Title: Rigorous Verification of Stability of Traveling Waves Via Computer Assisted Proof
Abstract: We discuss various aspects of numerical stability analysis of
periodic roll wave solutions arising in equations of inclined thin film flow,
with an eye toward the development of guaranteed error bounds. In particular,
we rigorously verify stability of a family of periodic wave solutions arising
in a generalized Kuramoto-Sivashinsky equation in the Korteweg-de Vries limit,
that is, in the limit viscosity goes to zero.
Abstract: A very classical question is to consider a polygon whose angles are
rational multiples of pi and study the trajectories under the billiard flow
(straight lines inside the polygon with optical reflection in the edges). An
obvious question is which directions have periodic trajectories or, at the
other extreme, are uniquely ergodic. Billiards tie in to Teichmüller theory
(the study of varying geometric structures on a fixed topological surface) in
a beautiful way, and some recent work in Teichmüller dynamics has been much
celebrated (for instance, it's a primary interest for two of last summer's
four Fields medalists!). I'll survey a small corner of this research area and
discuss some of my own work in this direction.
Title: Quantitative Systems Pharmacology: Enabling drug invention from Research
through Clinical Trials
Abstract: Quantitative Systems Pharmacology (QSP) modeling has been used
successfully in pharma. The cases include: feasibility assessments; prediction
of best-in-class drug properties; and clinical trial support. In each case,
systems approaches enabled decisions that traditional methods could not
address as easily or as early in the pipeline. Unlike traditional PK/PD
models, QSP models leverage known biophysical interactions and integrate
data from a variety of sources (in vitro, in vivo, and clinical)and act as
a central repository of data and knowledge of human pathomechanisms,
allowing for the exploration of hypotheses that cannot be fully tested
prior to dosing patients.
These case studies show the potential of QSP approaches to enable earlier
quantitative decision making with the end goal of reducing late stage
attrition and accelerating premier therapeutics to address unmet medical
need.
Abstract: 2015 is the international year of light. What better illustrates the
use of light than a sundial? Sundial caches the apparent position of the Sun
in the sky. Like a Pinhole Camera the light from the Sun passes through the
gnomon, often a thin rod, and projects the shadow on a surface. All knowledge
about Earth orbit (Kepler laws) and spherical astronomy or positional
astronomy are used. After presenting some terminology and different types of
sundials, we will talk about time (apparent solar time, mean solar time, ...),
Celestial sphere (apparent motion of the Sun, ...) and Celestial coordinate
(Horizontal, Equatorial, ...). All well mixed leading to the theory of
sundials.
Title: Hypoelliptic multiscale Langevin diffusions: Large deviations, invariant measures and small mass asymptotics
Abstract: We consider a general class of hypoelliptic Langevin diffusions and
study two related questions. The first one is large deviations for
hypoelliptic multiscale diffusions. The second one is small mass asymptotics
of the invariant measure corresponding to hypoelliptic Langevin operators and
of related hypoelliptic PDEs. The invariant measure corresponding to the
hypoelliptic problem and appropriate hypoelliptic PDEs enter the large
deviations rate function due to the multiscale effects. Based on the small
mass asymptotics we derive that the large deviations behavior of the
multiscale hypoelliptic diffusion is consistent with the large deviations
behavior of its overdamped counterpart. Additionally, we rigorously obtain an
asymptotic expansion of the solution to the related hypoelliptic PDEs with
respect to the mass parameter, characterizing the order of convergence. The
proof of convergence of invariant measures is of independent interest, as it
involves an improvement of the hypocoercivity result for the kinetic
Fokker-Planck equation. We do not restrict attention to gradient drifts and
our proof provides explicit information on the dependence of the bounds of
interest in terms of the mass parameter.
Title: On the measure of the Feigenbaum Julia set.
Abstract: We discuss a computer-assisted proof that the Julia Set of the map
$z \mapsto z^2 -1.4011551890\cdots$, which is the limit of period doubling
bifurcations in the quadratic family, has zero Lebesgue measure. This is joint
work with Artem Dudko.
Title: On the dynamics of perturbed Weierstrass P functions over real square lattices
Abstract: Consider the Weierstrass P function defined over a fixed real square
lattice. The dynamical system obtained by iteration of P can be completely
characterized by the behavior of its single free critical orbit. In contrast,
as soon as the function is perturbed by the addition of a complex parameter,
the elliptic function P+b exhibits two free critical orbits, which in
principle, complicates the study of the connectedness and bifurcation loci.
This talk presents results that explain some of the rich structures found in
parameter plane. In particular, we concentrate our study along horizontal
lines of b-values where the family P+b exhibits a variety of dynamical
behavior. Indeed, these lines may contain parameters for which the Julia set
of P+b is either the whole complex plane, or it is connected with nonempty
Fatou set, or it is totally disconnected. We also show that in the instance of
a connected Julia set with nonempty Fatou set, varying the parameter along a
particular horizontal line gives rise to both period-doubling and reversed
period-doubling bifurcations.
This is a joint work with Jane M. Hawkins, UNC-Chapel Hill.
November 16: Russell Lodge (Jacobs University, Bremen)
Title: A combinatorial classification of postcritically finite Newton maps
Title: The mean encounter time between two polymer sites: a Brownian search process in high dimensional manifolds
Abstract: The encounter between two sites on chromosomes in the cell nucleus
can trigger gene regulation, exchange of genetic material and repair of DNA
breaks. By forming a DNA loop, a protein located on the DNA can trigger the
expression of a gene located far along the chain. We computed asymptotically
the mean time encounter time (MFET) between two polymer sites, using the
classical Rouse polymer model, in which the polymer is described as a
collection of bead monomers connected by harmonic springs. When two monomers
come closer than a distance epsilon, the search process ends.
The novel asymptotic relies on the expansion of the spectrum of the
Fokker-Planck operator as a function of the small parameter epsilon. The key
of the asymptotic is the explicit computation of the Riemannian volume for
Chavel-Feldman formula, which gives the shift in the spectrum of the Laplacian
operator when Dirichlet boundary conditions are imposed on the boundary of
tubular neighborhood of a constraint manifold (removal of a manifold with a
small volume). We have shown that the encounter time is Poissonian and
obtained an asymptotic expression of the MFET as function of the polymer
length N and epsilon. The MFET is inversely proportional to the first
eigenvalue of the FP-operator but the expansion is not uniform in epsilon and
N. Finally, we show that the asymptotic estimation of the eigenvalue can be
obtained by a generalized boundary layer analysis in high dimension.
Title: Numerical Multiscale Methods and Lagrangian Data Assimilation
Abstract: Many problems in science and engineering are modelled by time scale
separated differential equations. These systems can rarely be solved
analytically; at the same time, the fast processes restrict the time step of
numerical integrators so that the (high dimensional) system can only be
simulated on short time scales. We are interested in the dynamics of slow
processes that evolve on long time scales and so cannot be approximated by
standard analytical or numerical methods. This talk focusses on the
heterogeneous multiscale and projective integration methods, which efficiently
integrate such multiscale systems to simulate the behaviour of the slow
variables. I will present convergence results for various formulations of both methods
applied to stiff dissipative ordinary differential equations, highlighting
recent contributions to the field. I conclude by discussing potential
applications of these methods to Lagrangian Data Assimilation, and the
pitfalls that must be overcome in order to do so.
Title: The Maslov index and singularities of a matrix Riccati equation
Abstract: In this talk we use the Maslov index as a tool to study stability of
steady state solutions to reaction-diffusion equations in one spatial
dimension. A steady state solution naturally defines a path of Lagrangian
subspaces in $\mathbb{R}^{2n}$, the path of unstable subspaces, and the Maslov
index of this path detects spectral instability of the steady state solution.
Moreover, we show that this path of subspaces is governed by a matrix Riccati
equation. We discuss recent results suggesting the Maslov index can be
computed by the singularities of solutions to this matrix Riccati equation.