
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 45minute 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
JeanClaude Zambrini, Math Dep, GFMUL, Universidade de Lisboa
All organized events are in the Applied Computational Engineering and Sciences Building (ACES or equivalently ACE)

Room  ACE 6.304  ACE 4.304  ACE 6.304  ACE 2.402 
Chair  Diogo Gomes  JeanClaude Zambrini  Rafael de la Llave  Atle Hahn 
8:45  9:45  Irene Gamba: Extensions of the Kac Nparticle model to multi linear interactions  David Nualart: Stochastic partial differential equations: Regularity of the probability law of the solution  Ricardo Coutinho: Planar fronts in bistable coupled map lattices  Vladimir Dragović: Integrable billiards, PonceletDarboux grids and Kowalevski top 
9:45  10  coffee break  coffee break  coffee break  coffee break 
10  11  Helder Vilarinho: Strong stochastic stability for nonuniformly expanding maps  Lawrence E. Thomas: Stochastic wave equation model for heatflow in nonequilibrium statistical mechanics  Jay Mireles James: Homoclinic Tangle Dynamics in a VortexBubble  Rafael de la Llave: Invariant objects in coupled map lattices 
1111:15  break  break  break  break 
11:15  12:15  Thomas Chen: On the Boltzmann limit for a Fermi gas in a random medium with dynamical HartreeFock interactions  Kenji Nakanishi: Equivariant LandauLifshitz equation of degree two  Radosław Wojciechowski: Stochastic completeness of graphs  Hans Koch: Shadowing orbits for dissipative PDEs 
12:15  2  lunch break  lunch break  lunch break  adjourn 



José Ferreira Alves:
Recurrence times toward mixing rates and viceversa
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 HartreeFock 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 selfinteractions modeled in dynamical
HartreeFock 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 onedimensional extended dynamical system that
generalises onedimensional 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.
→ link to presentation
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, PonceletDarboux grids and Kowalevski top
A progress in a thirty years old programme of Griffiths and
Harris of understanding of higherdimensional analogues of Poncelet
porisms and synthetic approach to higher genera addition theorems is
presented. A set T of lines tangent to d1 quadrics from a given
confocal family in a ddimensional space is equipped with an
algebraic operation. Using it, wellknown 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 d1
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 higherdimensional and highergenera 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 nvalued BuchstaberNovikov 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 Nparticle model to multi linear interactions
We look at extensions Kac Nparticle model of pair interactions
to an Nparticle model which includes multiparticle 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 multiagent 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 AubryMather Measures
In this talk we use the adjoint method introduced by Evans to construct
analogs to the AubryMather measures for nonconvex Hamiltonians.
In particular we prove the existence of AubryMather measures
for a class of strictly quasiconvex Hamiltonians.
Atle Hahn:
A rigorous approach to the nonAbelian ChernSimons path integral
The study of the heuristic ChernSimons 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 Rmatrices or quantum 6jsymbols
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 VortexBubble
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
quasinumerical 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.
→ link to presentation
Hans Koch:
Shadowing orbits for dissipative PDEs
We describe a computerassisted technique for constructing
and analyzing orbits of dissipative evolution equations.
As a case study, the methods are applied to the KuramotoSivashinski 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 LandauLifshitz equation of degree two
This is recent progress in the joint work with Stephen Gustafson and
TaiPeng Tsai on the global dynamics of the LandauLifshitz 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
FeynmanKac formula will be also presented.
→ link to presentation
Ambar N. Sengupta:
YangMills in 2 dimensions for U(N) and its largeN limit
We will present a description of quantum YangMills theory on the plane
with gauge group U(N), and the limiting behavior of this theory as
N goes to infinity.
→ link to presentation
Lawrence E. Thomas:
Stochastic wave equation model for heatflow in nonequilibrium statistical mechanics
We consider a onedimensional nonlinear
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
steadystate 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 noncompact extensions of hyperbolic systems
Consider the restriction to a hyperbolic basic set of a smooth
diffeomorphism. We are interested in the transitivity of Hölder
skewextensions with fiber a noncompact connected Lie group.
In the case of compact fibers, the transitive extensions contain an open
and dense set. For the noncompact 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 Euclideantype groups G ∝ R^{n}
where G is a compact connected Lie group acting linearly on R^{n}.
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 nonuniformly expanding maps
We address the strong stochastic stability of a broad class
of discretetime dynamical systems  nonuniformly 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 L1norm, and in a more general framework of
random perturbations. We illustrate our main result for two examples
of nonuniformly 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.
JeanClaude 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) 2583826, or 1800BLUE 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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