William Feldman, University of Chicago
"Interfaces in inhomogeneous media: pinning, hysteresis, and facets"
I will discuss some models for the shape of liquid droplets on rough solid surfaces. The framework of homogenization theory allows to study the large scale effects of small scale surface roughness, including interesting physical phenomena such as contact line pinning, hysteresis, and formation of facets.
William Feldman, University of Chicago
"Recent developments in stochastic homogenization of Hamilton-Jacobi equations "
I will describe some background and some new developments in the theory homogenization of Hamilton-Jacobi equations in random environments. The primary difficulties in this field have been around understanding the roles of convexity and coercivity (controllability). Many interesting problems, especially involving interface motions, lack one or both of these properties. I will discuss some positive results and some counter-examples where homogenization does not hold.
Fei Pu, University of Utah
"The stochastic heat equations: optimal lower bound on hitting probabilities and the probability density function of the supremum "
The main topic of this talk is the study of hitting probabilities
for solutions to systems of stochastic heat equations. We establish
an optimal lower bound on hitting probabilities in the non-Gaussian case,
which is as sharp as that in the Gaussian case. This is achieved by
a sharp Gaussian-type upper bound on the two-point probability density
function of the solution. Motivated by the study of upper bounds on
the hitting probabilities, we establish a smoothness property and
a Gaussian-type upper bound for the probability density function
of the supremum of the solution over a space-time rectangle touching the t = 0 axis.
This talk is based on joint work with Robert C. Dalang.
Gregory Rice, University of Waterloo
"Change point analysis with functional time series "
We consider methods for detecting and dating changes in both the
level and variability of a time series of curves or functional data
objects. Regarding level shifts, we propose a new detection and dating
procedure that is ``fully functional", in the sense that it does not rely
on dimension reduction techniques. To test for changes in variability, we
consider methods based on measuring the fluctuations of eigenvalues of the
sequential estimates of the empirical covariance operator. A thorough
asymptotic theory is developed for each procedure that highlights their
relative strengths and weaknesses when compared to existing methods. An
application to annual temperature curves illustrates the practical use of
the proposed methods.
In this talk, I will discuss recent work with Janosch Ortmann and Balint Virag on constructing the four-parameter scaling limit of (certain integrable models of) last passage percolation. The limit is the directed landscape, a random `directed' metric on the plane. Rescaled last passage paths converge to geodesics in this metric. These geodesics double as the limit of both the longest increasing subsequence path in a random permutation and the trajectory of a second class particle in TASEP.
Lai-Sang Young, Courant Institute, NYU
"Chaotic vs random dynamical systems"
In this talk I will compare and contrast (deterministic) chaotic
dynamical systems and their stochastic counterparts, i.e. when small
random perturbations are added to such systems to model uncontrolled
fluctuations. Three groups of results, some old and some new, will
be discussed. The first has to do with how deterministic systems,
when sufficiently chaotic, produce observations resembling those from
genuinely random processes. The second compares the ergodic theories
of chaotic systems and of random maps (as in stochastic flows of
diffeomorphisms generated by SDEs). One will see that results on
SRB measures, Lyapunov exponents, entropy, fractal dimension, etc.
are nicer in the random setting. I will finish by suggesting that
to improve the applicability of existing theory of chaotic systems,
a little bit of random noise can go a long way.
Samy Tindel, Purdue University
"Moment estimates for some renormalized parabolic Anderson models. "
The theory of regularity structures enables the definition parabolic Anderson models (that is linear stochastic heat equations) in a very rough environment (that is a noise with very singular space-time covariance function). If we call u(t,x) the solution to our renormalized stochastic heat equation, in this talk we shall give some information about the moments of u(t,x) when the stochastic heat equation is interpreted in the Skorohod as well as the Stratonovich sense. Of special interest is the critical case, for which one observes a blowup of moments for large times.
David Levin, University of Oregon
"Estimating the spectral gap of a reversible Markov chain from a single trajectory. "
The spectral gap is an important parameter of a Markov chain which
governs the asymptotic rate of convergence. I discuss (solutions to) the following problem:
how many steps are required from a single trajectory of a reversible ergodic Markov chain to
accurately estimate the gap?