) transformations? In order to address this question, I shall describe a few results that illustrate how one can use p-adic or finite fields for problems which are initially phrased in terms of complex numbers. 

Let Xk,d denote the space of rank-k lattices in Rd. Topological and statistical properties of the dynamics of discrete subgroups of G=SL(d,R) on Xd,d were described in the seminal works of Benoist-Quint. A key step/result in this study is the classification of stationary measures on Xd,d. Later, Sargent-Shapira initiated the study of dynamics on the spaces Xk,d. When k ≠ d, the space Xk,d is of a different nature and a clear description of dynamics on these spaces is far from being established. Given a probability measure μ which is Zariski-dense in a copy of SL(2,R) in G, we give a classification of μ-stationary measures on Xk,d and prove corresponding equidistribution results. In contrast to the results of Benoist-Quint, the type of stationary measures that μ admits depends strongly on the position of SL(2,R) relative to parabolic subgroups of G. I will start by reviewing preceding major works and ideas. The talk will be accessible to a broad audience. Joint work with Alexander Gorodnik and Jialun Li. 

We approximate boundaries of convex polytopes  by smooth hypersurfaces with positive mean curvatures and, by using basic geometric relations between the scalar curvatures of Riemannian manifolds and the mean curvatures of their boundaries, establish lower bound on the dihedral angles of these polytopes.

Port du masque obligatoire / Mask mandatory

Institut Des Hautes Etudes Scientifiques vous invite à une réunion Zoom planifiée.

Sujet : Séminaire de mathématique : M. Gromov
Heure : 27 juin 2022 02:00 PM Paris

Participer à la réunion Zoom
https://us02web.zoom.us/j/87455718639?pwd=oDF-dSYQd8Sevw2yEd1AVx4-HzxrBF.1

ID de réunion : 874 5571 8639
Code secret : 052856

 

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 »
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LSC, which stands for Life, Structure, and Cognition, is an initiative with the aim of learning about the latest advancements and of exploring the modalities for cooperative progress between Biology and Artificial Intelligence, as is already starting to take place under our eyes. The goal is to investigate how biological systems evolved cognitive functions, using AI both as a concept toolbox and as a tool, and in turn, investigate how biological systems can serve as models to understand and stimulate new progress in AI.

The idea of LSC is an annual evaluation, expert discussion, and lecture series on the key topics and latest advancements in these directions, thus the Life and Cognition components of LSC. Structure in this LSC epithet refers to Mathematics and how it might and must help process and formalize aspects of these advancements. 

LSC 2022 Webinars

LSC 2022 Meeting

ORGANIZING COMMITTEE:

– Yves Barral (ETH Zürich)
– Mikhail Gromov (IHES, Université Paris-Saclay)
– Robert Penner (IHES, Université Paris-Saclay)
– Vasily Pestun (IHES, Université Paris-Saclay)

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