Speaker: Dario Martelli
Title: From contact structures to black hole entropy via volume extremization
Abstract: Contact structures arise naturally in supergravity as geometric data characterising supersymmetric solutions. In particular, large classes of supersymmetric black holes and their horizons admit canonical contact structures. I will discuss how the entropy of supersymmetric black holes, and related gravitational objects, is obtained by extremizing appropriate contact volume functionals and their generalisations.
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Speaker: Jun Zhang
Title: Rigidities via contact Hamiltonian Floer homology
Abstract: This talk introduces a contact-geometric analogue of Hamiltonian Floer theory, based on the maximum principle established by Merry and Uljarević. This Floer theory can be regarded as an intermediate step leading to Viterbo’s symplectic homology, but it is built upon a general contact Hamiltonian dynamics that extends beyond classical Reeb dynamics. Various rigidity phenomena in contact geometry can be captured through quantitative invariants derived from this Floer theory. In particular, we will show how it can be used to detect a contact big fiber—a result parallel to the celebrated rigidity phenomenon discovered by Entov and Polterovich in symplectic geometry. This talk is based on joint work with Danijel Djordjević and Igor Uljarević.
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Abstract: In these lectures I will give an overview of the non-perturbative S-matrix Bootstrap program. In the first lecture, I will focus on gapped systems and discuss basic concepts such as dispersion relations and unitarity constraints and explain what we can learn by asking Bootstrap questions. In the second lecture I will consider gapless scattering such as Goldstones and gravitons. First, I will explain how unitarity constraints relax into positivity bounds and how to connect them. Then, I will conclude by reviewing what we have learned so far about graviton scattering in dimensions greater than four.
Speaker: Philip Morrison
Title: A potpourri of geometric structures for fluids and plasmas
Abstract: In this talk I will give a survey of geometric structures and concomitant foliations for a variety of classical dynamical systems that are intended for classical purposes. These include the following:
symplectic maps of planar regions that may or may not satisfy Moser’s twist condition, which describe magnetic field lines in plasma devices and advective transport in fluid flows of geophysical fluid dynamics; flows on finite- and infinite-dimensional Poisson manifolds, applicable, e.g., to ideal fluid flow, magnetohydrodynamics, and collisionless kinetic theories; and the metriplectic formulation, which is the natural geometric setting for thermodynamically consistent systems. A relationship between metriplectic flows and contact structures will be described and means for preserving structure in numerics, e.g., so-called symplectic and Poisson integrators, and recent work on metriplectic integrators.
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Speaker: Friedrich Bauermeister
Title: Topological consequences of null-geodesic refocusing and applications to $Z^x$ manifolds
Abstract: Let $(M,h)$ be a connected, complete Riemannian manifold, let $x\in M$. Then $M$ is called a $Z^x$ manifold if all geodesics starting at $x$ return to $x$. We define a class of globally hyperbolic spacetimes (called observer-refocusing) such that the spacetime $(M\times \R, h - dt^2)$ is observer-refocusing for every $Z^x$ manifold $(M,h)$. Studying the Cauchy surfaces of observer-refocusing spacetime enables the use of Lorentzian techniques to study $Z^x$ manifolds as a special case. This is analogous to the past study of $Y^x_l$ manifolds as Cauchy surfaces of strongly refocusing spacetimes. We end by stating contact-theoretic conjectures analogous to the results in Riemannian and Lorentzian geometry.
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Speaker: Mohammed Abouzaid
Title: TBA
Abstract: TBA
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Speaker: Jakob Hedicke (Radboud University, Nijmegen)
Title: On the non-orderability of contact manifolds with sub-critical Weinstein filling
Abstract: Since the seminal work of Eliashberg and Polterovich it is known, that the universal cover of the group of contactomorphisms of a closed contact manifold admits a natural invariant cone structure. An important question in contact topology concerns the causality of this cone structure, i.e., the non-existence of contractible positive loops of contactomorphisms. In this talk we show that contractible positive loops always exist in the case of contact manifolds with a sub-critical Weinstein filling. The talk is based on joint work with Egor Shelukhin.
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Speaker: Nikita Nekrasov
Title: Geometric ideal hydrodynamics
Abstract: We reformulate the equations of Lichnerowicz and Carter describing ideal relativistic fluid as intersection problem in the auxiliary infinite dimensional symplectic manifold associated with four dimensional manifold (spacetime), making contact with the Poisson sigma model. Kontsevich used the latter to solve another problem originally posed by Lichnerowicz and collaborators: deformation quantization of Poisson manifolds.
Our approach separates the smooth structure of spacetime from its metric structure and the chemical composition of the fluid, similar to the way conformal blocks in two dimensional conformal field theory separate left- and right-moving degrees of freedom, complex structure and monodromy data. Joint work with P.Wiegmann
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Title: Symmetry origin of quantum-classical transition, hydrodynamics, decodability
Abstract: We address the following question: when a quantum system evolves into a classical one, can this evolution exhibit a sharp transition? We will show that strong-to-weak spontaneous symmetry breaking (SW-SSB) provides a sharp onset of classical behavior. We will present the theoretical framework for SW-SSB and summarize ongoing experimental progress. Finally, we will discuss key consequences of SW-SSB, including the emergence of hydrodynamics and its information-theoretic implications, such as a transition in decodability and distinguishability.
Speaker: Filip Brocic
Title: Arnold’s chord conjecture in cotangent bundles and the three-body problem
Abstract: In this talk, I will present two results of similar flavor. The first concerns Arnold’s chord conjecture for conormal lifts of submanifolds of the base of a cotangent bundle, in joint work with Dylan Cant and Egor Shelukhin. The second concerns periodic orbits and Reeb chords in the circular restricted three-body problem, in joint work with Urs Frauenfelder. Both results are obtained via an analysis of wrapped Floer cohomology. In the first case, we choose a suitable local system to extract homotopical information from homology and deduce the existence of Reeb chords. In the second case, we use the invariance of wrapped Floer cohomology under subcritical handle attachment, together with results on the regularization of the energy hypersurface below and slightly above the first critical value of the Hamiltonian in rotating coordinates.
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Speaker: Pierre-Alexandre Arlove
Title: Contact non-squeezing in various closed prequantizations
Abstract: I will describe and argue the existence of contact non-squeezing phenomena in contact lens spaces and in strongly orderable prequantizations.
The proof is based on the construction of contact capacities coming from spectral selectors defined on the contactomorphisms group of the latter contact manifolds. I will define all these notions during my talk.
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Speaker: Igor Uljarevic (Belgrade)
Title: Contact big fibre theorem and non-squeezing
Abstract: This talk will provide an overview of recent results on contact rigidity, focusing on contact non-squeezing and contact big fiber theorems. I will describe the main ideas and techniques underlying these results. The talk is based on joint work with Yuhan Sun and Umut Varolgunes, and with Danijel Djordjevic and Jun Zhang.
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Abstract: In these lectures I will give an overview of the non-perturbative S-matrix Bootstrap program. In the first lecture, I will focus on gapped systems and discuss basic concepts such as dispersion relations and unitarity constraints and explain what we can learn by asking Bootstrap questions. In the second lecture I will consider gapless scattering such as Goldstones and gravitons. First, I will explain how unitarity constraints relax into positivity bounds and how to connect them. Then, I will conclude by reviewing what we have learned so far about graviton scattering in dimensions greater than four.
Speaker: Lukas Nakamura
Title: The size of an overtwisted disk.
Abstract: In this talk, I will introduce a notion of size of overtwisted disks in strict contact manifolds that is defined via the contact Hofer distance of Legendrians in the complement. I will show that one can obtain quantitative refinements of known flexibility results in contact topology in terms of the size of an overtwisted disk: I will explain how to obtain upper bounds on the contact Hofer distance of Legendrians in the complement of overtwisted disks, and I will discuss when a product of an overtwisted contact manifold with an exact symplectic manifold is again overtwisted. Part of this talk is based on joint work with Álvaro del Pino Gómez.
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Speaker: Michael Sullivan
Title: Higher Legendrian Torsion
Abstract: With the language of generating families, I'll use the higher
torsion from algebraic topology to construct a Legendrian submanifold
invariant in the standard contact one-jet space. I'll also propose a
``restricted" higher-dimensional version of rulings for such Legendrians.
If the Legendrian is the lift of a nearby Lagrangian with such a ruling,
then the torsion is trivial. (Although the torsion is non-trivial in
general, any Legendrian which is the lift of a nearby Lagrangian and with
non-trivial torsion would disprove the Nearby Lagrangian Conjecture.) This
is joint work in progress with Dani Alvarez-Gavela and Kiyoshi Igusa.
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Schedule of talks
*All talks will take place in SCGP Room 313*
11:100AM Opening Discussion
11:15AM: Talk by Dominik Schmid
12:00PM: Talk by Hindy Drillick
1:00PM: Lunch Break
2:00PM: Talk by Roger Van Peski
2:45PM: Closing Discussion
Speaker: Olaf Muller
Title: Holography at Cauchy sets
Abstract: This talk explores the question to which extent data at a Cauchy set encode the geometry of Lorentzian spaces. If we restrict to classical Cauchy developments of spacetimes, this question has been settled a long time ago, but if we relax one of the hypotheses, many questions remain unsolved as of today. Relaxing the condition of classical Cauchy development leads to the question whether we can identify general globally hyperbolic spacetimes with subsets of a Cauchy set or its tangent bundle. A fascinating partial answer in terms of contact geometry has been found by the answer to the Low conjecture by Chernov and Nemirowski, but several questions remain open in this context. A second line of research can be opened by relaxing the
spacetime hypothesis, admitting synthetic Lorentzian spaces instead. Here there are several caveats in the definition of maximal Cauchy developments, some of which we will discuss in the talk. We will also consider interesting cross-connections between these two lines of research, and, finally, an application to Kruskal spacetime.
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Speaker: Ivan Dynnikov
Title: Bi-Lipschitz contactomorphisms
Abstract: I will speak about Maxim Prasolov's work on a generalization of the notion of a contactomorphism to non-smooth continuous maps. In three-dimensional contact topology one often deals with piecewise smooth objects like Legendrian links and graphs. The notion of a Legendrian curve is readily generalized to the case of a piecewise smooth curve, and it is natural to ask for a respective generalization of the notion of a contactomorphism. Piecewise smooth settings do not seem to work for that because piecewise smooth homeomorphisms of a manifold do not form a group. It sounds plausible that the Lipschitz category is the one to which all basic notions of contact topology can be extended without loosing their crucial properties. What is proven to date is that the concepts of a Legendrian link and Legendrian isotopy can be extended to the Lipschitz category so that, after the extension, the set of equivalence classes of Legendrian links in a contact manifold remains unchanged. Contactomorphisms are then defined as bi-Lipschitz homeomorphisms that take Legendrian curves to Legendrian curves. With this definitions, it is proven that every bi-Lipschitz Legendrian knot can be taken to a smooth one by a bi-Lipschitz contactomorphism defined on its tubular neighborhood. It is yet to be proved that any
bi-Lipschitz Legendrian isotopy can be extended to an ambient isotopy in the class of bi-Lipschitz contactomorphisms.
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Speaker: Yakov Eliashberg
Title: Invariants of open contact manifolds
Abstract: I will introduce contact homology for all open contact manifolds without any control of contact structure at infinity, and discuss geometric applications. This is a joint work with Kiran Ajij, Mahan Mj, Dishant Pancholi and Leonid Polterovich.
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Observance
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| Title: | TBA |
| Speaker: | Egor Shelukhin, University of Montreal |
| Location: | Math P-131 |
| Abstract: | |
| TBA |
Title: Topological solitons and their role in the long-term dynamics of classical field theories
Abstract: Following the 2004 Cambridge book by Manton and Sutcliffe Topological Solitons, we recall three classical field theories via their Lagrangians: i) scalar fields on the line associated with kinks (the topological invariant is the charge) ii) wave maps in the energy critical regime associated with harmonic maps (the invariant is the degree) iii) Ginzburg-Landau (non-magnetic or non-gauged) and abelian Yang-Mills-Higgs models (magnetic or gauged) associated with vortices (the invariant is the degree). The gradient flow associated with the underlying Hamiltonian leads to dissipative dynamics in the form of a heat equation, while the most common conservative time evolutions are the Schrödinger flow (with Galilei symmetry), respectively the wave or Klein-Gordon flow (with Lorentz symmetry). In each of these infinite-dynamical systems we would like to describe or even classify the long-term behavior of trajectories. This is a deeply challenging problem as a multitude of phenomena might arise (breathers and wobbling kinks in the sine-Gordon equation, multi-kink/antikink solutions in the phi-4 model, bubbling in the harmonic map heat flow, vortex splitting and vortex collisions in Ginzburg-Landau). As stationary solutions, solitons and their moduli space play a fundamental role in the complicated dynamics. A starting point here is the question of asymptotic stability of these equilibria. While the past 20 years have seen dramatic advances, we are still far from a complete understanding. We will survey some of the work in this direction - both past and ongoing - which combines techniques from elliptic PDEs, the spectral and scattering theory of linear differential equations, harmonic analysis and linear dispersive estimates, and nonlinear dispersive equations (space-time resonances, normal forms, vector fields, Fermi Golden Rule).