Nov 26, 2018
Monday

09:15 AM  09:30 AM


Welcome

 Location
 SLMath: Eisenbud Auditorium
 Video


 Abstract
 
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09:30 AM  10:30 AM


Infinitely many coplanarities
Richard Montgomery (University of California, Santa Cruz)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
Consider the Newtonian 4body problem in 3space with zero angular momentum and any mass ratios. Theorem: every bounded solution defined on an infinite time interval suffers infinitely many coplanar instants: instants where all four bodies lie in the same plane. This generalizes a theorem asserting that for the zero angular momentum 3body problem in the plane, every bounded solution suffers infinitely many collinear instants, or ``syzygies''. The obvious generalization holds for any dimension $d$, that is, for the zero angular momentum $d+1$body problem in $d$space. After stating the results, I give a heuristic physical ``proof'' of this theorem inspired by Mark Levi. I then proceed to sketch the mathematical proof. Among its main ingredients are that (i) the translationreduced problem has for its configuration space the space of $d \times d$ matrices (ii) the signed distance from the degeneration locus satisfies a HamiltonJacobi equation, (iii) the oriented shape space is homeomorphic to a Euclidean space (iv) the oriented shape space has nonnegative sectional curvature where smooth. The theorem gives a ``hunting license'' to set up symbolic dynamics for the spatial 4body problem, in analogy with the idea of syzygy sequences for the planar threebody problem.
 Supplements

Notes
863 KB application/pdf


10:30 AM  11:00 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Shearless Invariant Curves In Confined Plasmas
Ibere Caldas (University of Sao Paulo)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
Nontwist Hamiltonian systems have shearless invariant curves that act like barriers in phase space [1, 2]. Recently, secondary shearless curves have also been identified in the phase space of twist maps, in the neighbourhood of peculiar bifurcations of elliptic fixed points [3]. We use Slater’s theorem to develop a qualitative and quantitative numerical approach to determine the breakup of invariant curves in the phase space of areapreserving maps [4]. We also determine the breakup critical parameters, of the shearless curves, with a procedure based on the determinism analysis performed on the recurrence plot of orbits near the critical transition [5]. Finally, we present evidences of transport barriers in plasmas confined in the tokamak TCABR [6] and in the Texas Helimak [7].
References 1 P. J. Morrison, Physics of Plasmas 7, 2279 [2000]. 2 D. DelCastilloNegrete, Physics of Plasmas 7, 1702 [2000]. 3 C. V. Abud, I. L. Caldas. Chaos 22, 033142 (2012). 4 C. V. Abud, I. L. Caldas. Physica D 308, 34 (2015) 5 M. S. Santos, M. Mugnaine, J. D. Szezech Jr, A. M. Batista, I. L. Caldas, M. S. Baptista, R. L. Viana. Chaos 28, 085717 (2018). 6A. F. Marcus, I. L. Caldas, Z. O. GuimarãesFilho, P. J. Morrison, W. Horton, I. C. Nascimento, Yu. K. Kuznetsov. Phys. Plasmas 15, 112304 (2008). 7 D. L. Toufen, Z. O. GuimarãesFilho, I. L. Caldas, F. A. Marcus, K. W. Gentle. Phys. Plasmas (2012).
 Supplements

Notes
7.46 MB application/pdf


12:00 PM  02:00 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Hamiltonian ODE and HamiltonJacobi PDE with Stochastic Hamiltonian Function
Fraydoun Rezakhanlou (University of California, Berkeley)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
In this talk I give an overview on the existing results for two questions associated with a Hamiltonian function that is selected randomly according to a probability measure that is translation invariant (this includes Hamiltonian functions that are almost periodic in position or in both position and momentum) 1. Consider a Hamiltonian ODE with a Hamiltonian function that is periodic in time. As in Arnold's conjecture we may wonder whether or not such an ODE has periodic orbits. 2. Analogously, we may study HamiltonJacobi PDEs and examine the question of homogenization that is closely related to the long time behavior of the solutions to the corresponding Hamiltonian ODE.
 Supplements

Notes
781 KB application/pdf


03:00 PM  03:30 PM


Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Rigorous HighOrder Methods in the Description of Large Particle Accelerators
Martin Berz (Michigan State University)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
Large particle accelerators and storage rings are represented by exceedingly complicated Hamiltonians. On the one hand the relativistic motion in electromagnetic fields leads to nonlinear connections between positions and momenta due to the presence of a square root. On the other hand, different from other examples including typical many body problems, an accelerator has thousands of separate sections with different localized fields. Finally it is often necessary to predict orbit stability for around billions of revolutions. All these make even attempts at symplectic tracking very difficult.
One method to describe the nonlinear motion is via perturbation theory describing the motion on a Poincare section for one revolution. Recent advances based on differential algebraic methods have allowed the generalization of such perturbation analysis from typically orders two and three to in principle arbitrary order, and in practice to orders around ten. Such representations of the Poincare map allow symplectic tracking in various ways.
More importantly, the high order Poincare maps can be used to perform rigorous stability analysis. For this purpose, the differential algebraic approach allows performing automated high order normal form analysis of the motion, which upon adjusting system parameters of the accelerator leads to dynamics that is nearly integrable. Utilizing modern methods of rigorous global optimization based on highorder Taylor model methods, it is possible to perform rigorous Nekhoroshevtype stability estimates. Results are shown about such stability estimates, which can establish guaranteed stability within the regions typically occupied by the particle beam for operationally required times.
 Supplements

Notes
7.76 MB application/pdf



Nov 27, 2018
Tuesday

09:15 AM  10:15 AM


Plasma physics inspired Hamiltonian dynamics problems
Diego delCastilloNegrete (Oak Ridge National Laboratory)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
Since the early days of the controlled nuclear fusion program, plasma physics in general, and magnetically confined plasmas in particular, have been a source of Hamiltonian problems. In the spirit of the workshop that aims to bring together the plasma physics and mathematics communities, in this talk we present an overview of past and recent developments of plasma physics problems that have led to novel and challenging mathematical problems in Hamiltonian dynamical systems. Among the problems discussed are: nontwist systems, meanfield coupled Hamiltonian systems, nonautonomous symplectic maps, and gyroaveraged symplecticmaps. Some recent developments on relativistic dynamics of highenergy (runaway) electrons in fusion plasmas will also be discussed.
 Supplements

Notes
9.58 MB application/pdf


10:15 AM  10:30 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



10:30 AM  11:30 AM


Two dimensional examples of the JacobiMaupertuis metric
Richard Moeckel (University of Minnesota Twin Cities)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
The orbits of a Hamiltonian system on a fixed energy level can be viewed as geodesics of the corresponding JacobiMauptertuis metric on the configuration space. For systems of two degrees of freedom, this is a metric on the twodimensional configuration space. In this talk I will look at some simple examples from celestial mechanics, starting with the Kepler problem and moving on to the collinear and isosceles threebody problems. I will look at the problem of visualizing the Kepler surface by embedding it in Euclidean space and discuss questions about lengthminimizing geodesics for the threebody problems.
 Supplements

Notes
5.96 MB application/pdf


11:30 AM  12:30 PM


Geometric and Hamiltonian hydrodynamics via Madelung transform
Boris Khesin (University of Toronto)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
We introduce a geometric framework to study Newton's equations on infinitedimensional configuration spaces of diffeomorphisms and smooth probability densities. It turns out that several important PDEs of hydrodynamical origin can be described in this framework in a natural way. In particular, the socalled Madelung transform between the Schrödingertype equations on wave functions and Newton's equations on densities turns out to be a Kähler map between the corresponding phase spaces, equipped with the FubiniStudy and FisherRao information metrics. This is a joint work with G.Misiolek and K.Modin.
 Supplements

Notes
764 KB application/pdf


12:30 PM  02:00 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Head and Tail speeds of Mean curvature flow with periodic forcing
Inwon Kim (University of California, Los Angeles)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
We study large time behavior of mean curvature flow with periodic forcing. It turns out that the behavior of the interface can be characterized by its head and tail speeds, which depend continuously on its overall direction of propagation. We will also discuss formulation of localized traveling waves in the graph setting.
 Supplements

Notes
939 KB application/pdf


03:00 PM  03:30 PM


Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Lecture
Tadashi Tokieda (Stanford University)

 Location
 SLMath: Eisenbud Auditorium
 Video


 Abstract
 
 Supplements

Notes
339 KB application/pdf


04:30 PM  06:20 PM


Reception

 Location
 SLMath: Atrium
 Video


 Abstract
 
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Nov 28, 2018
Wednesday

09:30 AM  10:30 AM


Singularities in Poisson manifolds: bifurcation and symmetry breaking
Zensho Yoshida (University of Tokyo)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
A variety of interesting structures and dynamics stem from singularities of Poisson operators. As well known, the kernel of a finitedimensional Poisson operator can be integrated “locally” to define Casimirs, and their levelsets foliate the phase space yielding symplectic submanifolds (LieDarboux theorem). However, the global structure can be more complex when the Poisson operator has “singularities” where its rank changes. Invoking some physical examples (both finite dimensional and infinite dimensional), we discuss how singularities cause seemingly nonHamiltonian phenomena. Some theoretical methods for probing into singularities will be also introduced.
 Supplements

Notes
1.29 MB application/pdf


10:30 AM  11:00 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


From Celestial Mechanics to Fluid Dynamics: Contact structures with singularities
Eva Miranda (Polytechnical University of Cataluña (Barcelona))

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
Taking as starting point several examples from Celestial mechanics where regularization techniques bring singularities in, we will introduce the geometry of contact structures where the regularity of the contact 1form is relaxed. Contact structures also show up modelling problems in Fluid Dynamics and singularities also appear naturally in this context (ongoing joint work with Robert Cardona and Daniel PeraltaSalas).
Two main geometrical problems will be addressed in this talk: The existence problem of contact structures with singularities on a given manifold and the study of its Reeb Dynamics, in particular, the existence of periodic orbits (Weinstein conjecture).
This is joint work with Cédric Oms.
 Supplements

Notes
1.22 MB application/pdf



Nov 29, 2018
Thursday

09:30 AM  10:30 AM


Drift and Diffusion in Symplectic and VolumePreserving Maps
James Meiss (University of Colorado at Boulder)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
A nearlyintegrable dynamical system has a natural formulation in terms of actions that are nearly constant and angles that nearly rigidly rotate. Such maps arise naturally in the Hamiltonian, or symplectic case where the variables appear in canonical pairs. We study angleaction maps that are close to symplectic and have a twist that is positivedefinite. When the map is symplectic, Nekhoroshev's theorem implies that the actions are confined for exponentially long times: the drift is exponentially small and numerically appears to be diffusive. We show that when the symplectic condition is relaxed, but the map is still volumepreserving, the actions can have a strong drift along resonance channels. Averaging theory is used to compute the drift for the case of rankr resonances. A comparison with computations for a generalized Froeschlé map in fourdimensions, shows that this theory gives accurate results for the rankone case.
 Supplements

Notes
7.71 MB application/pdf


10:30 AM  11:00 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


From Particle Noise to Coherent XRays: Beam Dynamics of Xray FreeElectron Lasers
Kwang Je Kim (Enrico Fermi Institute (EFI))

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
The successful development of Xray freeelectron lasers (XFELs) as a major scientific instrument is one of the most significant advances in accelerator physics during the last several decades. Central to this development was the accurate prediction of the process in which the initially incoherent radiation from individual electrons evolves to an intense coherent radiation in a long undulator. The calculation involves Klimontovich equation describing the evolution of the electrons’ motion in 6D phase space and 3D Maxwell equation. These coupled equations can be solved in the exponential gain regime by treating the high frequency part of the Klimontovich distribution, which contains electrons’ discreteness and modulations from the FEL interaction, to be small compared to the smooth background. A formal solution can be written down in terms of the VanKampen modes, and the resulting equations can then be solved numerically. The excellent agreement of this analysis with the results of elaborate simulation codes gave a solid basis for a practical design of an XFEL.
 Supplements

Notes
2.2 MB application/pdf


12:00 PM  02:00 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


KAM theory for ultradifferentiable Hamiltonians
Abed Bounemoura (Université de Paris IX (ParisDauphine))

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
We propose an extension of the KAM theorem to a class of ultradifferentiable Hamiltonians (including analytic and Gevrey Hamiltonians) under an adapted arithmetic condition (corresponding to the BrunoRüssmann condition in the analytic case). This is joint work with Jacques Féjoz.
 Supplements

Notes
772 KB application/pdf


03:00 PM  03:30 PM


Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Gyroscopic analogy of a rotating stratified flow confined in a tilted spheroid with a heavy symmetrical top with the top axis misaligned from the axis of symmetry
Yasuhide Fukumoto (Institute of Mathematics for Industry)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
We address the suppression of the gravitational instability of rotating stratified flows in a confined geometry. A rotating flow of a stratified fluid confined in an ellipsoid, subject to gravity force, whose velocity and density fields are linear in coordinates, bears an analogy with a mechanical system of finite degrees of freedom, that is, a heavy rigid body. An insight is gained into the mechanism of system rotation for the ability of a lighter fluid of sustaining, on top of it, a heavier fluid when the angular velocity is greater than a critical value. The sleeping top corresponds to such a state. We show that a rotating stratified flow confined in a tilted spheroid is equivalent to a heavy symmetrical top with the symmetric axis tilted from the top axis. The effect of this misalignment on the linear stability of the sleeping top and its bifurcation is investigated in some detail.
 Supplements

Notes
3.01 MB application/pdf



Nov 30, 2018
Friday

09:30 AM  10:30 AM


A weaker notion of convexity for Lagrangians not depending solely on velocities and positions.
Wildrid Gangbo (University of California, Los Angeles)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
In dynamical systems, one often encounters actions A ≡ RΩLx, v(x)%dx which
depend only on v, the velocity of the system and on % the distribution of the particles. In
this case, it is well–understood that convexity of L(x, ·) is the right notion to study variational
problems. In this talk, we consider a weaker notion of convexity which seems appropriate
when the action depends on other quantities such as electro–magnetic fields. Thanks to the
introduction of a gauge, we will argue why our problem reduces to understanding the relaxation
of a functional defined on the set of differential forms (Joint work with B. Dacorogna).
 Supplements

Notes
943 KB application/pdf


10:30 AM  11:00 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Hamiltonian Instability via Geometric Method
Marian Gidea (Yeshiva University)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
We will describe a geometric method to prove instability in nearly integrable Hamiltonian systems of ndegrees of freedom. The approach is based on tracking the `outer dynamics’ along homoclinic orbits to a normally hyperbolic invariant manifold (NHIM). Only little information is needed on the `inner dynamics' restricted to the NHIM, so this applies to rather general situations; for instance, the unperturbed Hamiltonian does not need to be convex. The conditions needed for this approach are transversality conditions and hence generic. Moreover, these conditions can be verified in concrete systems, such as in celestial mechanics.
 Supplements

Notes
891 KB application/pdf


12:00 PM  02:00 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Erratic behaviour for onedimensional random walks in a generic quasiperiodic environment
Maria Saprykina (Royal Institute of Technology (KTH))

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
Consider a Markov chain on a onedimensional torus $\mathbb T$, where a moving point jumps from $x$ to $x\pm \alpha$ with probabilities $p(x)$ and $1p(x)$, respectively, for some fixed function $p\in C^{\infty}(\mathbb T, (0,1))$ and $\alpha\in\mathbb R\setminus \mathbb Q$. Such Markov chains are called random walks in a quasiperiodic environment. It was shown by Ya.~Sinai that for Diophantine $\alpha$ the corresponding random walk has an absolutely continuous invariant measure, and the distribution of any point after $n$ steps converges to this measure. Moreover, the Central Limit Theorem with linear drift and variance holds. In contrast to these results, we show that random walks with a Liouvillian frequency $\alpha$ generically exhibit an erratic statistical behaviour. In particular, for a generic $p$, the corresponding random walk does not have an absolutely continuous invariant measure, both drift and variance exhibit wild oscillations (being logarithmic at some times and almost linear at other times), Central Limit Theorem does not hold. These results are obtained in a joint work with Dmitry Dolgopyat and Bassam Fayad.
 Supplements

Notes
907 KB application/pdf


03:00 PM  03:30 PM


Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Recurrence on abelian coverings
Albert Fathi (Georgia Institute of Technology; École Normale Supérieure de Lyon)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
 
 Supplements

Notes
755 KB application/pdf


