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

For more information visit: https://scgp.stonybrook.edu/archives/46936

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).

Last day to submit an approved adjustment form for selected AMS, MAT, MAP and PHY courses to the Office of Registrar. 

Title:   Lyapunov exponents, Schrödinger cocycles, and Avila’s global theory.

Abstract:   In the 1950s Phil Anderson made a prediction about the effect of random impurities on the conductivity properties of a crystal. Mathematically, these questions amount to the study of solutions of differential or difference equations and the associated spectral theory of self-adjoint operators obtained from an ergodic process. With the arrival of quasicrystals, in addition to random models, nonrandom lattice models such as those generated by irrational rotations or skew-rotations on tori have been studied over the past 30 years.

By now, an extensive mathematical theory has developed around Anderson’s predictions, with several questions remaining open. This talk will attempt to survey certain aspects of the field, with an emphasis on the theory of SL(2,R) cocycles with an irrational or Diophantine rotation on the circle as base dynamics. In this setting, Artur Avila discovered about 15 years ago that the Lyapunov exponent is piecewise affine in the imaginary direction after complexification of the phase. In fact, the slopes of these affine functions are integer valued. This is easy to see in the uniformly hyperbolic case, which is equivalent to energies falling into the gaps of the spectrum, due to the winding number of the complexified Lyapunov exponent. Remarkably, this property persists also in the non-uniformly hyperbolic case, i.e., on the spectrum of the Schrödinger operator. This requires a delicate continuity property of the Lyapunov exponent in both energy and frequency. Avila built his global theory (Acta Math. 2015) on this quantization property. I will present some results with Rui HAN (Louisiana State) connecting Avila’s notion of acceleration (the slope of the complexified Lyapunov exponent in the imaginary phase direction) to the number of zeros of the determinants of finite-volume Hamiltonians relative to the complex phase. This connection, which builds on the machinery developed by Michael Goldstein and the author (GAFA 2008, Annals of Math. 2011),  allows one to answer questions arising in the supercritical case of Avila’s global theory concerning the measure of the second stratum, Anderson localization on this stratum, as well as settle a conjecture on the Hölder regularity of the integrated density of states.  We will also describe applications to the block Jacobi matrix setting which shed light on almost reducibility questions via duality.