Oct 08, 2018
Monday

09:15 AM  09:30 AM


Welcome

 Location
 SLMath: Eisenbud Auditorium
 Video


 Abstract
 
 Supplements



09:30 AM  10:30 AM


On some nearly separable impact systems
Vered RomKedar (The Weizmann Institute)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
Nearintegrability is usually associated with smooth small perturbations of smooth integrable systems. We show that studying integrable mechanical Hamiltonian flows with impacts that respect the symmetries of the integrable structure provide an additional rich class of nonsmooth systems that can be studied by perturbation methods. Moreover, the analysis can be extended to systems with soft steep potentials that limit to the impact systems. For example, for some of these systems, we show that KAM theory may be applied, proving that for a large portion of phase space the perturbed motion is conjugate to rotations on a torus [1]. On the other hand, other simple impact systems have inherently nonrotational motion – we show cases in which the motion is conjugate to geodesic flow on a flat torus with several handles [2]. [1] M. Pnueli & V. RomKedar “On near integrability of some impact systems”, SIAMDS, 2018 to appear. [2] L. Becker, S. Elliott, B. Firester, S. Gonen Cohen, M. Pnueli & V. RomKedar, in preparation.
 Supplements


10:30 AM  11:00 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Minimum Energy Configurations in the NBody Problem and the Celestial Mechanics of Granular Systems
Daniel Scheeres (University of Colorado at Boulder)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
Celestial Mechanics systems have two fundamental conservation principles: conservation of momentum and conservation of (mechanical) energy. Of the two, conservation of momentum provides the most constraints on a general system, with three translational symmetries (which can be trivially removed) and three rotational symmetries. If no external force acts on the system, these quantities are always conserved independent of the internal interactions of the system. In contrast, conservation of energy involves assumptions on both the lack of exogenous forces and on the nature of internal interactions within the system. For this reason energy is often not conserved for “real” systems that involve internal interactions, such as tidal deformations or impacts, even though such systems conserve total momentum. Thus, mechanical energy generally decays through dissipation until the system has found a local or global minimum energy configuration that corresponds to its constant level of angular momentum. This observation motivates a fundamental question for celestial mechanics: What is the minimum energy configuration of a Nbody system with a fixed level of angular momentum? It can be shown that this is an illdefined question and has no answer for traditional pointmass celestial mechanics systems. If, instead, the system and problem are formulated accounting for finite density distributions this question becomes well posed and we can prove the existence of minimum energy configurations for all suitably formulated celestial mechanics systems (Scheeres, CMDA 113(3): 291320, 2012). This small change also leads to fundamental changes in the nature and stability properties of relative equilibria and, ultimately, the dynamics of these systems. Finally, we can also show that this naturally leads to a “granular mechanics” extension of celestial mechanics, with fundamental links between this topic and the science of small solar system bodies.
 Supplements

Notes
10.3 MB application/pdf


12:00 PM  02:00 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Symplectic gyrokinetic VlasovMaxwell theory
Alain Brizard (Saint Michael's College)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
We consider a general form of electromagnetic gyrokinetic VlasovMaxwell theory in which the gyrocenter symplectic structure contains electric and magnetic perturbations that are necessary to cause the firstorder gyrocenter polarization displacement to vanish. The gyrocenter Hamilton equations, which are expressed in terms of a gyrocenter Poisson bracket that contains electromagnetic perturbations and a gyrocenter Hamiltonian, satisfy the Liouville property exactly with a timedependent gyrocenter Jacobian. The gyrokinetic VlasovMaxwell equations are derived from a variational principle, which also yields exact conservation laws through the Noether method. We show that the new symplectic gyrokinetic VlasovMaxwell equations retain all firstorder polarization and magnetization effects without the need to consider secondorder contributions in the gyrocenter Hamiltonian.
 Supplements

Notes
1.12 MB application/pdf


03:00 PM  03:30 PM


Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Jumps of energy near a separatrix in slowfast Hamiltonian systems
Sergey Bolotin (University of WisconsinMadison)

 Location
 SLMath: Eisenbud Auditorium
 Video


 Abstract
We consider a Hamiltonian system slowly depending on time: $$ H=H(x,\tau),\qquad \dot\tau=\epsilon\ll 1. $$ For small $\epsilon$ the energy $H$ changes slowly: $\dot H=O(\epsilon)$. If the frozen autonomous system with Hamiltonian $H(\cdot,\tau)$ has one degree of freedom and energy levels are closed curves, there is an adiabatic invariant $I$ which changes much slower. Then the energy $H=h(\tau,I)$ changes gradually. However, the adiabatic invariant is destroyed for trajectories passing near equilibria. Neishtadt showed that if the frozen system has a hyperbolic equilibrium with a figure eight separatrix, then generically the energy will have quasirandom jumps of order $\epsilon$ with frequency of order $1/\ln\epsilon$. We partly extend Neishtadt's result to multidimensional systems such that for each $\tau$ the frozen system has a hyperbolic equilibrium possessing several transverse homoclinics. The trajectories we construct have quasirandom jumps of energy of order $\epsilon$ with frequency $1/\ln\epsilon$ while staying distance of order $\epsilon$ away from the homoclinic set. Gelfreigh and Turayev showed that if the frozen system has compact uniformly hyperbolic chaotic invariant sets on energy levels, then generically there exist trajectories with energy having quasirandom jumps of order $\epsilon$ with frequency of order 1. Thus the energy may grow with rate of order $\epsilon$. However, this result does not work near a homoclinic set, where dynamics of the frozen system is slow, so there is no uniform hyperbolicity.
 Supplements

Notes
1.73 MB application/pdf



Oct 09, 2018
Tuesday

09:15 AM  10:15 AM


Optimal rate of convergence in periodic homogenization of HamiltonJacobi equations
Yifeng Yu (University of California, Irvine)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
In this talk, I will present some recent progress in obtaining the optimal rate of convergence $O(\epsilon)$ in periodic homogenization of HamiltonJacobi equations. Our method is completely different from previous pure PDE approaches which only provides $O(\epsilon^{1/3})$. We have discovered a natural connection between the convergence rate and the underlying Hamiltonian system. This allows us to employ powerful tools from the AubryMather theory and the weak KAM theory. It is a joint work with Hiroyashi Mitake and Hung V. Tran.
 Supplements

Notes
2.58 MB application/pdf


10:15 AM  10:30 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



10:30 AM  11:30 AM


Stability for PDEs, the Maslov Index, and Spatial Dynamics
Margaret Beck (Boston University)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
Understanding the stability of solutions to PDEs is important, because it is typically only stable solutions which are observable. For many PDEs in one spatial dimension, stability is wellunderstood, largely due to a formulation of the problem in terms of socalled spatial dynamics, where one views the single spatial variable as a timelike evolution variable. This allows for many powerful techniques from the theory of dynamical systems to be applied. In higher spatial dimensions, this perspective is not clearly applicable. In this talk, I will discuss recent work that suggests both that the Maslov index could be a important tool for understanding stability when the system has a symplectic structure, particularly in the multidimensional setting, and also suggests a possible analogue of spatial dynamics in the multidimensional setting.
 Supplements

Notes
6.18 MB application/pdf


11:30 AM  12:30 PM


A topological mechanism for diffusion, with application to the elliptic restricted three body problem
Maciej Capinski (AGH University of Science and Technology)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
We present a topological mechanism of diffusion in a priori chaotic systems. The method leads to a proof of diffusion for an explicit range of perturbation parameters. The assumptions of our theorem can be verified using interval arithmetic numerics, leading to computer assisted proofs. As an example of application we prove diffusion in the NeptuneTriton planar elliptic restricted three body problem. Joint work with Marian Gidea.
 Supplements

Notes
3.14 MB application/pdf


12:30 PM  02:00 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Hamiltonian reduced fluid models for nondissipative plasmas
Emanuele Tassi (Laboratoire Lagrange)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
Progress in the understanding of several phenomena occurring in plasmas greatly benefited from the use of continuum models based on a fluid description of plasmas.
In the absence of dissipative effects, all such models are supposed to possess a Hamiltonian structure. The existence of such structure for a given model is, however, not guaranteed, in general,
unless it is implied in its derivation or shown a posteriori.
In this talk I will consider a class of socalled reduced fluid models for plasmas, which are applicable in the situation where the magnetic field can be written as the sum of
a uniform
and constant component (guide field) with a fluctuating contribution depending on space and time. The amplitude of the fluctuating contribution is also assumed to be much
smaller than that of the guide field. Such very commonly adopted assumption has led, together with further assumptions, to the derivation, over the last decades, of a number of reduced fluid models, a considerable part of which were shown to possess a Hamiltonian structure.
In this context, I will first recall earlier results on the derivation of a class of reduced fluid models, which guarantees the existence of a Hamiltonian structure. Such derivation is based on a closure
of the hierarchy of fluid equations evolving moments of the perturbation of the distribution function satisfying a Hamiltonian driftkinetic equation. In the remaining part of the talk I will consider recent extensions of this procedure, addressed to applications to collisionless space plasmas.
 Supplements

Notes
1.69 MB application/pdf


03:00 PM  03:30 PM


Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Variational discretizations of gauge field theories using groupequivariant interpolation spaces
Melvin Leok (University of California, San Diego)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
Variational integrators are geometric structurepreserving numerical methods that preserve the symplectic structure, satisfy a discrete Noether's theorem, and exhibit exhibit excellent longtime energy stability properties. An exact discrete Lagrangian arises from Jacobi's solution of the HamiltonJacobi equation, and it generates the exact flow of a Lagrangian system. By approximating the exact discrete Lagrangian using an appropriate choice of interpolation space and quadrature rule, we obtain a systematic approach for constructing variational integrators. The convergence rates of such variational integrators are related to the best approximation properties of the interpolation space. Many gauge field theories can be formulated variationally using a multisymplectic Lagrangian formulation, and we will present a characterization of the exact generating functionals that generate the multisymplectic relation. By discretizing these using groupequivariant spacetime finite element spaces, we obtain methods that exhibit a discrete multimomentum conservation law. We will then briefly describe an approach for constructing groupequivariant interpolation spaces that take values in the space of Lorentzian metrics that can be efficiently computed using a generalized polar decomposition. The goal is to eventually apply this to the construction of variational discretizations of general relativity, which is a secondorder gauge field theory whose configuration manifold is the space of Lorentzian metrics.
 Supplements

Notes
4.26 MB application/pdf


04:30 PM  06:20 PM


Reception

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements




Oct 10, 2018
Wednesday

09:15 AM  10:15 AM


Integrable Systems and Optimal Control
Anthony Bloch (University of Michigan)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
In this talk we discuss recent work on a geometric approach to certain optimal control problems and the relationship of the solutions of these problems to some classical integrable dynamical systems. These systems include the rigid body equations, geodesic flows on the ellipsoid and the Toda lattice flows. We discuss the Hamiltonian structure of these systems and relate our work to some classical work of Moser. We also discuss the link to discrete dynamics and symplectic integration. The work is joint with Francois GayBalmaz and Tudor Ratiu.
 Supplements

Notes
1.02 MB application/pdf


10:15 AM  10:30 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



10:30 AM  11:30 AM


Dynamics of Water Waves
Massimiliano Berti (International School for Advanced Studies (SISSA/ISAS))

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
I shall present recent results about the complex dynamics of the water waves equations of a bidimensional fluid under the action of gravity and eventually capillary forces, with space periodic boundary conditions. This is an infinite dimensional Hamiltonian system. We shall discuss both long time existence results as well as bifurcation of small amplitude time quasiperiodic solutions. Major difficulties are the quasilinear nature of the water waves equations and complex resonance phenomena.
 Supplements

Notes
3.07 MB application/pdf


11:30 AM  12:30 PM


Noether theorem for magnetised plasmas.
Natalia Tronko (MaxPlanckInstitut für Plasmaphysik (EURATOM))

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
How the fundamental mathematical tools such as Noether's Theorem can be implemented for practical purposes, such as control of quality of prediction of fusion plasmas discharges? In this talk I will try to respond on this intricate question. Below are some motivations. In fusion plasmas the strong magnetic field allows the fast gyro motion to be systematically removed from the description of the dynamics, resulting in a considerable model simplification and gain of computational time. The dynamically reduced kinetic model obtained in the given setup is called gyrokinetic model. Performing dynamical reduction in the framework of Lagrangian formalism allows one to control the exactness of reduction procedure and derive robust models for numerical implementations. In their turn, specific results of gyrokinetic (GK) simulations performed prior to consideralby costly fusion plasma experiments allows to predict and optimise experimental setup. In particular, numerical simulations of fusion plasma discharges can impact our predictions of the plasma behavior in ITER and future fusion reactors. Confidence in these predictions requires a rigorous and systematic verification of the underlying model, which should be regarded as an indispensable step before any validation of the numerical results against experiments can be considered meaningful. A new and generic theoretical framework and specific numerical applications to test the validity and the domain of applicability of existing GK codes will be presented. In particular the role of the energy invariants issued from the Noether theorem will be explicitly highlighted throughout the analysis of energy balance and generic plasma instabilities mechanisms providing the ultimate connection between the fundamental mathematical tools and practical implementations.
 Supplements

Notes
10.6 MB application/pdf



Oct 11, 2018
Thursday

09:30 AM  10:30 AM


Momentum maps for automorphism groups
Tudor Ratiu (Shanghai Jiaotong University)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
In mathematical physics there are conservation laws that are discrete in nature, such as topological information. These conservation laws cannot be captured by the usual momentum map and many of its extensions. I will present an enlarged definition of the momentum map with values in groups, that works also in infinite dimensions. It is inspired by the LuWeinstein momentum map for Poisson Lie group actions, but the groups involved do not necessarily have a Poisson Lie group structure on it. The most interesting applications include momentum maps for various automorphism groups which take values in CheegerSimons differential characters of the underlying manifolds. I will concentrate on Clebsch variables for fluids and show how this extended momentum map lads to new Clebsch variables admitting nonvanishing, but integer valued helicity.
 Supplements

Notes
957 KB application/pdf


10:30 AM  11:00 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Effective bounds for the measure of rotations
Alex Haro (University of Barcelona)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
A fundamental question in Dynamical Systems is to identify regions of
phase/parameter space satisfying a given property (stability, linearization, etc). In this talk, given a family of analytic circle diffeomorphisms depending on a parameter, we obtain effective (almost optimal) lower bounds of the Lebesgue measure of the set of parameters
that are conjugated to a rigid rotation. We estimate this measure using an aposteriori KAM scheme that relies on quantitative conditions that are checkable using computerassistance. We carefully describe how the hypotheses in our theorems are reduced to a finite number of computations, and apply our methodology to the case of the Arnold family. Hence we show that obtaining nonasymptotic lower bounds for the applicability of KAM theorems is a feasible task provided one has an aposteriori theorem to characterize the problem. Finally, as a direct corollary, we produce explicit asymptotic estimates in the so called local reduction setting which are valid for a global set of rotations.
This is joint work with Jordi Lluis Figueras and Alejandro Luque.
 Supplements

Notes
679 KB application/pdf


12:00 PM  02:00 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Dark Plasma
Joshua Burby (New York University, Courant Institute)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
Fully ionized plasmas emit both incoherent light and coherent light. Incoherent emission occurs whenever plasma particles suffer Coulomb collisions, and is therefore ubiquitous. On the other hand, coherent emission involves macroscopic collections of particles moving in concert. This incoherent emission is well described by the VlasovMaxwell system of equations. In this talk I will argue for the existence of plasmas that are "dark" in the sense that their coherent emission is extremely weak. The dark plasma motions will be identified with a slow manifold in the VlasovMaxwell phase space. In the case where collisions are extremely rare, I will give a complete description of the Hamiltonian formulation of dynamics on the slow manifold. In the leadingorder approximation, the dark motions are modeled by the VlasovPoisson system of equations. At the next order, which accounts for magnetostatic effects, the model also has a name: the VlasovDarwin system. Higherorder approximations account for nonradiative electromagnetic fields generated by collective acceleration of plasma particles. The dark motions may be modeled with any desired order of accuracy without sacrificing the problem's underlying Hamiltonian structure.
 Supplements

Notes
10.1 MB application/pdf


03:00 PM  03:30 PM


Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Exponential stability of Euler integral in the threebody problem
Gabriella Pinzari (Università di Padova)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
The first integral characteristic of the fixed twocentre problem is proven to be an approximate integral (in the sense of N.N.Nekhorossev) to the threebody problem, at least if the masses are very different and the particles are constrained on a plane. The proof uses a new normal form result, carefully designed around the degeneracies of the problem, and a new study of the phase portrait of the unperturbed problem. Applications to the prediction of collisions between the two minor bodies are shown.
 Supplements

Notes
11.1 MB application/pdf



Oct 12, 2018
Friday

09:30 AM  10:30 AM


The vanishing discount problem in a noncompact setting
Antonio Siconolfi (Università di Roma "La Sapienza'')

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
We study the asymptotic behavior of the solutions to a family of discounted HamiltonJacobi equations when the discount factor goes to zero.
The new point is that we tackle the problem in a noncompact setting. We prove that a distinguished critical solution of the equation with vanishing discount is selected at the limit. Our approach is based on some duality results between suitable cones of Lagrangian functions and families of probability measures defined on the tangent bundle of the ambient space.
 Supplements

Notes
1.65 MB application/pdf


10:30 AM  11:00 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Whiskered KAM Tori of Conformally Symplectic Systems
Renato Calleja (UNAM  Universidad Nacional Autonoma de Mexico)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
Many physical problems are described by conformally symplectic systems. We study the existence of whiskered tori in a family $f_\mu$ of conformally symplectic maps depending on parameters $\mu$. Whiskered tori are tori on which the motion is a rotation but having as many contracting/expanding directions as allowed by the preservation of the geometric structure. Our main result is formulated in an aposteriori format. Given an approximately invariant embedding of the torus for a parameter value $\mu_0$ with an approximately invariant splitting of the tangent space at the range of the embedding into stable/unstable/center bundles, there is an invariant embedding and invariant splittings for new parameters. Using the results of formal expansions as the starting point for the aposteriori method, we study the domains of analiticity of parameterizations of whiskered tori in perturbations of Hamiltonian Systems with dissipation. The proofs of the results lead to efficient algorithms that are quite practical to implement. Joint work with A. Celletti and R. de la Llave.
 Supplements

Notes
9.02 MB application/pdf


12:00 PM  02:00 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Multiphysics models for hybrid kineticfluid and classicalquantum systems
Cesare Tronci (University of Surrey)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
Many physical situations involve the interplay of different phenomena at different scales. The corresponding description requires the use of multiphysics models, whose mathematical formulation poses several challenges. Examples are found in the classicalquantum coupling in molecular dynamics or in the coupling between mean flow and fluctuation kinetics in turbulence. In plasma physics, the interaction of energetic particles (obeying kinetic theory) with a fluid bulk (obeying magnetohydrodynamics) requires formulating hybrid kineticfluid models, which are often obtained by making assumptions that destroy the correct energy balance. This talk shows how momentummap techniques in geometric mechanics provide a powerful unifying framework for both kineticfluid and classicalquantum coupling, thereby leading to new hybrid models in different contexts.
 Supplements

Notes
8.03 MB application/pdf

Notes
8.03 MB application/pdf


03:00 PM  03:30 PM


Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Local dynamics and invariant manifolds of traveling wave manifolds of Hamiltonian PDEs
Chongchun Zeng (Georgia Institute of Technology)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
Some Hamiltonian PDEs which are invariant under spatial translations possess traveling wave solutions which form finite dimensional invariant manifolds parametrized by their spatial locations. Extensive studies have been carried out for their stability analysis. In this talks we shall focus on local dynamics and invariant manifolds of the traveling wave manifolds for the GrossPitaevskii equation in $R^3$ and the gKdV equation as our main PDE models, while our approach works for a general class of problems. Noting that the symplectic operators of some of these models happen to be unbounded in the energy space, violating a commonly assumed assumption for the study of the linearized systems at these traveling waves, we could carry out linearized analysis in a general framework we developed recently. Nonlinearly our main results are the existence of local invariant manifolds of unstable traveling waving manifolds and the implications on the local dynamics. In addition to applying certain spacetime estimates, we use a bundle coordinate system to handle an issue of a seemingly regularity loss caused by the spatial translation parametrization.
 Supplements

Notes
2.8 MB application/pdf


