For the full schedule please visit: https://www.simonsfoundation.org/event/mps-conference-on-higher-dimensional-geometry-may-6-10-2024/
Title: Equivariant Lagrangian Floer Theory on Compact Toric Manifolds
Speaker: Yao Xiao [Stony Brook University]
Abstract: We define an equivariant Lagrangian Floer theory on compact symplectic toric manifolds for the subtorus actions. We prove that the set of Lagrangian torus fibers (with weak bounding cochain data) with non-vanishing equivariant Lagrangian Floer cohomology forms a rigid analytic space. We can apply tropical geometry to locate such Lagrangian torus fibers in the moment polytope. We prove, in certain cases, that the dimension of such a rigid analytic space is equal to that of the acting group. In addition, we apply equivariant theory to show that moment Lagrangian correspondences induced by symplectic reduction are unobstructed after bulk deformation, assuming the existence of certain equivariant Kuranishi structures and compatible equivariant CF-perturbations.
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Speaker: Assaf Naor (Princeton University)
Title: Unplanned consequences of the Ribe program, part I: Introduction
Abstract: Almost 50 years ago, Martin Ribe proved a remarkable geometric rigidity theorem for normed spaces. This inspired an intricate web of conjectures and analogies that aims to transfer phenomena, concepts and intuitions from the structured realm of linear spaces to the seemingly uncontrollably diverse world of general metric spaces. While research on this program has led to powerful, creative and deep discoveries, numerous mysteries remain. These talks will start by introducing the audience to the aforementioned research endeavor, which is known today as the “Ribe program,” assuming no prior knowledge of it. This program naturally enhanced our understanding of the structure of normed spaces, and we will indeed present examples of this, but our main focus will quickly shift to describing some of its unplanned applications. Namely, we will demonstrate how it informs us about objects that are further afield, such as groups, curvature, algorithms and probability.
Title: Lecture 1: Unplanned consequences of the Ribe program, part I: Introduction
Speaker: Assaf Naor [Princeton University]
Abstract: Almost 50 years ago, Martin Ribe proved a remarkable geometric rigidity theorem for normed spaces. This inspired an intricate web of conjectures and analogies that aims to transfer phenomena, concepts and intuitions from the structured realm of linear spaces to the seemingly uncontrollably diverse world of general metric spaces. While research on this program has led to powerful, creative and deep discoveries, numerous mysteries remain. These talks will start by introducing the audience to the aforementioned research endeavor, which is known today as the “Ribe program,” assuming no prior knowledge of it. This program naturally enhanced our understanding of the structure of normed spaces, and we will indeed present examples of this, but our main focus will quickly shift to describing some of its unplanned applications. Namely, we will demonstrate how it informs us about objects that are further afield, such as groups, curvature, algorithms and probability.
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Title: Wave turbulence and quantum field theory
Abstract: Wind blows over the ocean, exciting long wavelength waves, whose energy then cascades to shorter wavelength waves. The state is statistically stationary and the measured spectrum of energy per mode is a power law, over some range of scales. At very short scales there is sea foam (whitecaps), and the spectrum is believed to again be power law, but a different power. At long scales the nonlinearity is weak (if the wind is not too strong and the waves are small) and the spectrum can be derived analytically. At short scales the nonlinearity is strong, and one loses analytic control. Wave turbulence has been studied theoretically and experimentally in a wide range of systems for half a century. To date, all theoretical results have been at leading order in the nonlinearity.
We demonstrate how wave turbulence — a stochastic classical system — can be turned into a quantum field theory. The computation of the spectrum becomes a problem of computing correlation functions. This gives a scheme for computing beyond leading order in the nonlinearity. We consider wave turbulence in a large N system, allowing us to study strong wave turbulence. We develop the analog of the epsilon expansion, allowing us to go from one power law spectrum in the UV to a slightly different power law in the IR (the analog of flow between critical points).
Speaker: Assaf Naor (Princeton University)
Title: Unplanned consequences of the Ribe program, part II: Obstructions
Abstract: Almost 50 years ago, Martin Ribe proved a remarkable geometric rigidity theorem for normed spaces. This inspired an intricate web of conjectures and analogies that aims to transfer phenomena, concepts and intuitions from the structured realm of linear spaces to the seemingly uncontrollably diverse world of general metric spaces. While research on this program has led to powerful, creative and deep discoveries, numerous mysteries remain. These talks will start by introducing the audience to the aforementioned research endeavor, which is known today as the “Ribe program,” assuming no prior knowledge of it. This program naturally enhanced our understanding of the structure of normed spaces, and we will indeed present examples of this, but our main focus will quickly shift to describing some of its unplanned applications. Namely, we will demonstrate how it informs us about objects that are further afield, such as groups, curvature, algorithms and probability.
Title: Lecture 2: Unplanned consequences of the Ribe program, part II: Obstructions
Speaker: Assaf Naor [Princeton University]
Abstract: Almost 50 years ago, Martin Ribe proved a remarkable geometric rigidity theorem for normed spaces. This inspired an intricate web of conjectures and analogies that aims to transfer phenomena, concepts and intuitions from the structured realm of linear spaces to the seemingly uncontrollably diverse world of general metric spaces. While research on this program has led to powerful, creative and deep discoveries, numerous mysteries remain. These talks will start by introducing the audience to the aforementioned research endeavor, which is known today as the “Ribe program,” assuming no prior knowledge of it. This program naturally enhanced our understanding of the structure of normed spaces, and we will indeed present examples of this, but our main focus will quickly shift to describing some of its unplanned applications. Namely, we will demonstrate how it informs us about objects that are further afield, such as groups, curvature, algorithms and probability.
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ANNOUNCING THE 2023-2024 CONTEST WINNERS:
Katacala Nights by Gino Elia
STAGED READINGS ARE WEDNESDAY, MAY 8, 2024, 5:00 pm, Simons Center for Geometry & Physics
Speaker: Assaf Naor (Princeton University)
Title: Unplanned consequences of the Ribe program, part III: Structure
Abstract: Almost 50 years ago, Martin Ribe proved a remarkable geometric rigidity theorem for normed spaces. This inspired an intricate web of conjectures and analogies that aims to transfer phenomena, concepts and intuitions from the structured realm of linear spaces to the seemingly uncontrollably diverse world of general metric spaces. While research on this program has led to powerful, creative and deep discoveries, numerous mysteries remain. These talks will start by introducing the audience to the aforementioned research endeavor, which is known today as the “Ribe program,” assuming no prior knowledge of it. This program naturally enhanced our understanding of the structure of normed spaces, and we will indeed present examples of this, but our main focus will quickly shift to describing some of its unplanned applications. Namely, we will demonstrate how it informs us about objects that are further afield, such as groups, curvature, algorithms and probability.
Title: Lecture 3: Unplanned consequences of the Ribe program, part III: Structure
Speaker: Assaf Naor [Princeton University]
Abstract: Almost 50 years ago, Martin Ribe proved a remarkable geometric rigidity theorem for normed spaces. This inspired an intricate web of conjectures and analogies that aims to transfer phenomena, concepts and intuitions from the structured realm of linear spaces to the seemingly uncontrollably diverse world of general metric spaces. While research on this program has led to powerful, creative and deep discoveries, numerous mysteries remain. These talks will start by introducing the audience to the aforementioned research endeavor, which is known today as the “Ribe program,” assuming no prior knowledge of it. This program naturally enhanced our understanding of the structure of normed spaces, and we will indeed present examples of this, but our main focus will quickly shift to describing some of its unplanned applications. Namely, we will demonstrate how it informs us about objects that are further afield, such as groups, curvature, algorithms and probability.
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Title: TBA
Speaker: Mohamad Rabah [Stony Brook University]
Abstract: TBA
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Title: Singularities of geometric flows, a non-expert introduction
Speaker: Maxim Jeffs [Stony Brook University]
Abstract: Using a partial differential equation to 'flow' one geometric structure towards a 'more canonical' one is a powerful idea in many fields of differential geometry. Unfortunately solutions to these equations are usually not guaranteed to exist for all time: 'singularities' can form, which can then be 'removed' with some kind of 'surgery' operation. I'm not an expert in this field but I'll try to give a down-to-earth explanation of what the terms in quotation marks mean, illustrated with some examples I've managed to understand. I'll finish with some wild speculation about how this relates to various open problems in symplectic geometry.
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Title: SYMPLECTIC CAPACITIES, VITERBO ISOPERIMETRIC CONJECTURE, AND CONTACT MANIFOLDS ALL OF WHOSE REEB ORBITS ARE CLOSED
Speaker: Marco Mazzucchelli [Ecole normale superieure de Lyon]
Abstract: Symplectic capacities are fundamental invariants that govern many rigidity phenomena in symplectic and contact topology. Their introduction in the 1980s by Ekeland and Hofer was motivated by the celebrated Gromov's non-squeezing theorem: a round ball in the symplectic vector space does not symplectically embed into a symplectic cylinder of smaller radius. A conjecture due to Viterbo from the early 2000s asserts that, among the 2n-dimensional convex bodies of volume one, the round balls are the ones with the largest capacity. In this colloquium talk, I will provide an informal and general overview of some developments in symplectic geometry related to the Viterbo conjecture, including its application to convex geometry, and the study of contact manifolds all of whose Reeb orbits are closed.
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Title: GLOBAL SURFACES OF SECTION FOR REEB FLOWS ON CLOSED 3-MANIFOLDS
Speaker: Marco Mazzucchelli [ENS Lyon]
Abstract: Global surfaces of section are tools that allow to reduce the study of the dynamics of a nowhere vanishing vector field on a 3-manifold to the study of a surface diffeomorphism. Poincaré introduced global surfaces of sections in his study of the planar circular restricted 3-body problem from celestial mechanics. While certain vector fields (such as the one generating the horocycle flow) do not admit a global surface of section, it is conjectured that all Reeb flows of 3-dimensional closed manifolds do. In this talk, which is based on joint work with Gonzalo Contreras, I will sketch a proof of this conjecture for the Reeb flows of C^\infty-generic contact forms on any closed 3-manifold. This generic family includes all contact forms whose Reeb flows are non-degenerate and satisfy the Kupka-Smale transversality condition.
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Title: Separability as a window into many-body mixed-state phases
Abstract: Ground states as well as Gibbs states of many-body quantum Hamiltonians have been studied extensively for some time. In contrast, the landscape of mixed states that do not correspond to a system in thermal equilibrium is relatively less explored. In this talk I will explore a rather coarse classification of mixed quantum many-body states. The key idea I will employ is that of "separability", i.e., whether a mixed state can be expressed as an ensemble of short-range entangled pure states. I will discuss several examples of decoherence-driven phase transitions from a separability viewpoint, and argue that such a framework also provides a new view on Gibbs states.
Based on work with Yu-Hsueh Chen. References: 2309.11879, 2310.07286, 2403.06553.