This Workshop will be held under the auspices of the international partnership between the University of Texas at Austin and Portuguese Universities, a part of the International Collaboratory for Emerging Technologies.
The Workshop will consist of a series of 45-minute plenary session talks.

José Ferreira Alves, Math Dep, FC, Universidade do Porto

Thomas Chen, Math Dep, The University of Texas at Austin

Ricardo Coutinho, Math Dep, IST, Universidade Técnica de Lisboa

Rafael de la Llave, Math Dep, The University of Texas at Austin

Vladimir Dragović, Math Dep, GFMUL, Universidade de Lisboa

Irene Gamba, Math Dep, The University of Texas at Austin

Diogo Gomes, Math Dep, IST, Universidade Técnica de Lisboa

Atle Hahn, Math Dep, GFMUL, Universidade de Lisboa

Jay Mireles James, Math Dep, Rutgers University

Hans Koch, Math Dep, The University of Texas at Austin

Kenji Nakanishi, Math Dep, Kyoto University

David Nualart, Math Dep, University of Kansas

Ambar N. Sengupta, Math Dep, Louisiana State University

Lawrence E. Thomas, Math Dep, University of Virginia

Andrew Török, Math Dep, University of Houston

Helder Vilarinho, Math Dep, Universidade da Beira Interior

Radosław Wojciechowski, Math Dep, GFMUL, Universidade de Lisboa

Jean-Claude Zambrini, Math Dep, GFMUL, Universidade de Lisboa

All organized events are in the Applied Computational Engineering and Sciences Building (ACES or equivalently ACE)

José Ferreira Alves: Recurrence times toward mixing rates and vice-versa
One of the most efficient tools for studying the mixing rates of certain classes of dynamical systems is through Young towers: if a given system admits an inducing scheme whose tail of recurrence times decays at a given speed, then that system admits a physical measure with mixing rate of the same order. In this talk we shall consider the inverse problem: assume that a given dynamical system has a physical measure with a certain mixing rate; under which conditions does that measure come from an inducing scheme with the tail of recurrence times decaying at the same speed? We have optimal results for the polynomial case. The exponential case raises interesting questions on the regularity of the observables.

Thomas Chen: On the Boltzmann limit for a Fermi gas in a random medium with dynamical Hartree-Fock interactions
In this talk, we address the dynamics of a Fermi gas in a weakly disordered random medium. We first present some joint results with I. Sasaki (Shinshu University) on the Boltzmann limit for the thermal momentum distribution function, and on the persistence of quasifreeness, for the case of a free Fermi gas in a random medium. Subsequently, we present recent joint results with I. Rodnianski (Princeton University) on the derivation of the Boltzmann limit for a Fermi gas in a random medium with nonlinear self-interactions modeled in dynamical Hartree-Fock theory.
→ link to presentation

Ricardo Coutinho: Planar fronts in bistable coupled map lattices
Planar fronts in multidimensional coupled map lattices can be studied by reduction to an one-dimensional extended dynamical system that generalises one-dimensional coupled map lattices. This methodology is fully investigated and developed. Continuity of fronts velocity with the coupling strength and with the propagation direction is proven. Examples are provided and illustrated by some numerical pictures.
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Rafael de la Llave: Invariant objects in coupled map lattices
We consider infinite dimensional systems that consist of copies of a finite dimensional system at each point in the lattice coupled by interactions which decrease fast enough. These objects have appeared in applications under the name of "coupled map lattices", "oscillator networks" and in discretizations of PDE's. We consider in detail hyperbolic systems and their invariant manifolds. When the system is Hamiltonian, we also consider whiskered invariant tori and their invariant manifolds. The method allows to consider the persistence of tori with finitely many or infinitely many frequencies.
Joint work with E. Fontich, P. Martin, Y. Sire (previous work with M. Jiang)

Vladimir Dragović: Integrable billiards, Poncelet-Darboux grids and Kowalevski top
A progress in a thirty years old programme of Griffiths and Harris of understanding of higher-dimensional analogues of Poncelet porisms and synthetic approach to higher genera addition theorems is presented. A set T of lines tangent to d-1 quadrics from a given confocal family in a d-dimensional space is equipped with an algebraic operation. Using it, well-known results of Donagi, Reid and Knorrer are developed further. We derive a fundamental property of T: any two lines from T can be obtained from each other by at most d-1 billiard reflections at some quadrics of the confocal family. The interrelations among billiard dynamics, linear subspaces of intersections of quadrics and hyperelliptic Jacobians enabled us to obtain higher-dimensional and higher-genera generalizations of several classical genus 1 results. Among several applications, a new view on the Kowalevski top and Kowalevski integration procedure is presented. It is based on a classical notion of Darboux coordinates, a modern concept of n-valued Buchstaber-Novikov groups and a new notion of discriminant separability. Unexpected relationship ith the Great Poncelet Theorem for a triangle is established.

Irene Gamba: Extensions of the Kac N-particle model to multi linear interactions
We look at extensions Kac N-particle model of pair interactions to an N-particle model which includes multi-particle interactions in order to study the evolution of the corresponding probability density solution. Under the assumption of temporal invariance under scaling transformations of the phase space and contractive properties, we obtain a full description of existence, uniqueness and long time behavior from its spectral properties. This model can also be seen as an extension of the Boltzmann dynamics of Maxwell type for conservative or dissipative interactions and the formation of power tails for long time self similar behavior under very general conditions for the initial energy.
We will also focus on a couple of new examples of multi-agent dynamics and information percolation and some numerical simulations.
This is work is in collaboration with A. Bobylev, C. Cercignani. The Numerical simulations are in collaboration with Harsha Tharkabhushanam and the recent studies for information dynamics models with Ravi Srinivasan.

Diogo Gomes: Non Convex Aubry-Mather Measures
In this talk we use the adjoint method introduced by Evans to construct analogs to the Aubry-Mather measures for non-convex Hamiltonians. In particular we prove the existence of Aubry-Mather measures for a class of strictly quasiconvex Hamiltonians.

Atle Hahn: A rigorous approach to the non-Abelian Chern-Simons path integral
The study of the heuristic Chern-Simons path integral by E. Witten inspired (at least) two general approaches to quantum topology. Firstly, the perturbative approach based on the CS path integral in the Lorentz gauge and, secondly, the "quantum group approach" by Reshetikhin/Turaev. While for the first approach the relation to the CS path integral is obvious for the second approach it is not. In particular, it is not clear if/how one can derive the relevant R-matrices or quantum 6j-symbols directly from the CS path integral.
In my talk, which summarizes the results of a recent preprint, I will sketch a strategy that might lead to a clarification of this issue in the special case where the base manifold is of product form. This strategy is based on the "torus gauge fixing" procedure introduced by M. Blau and G. Thompson for the study of the partition function of CS models. I will show that the formulas of Blau & Thompson can be generalized to Wilson lines and that at least for the simplest types of links the evaluation of the expectation values of these Wilson lines leads to the same state sum expressions in terms of which Turaev's shadow invariant is defined.
Finally, I will sketch how - using methods from Stochastic Analysis or, alternatively, a suitable discretization approach - one can obtain a rigorous realization of the path integral expressions appearing in this treatment.
→ link to presentation

Jay Mireles James: Homoclinic Tangle Dynamics in a Vortex-Bubble
We consider a three dimensional, quadratic, volume preserving map, which is a normal form for quadratic diffeomorphisms with quadratic inverse. The map also serves as a toy model for a certain type of vortex dynamics which arises in fluid and plasma physics. We will discuss a quasi-numerical numerical scheme, based on the Parameterization Method, for accurately computing the one and two dimensional stable and unstable manifolds of the maps fixed points. Studying the embedding of the stable and unstable manifolds provides insights into the chaotic motions in the vortex.
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Hans Koch: Shadowing orbits for dissipative PDEs
We describe a computer-assisted technique for constructing and analyzing orbits of dissipative evolution equations. As a case study, the methods are applied to the Kuramoto-Sivashinski equation. In particular, we give a partial description of the bifurcation diagram for stationary solution, involving 23 bifurcations and 44 branches. More general orbits are obtained by solving the Duhamel equation for small time intervals, and then using shadowing techniques (covering relations). We will describe estimates on the flow, its derivative, Poincaré maps, and a proof for the existence of a hyperbolic periodic orbit.
This is joint work with Gianni Arioli (Politecnico di Milano).
→ link to presentation

Kenji Nakanishi: Equivariant Landau-Lifshitz equation of degree two
This is recent progress in the joint work with Stephen Gustafson and Tai-Peng Tsai on the global dynamics of the Landau-Lifshitz equation around the ground states under the equivariant symmetry. Previously we proved that in the degree higher than two, every solution with energy close to the ground states converges to a ground state of a fixed scaling at time infinity, whereas in the degree two, the family of the ground states is still asymptotically stable but the scaling parameter can blow up or oscillate at time infinity. For the latter result, however, we needed additional restrictions that the dispersion was absent (i.e. the heat flow), and the map modulo the equivariant rotation was confined in a great circle. I will show how we remove those restrictions for the asymptotic stability.

David Nualart: Stochastic partial differential equations: Regularity of the probability law of the solution
We will present some recent results on the regularity of the density of the solution of a general class of stochastic differential equations driven by a Gaussian white noise with an homogeneous spacial covariance. To show that the density of the solution is infinitely differentiable we apply the techniques of Malliavin calculus, and we require the diffusion coefficient to satisfy some non degeneracy conditions. We will also discuss the relation of this problem with the existence of negative moments for solutions to linear stochastic partial differential equations with random coefficients. A recent approach to this question using a stochastic version of Feynman-Kac formula will be also presented.
→ link to presentation

Ambar N. Sengupta: Yang-Mills in 2 dimensions for U(N) and its large-N limit
We will present a description of quantum Yang-Mills theory on the plane with gauge group U(N), and the limiting behavior of this theory as N goes to infinity.
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Lawrence E. Thomas: Stochastic wave equation model for heat-flow in non-equilibrium statistical mechanics
We consider a one-dimensional non-linear stochastic wave equation system modeling heat flow between thermal reservoirs at different temperatures. We will briefly review the problem of solving these equations in Sobolev spaces of low regularity. The system with ultraviolet cutoffs has, for each cutoff, a unique invariant measure exhibiting steady-state heat flow. We provide estimates on the field covariances with respect to the invariant measures which are uniform in the cutoffs.
→ link to presentation

Andrew Török: Transitivity of non-compact extensions of hyperbolic systems
Consider the restriction to a hyperbolic basic set of a smooth diffeomorphism. We are interested in the transitivity of Hölder skew-extensions with fiber a non-compact connected Lie group.
In the case of compact fibers, the transitive extensions contain an open and dense set. For the non-compact case, we conjectured that this is still true within the set of extensions that avoid the obvious obstructions to transitivity.
We will discuss results that support this conjecture.
For r > 0, we show that in the class of C^r-cocycles with fiber the special Euclidean group SE(n), those that are transitive form a residual set (countable intersection of open dense sets). This result is new for n ≥ 3 odd. More generally, we consider Euclidean-type groups G ∝ Rⁿ where G is a compact connected Lie group acting linearly on Rⁿ. When Fix G = {0}, it is again the case that the transitive cocycles are residual. When Fix G ≠ {0}, the same result holds on the subset of cocycles that avoid an obvious and explicit obstruction to transitivity.
We also prove such genericity results for a class of nilpotent groups.
This is joint work with Ian Melbourne and Viorel Nitica.

Helder Vilarinho: Strong stochastic stability for non-uniformly expanding maps
We address the strong stochastic stability of a broad class of discrete-time dynamical systems - non-uniformly expanding maps - when some random noise is introduced in the deterministic dynamics. A weaker form of stochastic stability for this systems was established by J.F. Alves and V. Arajo (2003) in the sense of convergence of the physical measure to the SRB probability measure in the weak* topology. We present a strategy to improve this result in order to obtain the strong stochastic stability, i.e., the convergence of the density of the physical measure to the density of the SRB probability measure in the L1-norm, and in a more general framework of random perturbations. We illustrate our main result for two examples of non-uniformly expanding maps: the first is related to an open class of local diffeomorphisms introduced by J.F. Alves, C. Bonatti and M. Viana (2000) and the second to Viana maps - a higher dimensional example with critical set introduced by M. Viana (1997).
This is a joint work with J.F. Alves.

Radosław Wojciechowski: Stochastic completeness of graphs
We introduce the heat kernel on graphs and give geometric conditions which imply the stochastic completeness or incompleteness of the underlying diffusion process. Furthermore, connections to the spectrum of the discrete Laplacian will be considered. The proofs will rely on studying the stability of solutions of difference equations.

Jean-Claude Zambrini: Stochastic reversible deformation of dynamical systems
We shall describe a program of symmetrization in time of Stochastic Analysis. Its main purpose is to deform stochastically the classical approaches to the theory of elementary dynamical systems, but it may be of interest more generally when random modeling of reversible phenomena is necessary.
→ link to presentation

Travel arrangements and booking are the responsibility of the individual participants. The workshop will begin at 8:45am on March 31 and end at 12:15pm on April 3.

Arrival and Departure

The Austin Bergstrom International Airport (AUS) is the closest airport to Austin and the University of Texas. For transportation to and from the airport, feel free to take a taxi or the Super Shuttle. The Super Shuttle can be reached at (512) 258-3826, or 1-800-BLUE VAN, or online; or you can book a ride upon arrival.

Lodging

We have reserved a block of rooms for participants of this workshop at the

To book a room at the Extended Stay America, please contact the hotel by phone or email.

The dates of the contract are March 30 (arrival) to April 4 (departure). If you arrive earlier and/or leave later (e.g. to get a better airfare) please book your room as soon as possible. Availability outside the dates of the contract is not guaranteed, and it decreases with time.

Local Transportation

For transportation to campus the Capitol Metro Airport Flyer #100 bus route (inbound) stops one block from the Extended Stay America hotel at 6th & Guadalupe, approximately every 35 minutes. The fare is $1 per trip or $1.50 for a 24 hour pass on any Capital Metro bus. The #100 bus stops on the University campus at Dean Keeton and Speedway, near the Math building (RLM) and one block north of the conference location (ACE). Several other bus routes service the University from Congress avenue, a few blocks east of the hotel, including routes 1L, 1M, 5, and 7.

Maps and directions for the UT campus are available here and here and here.

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