**Abstracts:**

**Plenary speakers:**

Niky Kamran (McGill University)

**Wave Equations in General Relativity**

**Adrian Nachman **(University of Toronto)

**Introduction to Inverse Problems in Partial Differential Equations
**

**Participants:**

M. Agueh (University of British
Columbia)

**Asymptotic Behavior for Doubly Degenerate Parabolic Equations**

We use mass transportation inequalities to study the asymptotic behavior
for a class of parabolic equations which contains the Heat,

Fokker-Planck, porous-medium, fast diffusion, and parabolic $p$-Laplacian
equations, and some generalizations of these equations. We establish
an exponential decay in "relative entropy" and in the $p$-Wasserstein
distance of solutions -- or self-similar solutions -- of these equations
to equilibrium, and we give the explicit rates of convergence. In particular,
we generalize to all $p>1$ the HWI inequalities obtained by Otto
and Villani (J.Funct.Anal.173 (2) (2002) 361-400) when $p=2$.

**N. Ahmad** (University
of Toronto)

*Boundary Matching via Optimal Transportation*

**R. Akila** (University
of Guelph)

*Hopf Bifurcation of Coupled Oscillator Systems*

Two-dimensional arrays of regularly spaced oscillators, with
symmetric nearest neighbor coupling, may synchronize in patterns that
are periodic in both space and time. This work presents a complete list
of invariants, equivariants, normal forms, isotropy subgroups and fixed-point
subspaces, for the cases with spatial periodicity N = 2,3,4, both with
and without a Z2-internal symmetry. The analysis includes all the generic
equivariant Hopf bifurcations in this setting, and determines the onset
and stability of the patterns. This study is motivated by flow- induced
vibrations of arrays of tubes in heat exchangers with cross-flow, and
leads to predictions of the spatio-temporal patterns in such arrays.

**W. Bahsoun** (Concordia
University)

**Ergodic Properties of Random Maps and Their Applications**

A random map is a discrete dynamical system consisting of a collection
of maps which are selected randomly by means of probabilities at each
iteration. We prove the existence of absolutely continuous invariant
measures for a general class of position dependent random maps under
mild conditions. Moreover, we prove that these measures are stable under
small stochastic perturbations. We also apply these results to forecasting
in financial markets.

** A. Biryuk** (The
Fields Institute & McMaster University)

**Flows Without Pressure, Geometry and Turbulence**

We start with the Cauchy problem for the multidimensional Burgers
type equation with periodic boundary conditions. We introduce the notion
of degeneracy for vector fields. Vector field $u$ is degenerate iff
one of the following equivalent conditions holds: (*) the Jacoby matrix
of $u$ is everywhere nilpotent (*) the pressureless Euler equation with
the initial state $u$ is globally solvable in the class of $C1$-continuous
functions. Non-degenerate initial states develops large spatial derivatives
(turbulence). In 2D there is a nice geometric criterion for degeneracy
due to Pogorelov cylinder theorem.

**M. Braverman**
(University of Toronto)

**On the Computability of Julia Sets**

While the computer is a discrete device, it is often used to solve problems
of a continuous nature. The field of Real Computation addresses the
issues of computability in the continuous setting. As in the discrete
case, we would like to define the notion of a computable subset of R^n.
The definition we use has a computer graphics interpretation (in the
case n=2), as well as a deeper mathematical meaning.

The Julia sets are particularly well studied sets arising from complex
dynamics. In the talk we will discuss efficient computability of real
sets, and describe some results about the computability of Julia sets.
Our computability results come in contrast to the Julia sets noncomputability
results presented by Blum/Cucker/Shub/Smale. This discrepancy follows
from the fact that we are using a different computability model.

**M. Cojocaru** (University
of Guelph)

**Projected Dynamical Systems: Overview and Recent Developments**

A projected dynamical system (PDS) is defined by an ODE with a nonlinear
and discontinuous righthand side whose solutions are constrained to
evolve only within a closed and convex subset K of a Hilbert space X.
A PDS was first introduced in 1992, however references to the ODE defining
it go back to the 70's. It is worth mentioning that it was first formulated
in the context of applied economic problems. In this talk, we present
a brief overview of this kind of systems, the mathematical questions
that were studied so far in its case and some recent developments on
the topic.

**F. Colin** (Universite
de Montreal)

*Decomposition Lemmas Applied to Minimization Problems*

**D. Gaidashev** (University
of Toronto)

**Renormalization of Isoenergetically Degenerate Hamiltonian Flows**

KAM proofs of existence of invariant surfaces for isoenergetically degenarate
(non-twist) Hamiltonian flows are known to present significant difficulties.
We will demonstrate how the renormalization group transformation, a
non-perturbative technique, can be used to

study problems which are traditionally the subject of the KAM theory.
We will also mention some recent results about universality and self-similarity
of critical Hamiltonian flows whose invariant tori are at the break-up.

**P. Guyenne***
*(The Fields Institute & Mcmaster University)

**Wave Turbulence in One-dimensional Models**

A two-parameter nonlinear dispersive wave equation proposed by Majda,
McLaughlin and Tabak (1997) is studied analytically and numerically
as a model for testing the validity of weak turbulence theory. Kolmogorov-type
solutions for the energy spectrum are determined explicitly and compared
with numerical results. These show a strong dependence on the sign of
the nonlinear term. In one case, there is agreement with the theory.
In the other, there is disagreement. Possible explanations for this
discrepancy will be given such as the emergence of coherent structures:
quasi-solitons and wave collapses.

**M. Harada** (University
of Toronto)

**Integrable systems, polytopes, and skew-hermitian matrices**

One of the main themes of symplectic geometry in the past few decades
has been the link between torus actions and polytopes. I will discuss
this phenomenon in cases where the symplectic manifolds are familiar
from linear algebra: spaces of skew-hermitian matrices (over either
\C or the quaternions \H) conjugate to a given diagonal matrix. When
one works over \C, a construction of Gel'fand-Cetlin, Guillemin-Sternberg
links, via a torus action, a maximal polytope to these spaces of matrices,
giving an integrable system. I will explain how to get an integrable
system also for the skew-hermitian matrices over \H.

**Z. Hu** (Entrust Inc. &
Carleton University)

**Favard's Theorem for Stepanov Almost Periodic Differential Equations**

In the theory of Bohr almost periodic differential equations, Bochner's
Theorem plays an important role in discussing the existence of almost
periodic solutions, in particular, in establishing Favard's Theorem.
In this paper, we, firstly, extend Bochner's Theorem to the case of
Stepanov almost periodic functions and then we establish a result about
the existence of Stepanov almost periodic solutions for

linear Stepanov almost periodic differential equations on $R^n$, that
is, we extend Favard's Theorem to the case of Stepanov almost periodic
differential equations.

**K. Kang** (University of
British Columbia)

**On Regularity to the Stokes System and Navier-Stokes Equations
Near Boundary**

We study regularity problem for the Stokes system (SS) and the Navier-Stokes
equations (NSE) near boundary. For the steady-state case we obtained
local estimates of the SS "without pressure" and as an application
we prove the partial regularity up to the boundary for the stationary
NSE in five dimension. For the non-stationary SS we constructed an example,
which shows, unlike in the interior case,

H\"{o}lder continuity does not imply smoothness in the spatial
variable near boundary. For the NSE We proved that weak solutions,

which is locally in $L^{p,q}$ with $3/p+n/q=1, q>n$ near boundary
are regular up to the boundary. In three dimension we also observed

that "suitable weak solutions" of the NSE are regular near
boundary if the scaled $L^{r,s}$-norm with $3/r+2/s=2, 2<s<\infty$
of the velocity is sufficiently small.

**T. Kolokolnikov** (University
of British Columbia)

**Stripe Instabilities in the Two-Dimensional Gray-Scott Model**

The Gray-Scott model is a reaction-diffusion system that is
known to exhibit complicated spatial patterns. These include: stripes,
rings, spots, domain-filling curves and any combination thereof. In
this talk, we consider the stripe solutions in two dimensions. Such
a solution can exhibit three different types of instability: a splitting
instability, whereby a stripe self-replicates into two parallel stripes;
a breakup instability, where a stripe breaks up into spots; and a zigzag
instability, whereby a stripe develops a wavy perturbation in the transversal
direction. We derive explicit thresholds for all three types of instability.
Some open problems will be discussed.

**A. Kuznetsov**
(University of Toronto)

**Solvable Markov Models with Stochastic Volatility and Jumps**

There are only few known examples of Markov processes for which
the transitional probability density can be computed in analytically
closed form. Our method gives a general framework for building a family
of analytically tractable models, which include all the previously known
processes, and enables us to construct new solvable martingale processes
having such important properties as state-dependent volatility and/or
stochastic volatility and/or jumps. The probability kernel is expressed
as expansion in orthogonal polynomials. The method can be summarized
as the change of measure, change of phase space for the process and
the spectrum deformation for the generator of the Markov semigroup.
I will also discuss the lattice approximations for these models and
some applications to Mathematical Finance.

**S. Lawi** (University
of Toronto)

**Generating Functions for Stochastic Integrals**

Generating functions for stochastic integrals have been known in analytically
closed form for just a handful of stochastic processes: namely, the
Ornstein-Uhlenbeck, the Cox-Ingerssol-Ross (CIR) process and the exponential
of Brownian motion. In virtue of their analytical tractability, these
processes are extensively used in modelling applications. In this paper,
we construct broad extensions of these process classes. We show how
the known models fit into a classification scheme for diffusion processes
for which generating functions for stochastic integrals and transition
probability densities can be evaluated as integrals of hypergeometric
functions against the spectral measure for certain self-adjoint operators.
We also extend this scheme to a class of finite-state Markov processes
related to hypergeometric polynomials in the discrete series of the
Askey-Wilson classification tree.

**B. Moore*** *(McGill
University)

**Numerical Methods that Preserve a Multi-Symplectic Conservation
Law**

A useful way to understand symplectic integration of Hamiltonian ODEs
is through the system of equations, known as the modified equations,
which are solved by the numerical solution. Here, the ideas of symplectic
integration are extended to Hamiltonian PDEs, such that the symplectic
structure in both space and time is exactly preserved. This paves the
way for the development of a local modified equation analysis solely
as a useful diagnostic tool for the study of these methods. In particular,
the modified equations are used to derive

modified conservation laws of energy and momentum that are preserved
to higher order along the numerical solution. For linear PDEs it is
possible to show that the modified equations converge to the numerical
scheme, and one is able to derive dispersion relations that completely
describe the numerical solution behavior.

**V. Panferov***
*(University of Victoria)

*Comparison Principles and Pointwise Upper Bounds for Solutions
of Boltzmann-type Equations*

I will describe a technique for obtaining pointwise upper bounds for
solutions of the Boltzmann equation for elastic and inelastic particles.
The results are relevant for description of tails of velocity distributions
in rarefied gases and granular media.

**A. Savu** (University
of Toronto)

**Hydrodynamic Scaling Limit of the Fourth Order Ginzburg-Landau
Model**

The fourth order Ginzburg-Landau model is a microscopic model for the
diffusion of particles on a surface relaxing to equilibrium. In my talk
I will discusss how the evolution of the surface, on the macroscopic
scale, given by a fourth order nonlinear evolution equation, emerges
as a scaling limit of the particle dynamics. Since the model is of non-gradient
type a major step in the computation of the limit is finding the right
decomposition of the Hilbert space of "closed functions".

**D. Slepcev*** *(University of Toronto)

**Gradient-Flow Structure and Stability of Selfsimilar Solutions
of Nonlinear Parabolic PDE's**

Nonlinear diffusion equations (porous-medium, fast-diffusion equation)
and thin-film equations (with certain types of nonlinearities) can be
recast as a gradient flows on an infinite-dimensional manifold.

The gradient-flow structure of these equations suggests a framework
in which to study linear stability of selfsimilar solutions. Furthermore,
in some cases it provides a way to show the asymptotic stability of
the selfsimilar solutions, with optimal rates of convergence.

Of particular interest to us will be the stability of blow-up profiles
of long-wave unstable thin-film equations. Future directions and open
problems will also be discussed.

**P. Tupper*** *(McGill
University)

**Ergodicity and Numerical Simulation**

An outstanding problem in numerical analysis is to justify long-time
simulations of molecular systems. Typically, these systems are chaotic
and numerically computed trajectories are accurate for only short periods
of time. Nevertheless, qualitative features of computed trajectories
are often quite accurate. I will discuss some recent work on this problem.
The main innovation is the introduction of a weak ergodicity that suffices
for the purposes of numerical simulation but is easier to work with
than the standard mathematical definition.

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