In this talk, we will discuss low Weissenberg number
effects on mathematical properties of solutions for several PDEs
governing different viscoelastic fluids.
An old conjecture by A. Zygmund proposes
a Lebesgue Differentiation theorem along a
Lipschitz vector field in the plane. E. Stein
formulated a corresponding conjecture about
the Hilbert transform along the vector field.
If the vector field is constant along
vertical lines, the Hilbert transform along
the vector field is closely related to Carleson's
operator. We discuss some progress in the area
by and with Michael Bateman and by my student
Shaoming Guo.
We consider equations with the simplest hysteresis operator at
the right-hand side. Such equations describe the so-called processes "with
memory" in which various substances interact according to the hysteresis
law. The main feature of this problem is that the operator at the
right-hand side is a multivalued.
We present some results concerning the optimal regularity of solutions.
Our arguments are based on quadratic growth estimates for solutions near
the free boundary. The talk is based on joint work with Darya
Apushkinskaya.
One dimensional analysis of Euler-Poisson system shows that when incoming supersonic flow is fixed,
transonic shock can be represented as a monotone function of exit pressure.
From this observation, we expect well-posedness of transonic shock problem for Euler-Poisson system
when exit pressure is prescribed in a proper range.
In this talk, I will present recent progress on transonic shock problem for Euler-Poisson system,
which is formulated as a free boundary problem with mixed type PDE system.
This talk is based on collaboration with Ben Duan, Chujing Xie and Jingjing Xiao
We will present a somewhat different proof of Agol's theorem that
3-manifolds
with RFRS fundamental group admit a finite cover which fibers over S^1.
This is joint work with Takahiro Kitayama.
The A-theory characteristic of a fibration is a
map to Waldhausen's algebraic K-theory of spaces which
can be regarded as a parametrized Euler characteristic of
the fibers. Regarding the classifying space of the cobordism
category as a moduli space of smooth manifolds, stable under
extensions by cobordisms, it is natural to ask whether the
A-theory characteristic can be extended to the cobordism
category. A candidate such extension was proposed by Bökstedt
and Madsen who defined an infinite loop map from the d-dimensional
cobordism category to the algebraic K-theory of BO(d). I will
discuss the connections between this map, the A-theory
characteristic and the smooth Riemann-Roch theorem of Dwyer,
Weiss and Williams.
I will introduce the physical phenomena of transonic shocks, and review the progresses on related boundary value problems of the steady compressible Euler equations. Some Ideas/methods involved in the studies will be presented through specific examples. The talk is based upon joint works with my collaborators.
The periodic KdV equation $u_t=u_{xxx}+\beta uu_x$ arises from a Hamiltonian system with infinite-dimensional phase space $L^2({\bf T})$. Bourgain has shown that there exists a Gibbs probability measure $\nu$ on balls $\{\phi :\Vert \phi\Vert^2_{L^2}\leq N\}$ in the phase space such that the Cauchy problem for KdV is well posed on the support of $\nu$, and $\nu$ is invariant under the KdV flow. This talk will show that $\nu$ satisfies a logarithmic Sobolev inequality. The seminar presents logarithmic Sobolev inequalities for the modified periodic KdV equation and the cubic nonlinear Schr\"odinger equation. There will also be recent results from Blower, Brett and Doust regarding spectral concentration phenomena for Hill's equation.
The question of how to derive useful bounds on
arbitrage-free prices of exotic options given only prices of liquidly
traded products like European call und put options has received much
interest in recent years. It also led to new insights about classic
problems in probability theory like the Skorokhod embedding problem. I
will take this as a starting point and show how this progress can be
used to give new results on general Monte-Carlo schemes.
I explore some new ideas on embedding problems for Brownian motion (and other Markov processes). I show how a (forward) Skorokhod embedding problem is transformed into an optimal stopping problem for the time-reversed process (Markov process in duality). This is deduced from the PDE (Variational Inequalities) interpretation of the classical results but then shown using probabilistic techniques and extended to give an n-marginal Root embedding. I also discuss briefly how to extend the approach to other embeddings such as the Azema-Yor embedding.