Speaker:   Kevin Ren, Princeton University
 

Abstract:

For any given n-point subset M in R^d, Charikar et. al. construct a randomized partition of M into clusters of diameter <= 1 such that any x, y in M belong to different clusters with probability <= O(1) * sqrt(min(log n, d)) * ||x - y||. We improve upon this result if M has small doubling constant lambda, replacing the sqrt(min(log n, d)) term with sqrt(log lambda). This new construction relies on a novel multiscale decomposition and localization analysis that exploits the doubling property of M. If time permits, I will discuss extensions of our techniques to L_p spaces and applications to metric embeddings. Joint work with Assaf Naor.

 

Speaker:   Vladimir Chernov
Title:   Contact Geometry, Causality and Smooth Structures on spacetimes Part III
Abstract:   Low and Penrose introduced contact structure on the space of light rays in a spacetime. The sphere S_p of all the light rays through the point p is called the sky of p and is Legendrian. Rudyak and myself started the work on generalized linking number and causality. Jointly with Nemirovski we proved that for almost all globally hyperbolic spacetimes two points p,q are causally related if and only if the Legendrian link S_p, S_q is nontrivial. This holds UNLESS the universal cover of the Cauchy surface S is compact AND has the same homology as a CROSS Compact Rank One Symmetric Spacetime. In particular this solved Low, Natario-Tod Conjectures and answered Penrose question on Arnold Problem list. Jointly with Rudyak, Nemrirovski and in a later work by Bauermeister it was proved that UNLESS both of these conditions hold the spacetime also has no strict refocusing. This is when all light rays through one of the two points pass through the other.
We also discuss our results with Nemirovski on the uniqueness of smooth structures on globally hyperbolic spacetimes and the conjectural generalizations of this results to all globally hyperbolic spacetimes of all dimensions. Conjecturally this smooth structure might be characterized by the contact manifolds of all the light rays or you might possibly need to add a few Legendrian sky submanifolds.

Speaker:   Jakob Hedicke
Title:   TBA
ABstract: TBA

Speaker:    Nataliya Goncharuk, Texas A & M
Title:   Complexified Marmi-Moussa-Yoccoz conjecture for high type

Abstract:

By classical Yoccoz's theorem, a neutral fixed point of a quadratic map z^2+c is linearizable if and only if the angle of rotation is a Brjuno number. Moreover, the conformal radius of the corresponding Siegel disc can be estimated via the Brjuno function of the angle. Marmi-Moussa-Yoccoz conjecture suggests that the difference between the logarithm of the conformal radius of the Siegel disc and the Brjuno function of the angle is 1/2-Holder continuous. X. Buff and A. Cheritat proved the continuity of this function. Later, D. Cheraghi and A. Cheritat proved the conjecture for high-type angles of rotation. I will talk on our joint results with S. Marmi and M. Yamplosky on the complexified version of the Marmi-Moussa-Yoccoz conjecture

  

Speaker:   Vladimir Chernov
Title:   Virtual Legendrian linking and causality in generalized spacetimes
Abstract:   A generalized spacetime is a singular Lorentz manifold that can be equipped with a singular timelike Morse function T such that the pieces cut by critical levels of T are globally hyperbolic. This can be thought of  as a universe for which the topology of the time-level set changes with time. A virtual Legendrian link is a Legendrian link L in ST*M up to isotopy and modifications of ST*M caused by surgery on M away from the front projection of L. Jointly with Sadykov we conjecture that virtual Legendrian isotopy class of the link of the Legendrian spheres through two points characterizes causality in generalized spacetimes and provide supporting evidence to this conjecture including positive virtual Legendrian isotopy conjecture. This can thought of the extension of our results with Nemirovski on Legendrian linking and causality in globally hyperbolic spacetimes that answered the question of Penrose on Arnold Problem List and solved Low and Natario-Tod Conjectures.
Hashtag: #workshop

Speaker:   Miguel Angel Javaloyes
Title:   Cone structure and metrics that depend on time
Abstract:   How can a metric that depends on time be studied? In General Relativity, space and time are intertwined in such a way that they cannot be understood separately, and moreover, time is relative, depending on a choice of reference frame. But our goal in this lecture will be different. We will show that it makes sense to work with an absolute time and still consider time-dependent metrics. The role of length will be played by time, and the Riemannian metric will determine the velocity of objects in each direction. Geodesics will be defined as the fastest trajectories, which, by applying the relativistic Fermat principle, can be calculated as light-like geodesics on a Finsler spacetime, most generally, from a cone structure. Finally, we will demonstrate that these structures can be applied to the study of forest fires and that we can also define a curvature that measures how geodesics diverge using the degenerate curvature introduced by Harris in Lorentzian manifolds. This curvature helps to identify focal points, which are crucial for firefighters. Moreover, we will give an interpretation of curvature in terms of Jacobi fields and will obtain some applications to compute flag curvature in Finsler manifolds.
Hashtag: #workshop

Speaker:   Shin-itiro Goto
Title:   Derivation of a contact Hamiltonian system from the Fokker-Planck equation
Abstract:   The Fokker-Planck equation is a well-known fundamental equation in nonequilibrium statistical mechanics, and it can be formulated on Riemannian manifolds. On the other hand, contact Hamiltonian systems are known to describe nonequilibrium thermodynamic systems. Because both equations describe relaxation processes, elucidating the relationship between these two equations is expected to lead to the development of a geometric theoryof nonequilibrium statistical mechanics. In this talk it is shown how a class of contact Hamiltonian systems is derived from the Fokker-Planck equation on Riemannian manifolds. A key step in this derivation is to focus on a few small eigenvalues of a symmetric operator
induced from the Fokker-Planck equation. This talk is based on Goto 2024 J. Phys. A: Math. Theor. 57 335005.
Hashtag: #workshop

Speaker:   Alberto Abbondandolo
Title:   Bi-invariant Lorentz-Finsler structures on the linear symplectic group and on contactomorphism groups
Abstract:   It has been known for a long time that the linear symplectic group and contactomorphism groups carry interesting bi-invariant causal structures. It turns out that these groups carry also natural bi-invariant Lorentz-Finsler structures. I will introduce these structures and discuss some of their properties. This talk is based on a joint work with Gabriele Benedetti and Leonid Polterovich.
Hashtag: #workshop

Speaker:   Gabriele Benedetti
Title:   Quantifying the flexibility of the Lorentz-Finsler geometry on contactomorphism groups
Abstract:   In contrast to the case of the linear symplectic group Sp(2n), measurements with respect to a natural Lorentz-Finsler structure on the group of contactomorphisms of the real projective space RP^{2n-1} are trivial. In this talk, we show in two examples that such triviality comes at the cost of considering paths with high complexity. Non-triviality can be recovered if (a) n=1 and the contact Hamiltonian are Fourier polynomials of fixed degree, or if (b) n is arbitrary and the contact Hamiltonians are pinched between some quadratic Hamiltonian with fixed pinching constant.
These results hinge on a refined Bernstein inequality due to Nazarov in a), and on Givental’s non-linear Maslov index together with the analysis on Sp(2n) in b). This is joint work with Alberto Abbondandolo and Leonid Polterovich.
Hashtag: #workshop

Speaker:   Yong-Geun Oh
Title:   Nonequilibrim thermodynamic, information entropy and contact geometry
Abstract:   Both statistical phase space (SPS), of many body particle system and kinetic theory phase space (KTPS), and the cotangent bundle of the probability space thereon, carry canonical symplectic structures. Starting from this first principle, we explain a canonical derivation of thermodynamic phase space (TPS) of nonequilibrium thermodynamics as a contact manifold using the Marsden-Weinstein reduction. Then we explain how the relative information entropy (or Kullbak-Leibler divergence) defines a generating function that provides a covariant construction of thermodynamic equilibrium as a Legendrian submanifold. In this regard, we interpret Maxwell’s construction via the equal area law as the procedure of
finding a continuous, not necessarily differentiable, Gibbs potential and explain the associated phase transition. This is a joint work with Jinwook Lim.
Hashtag: #workshop

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.
Hashtag: #workshop

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.
Hashtag: #workshop

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.
Hashtag: #workshop

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.
Hashtag: #workshop

Speaker:   Mohammed Abouzaid
Title:   TBA
Abstract:   TBA
Hashtag: #workshop

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.
Hashtag: #workshop

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
Hashtag: #workshop

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.
Hashtag: #workshop

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.
Hashtag: #workshop

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.
Hashtag: #workshop

Speaker:   Lukas Nakamura
Title:   TBA
Abstract:   TBA

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.
Hashtag: #workshop

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.
Hashtag: #workshop

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.
Hashtag: #workshop

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.
Hashtag: #workshop

Observance
To hide observances, go to Google Calendar Settings > Holidays in United States