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Current contact: Gerardo Mendoza
The seminar takes place Mondays 2:40 - 3:30 pm in Wachman 617. Click on title for abstract.
Martin Dindos, The University of Edinburgh
Abstract: The notion of an elliptic partial differential equation (PDE) goes back at least to 1908, when it appeared in a paper J. Hadamard.
The essence of ellipticity is described by L. Evans in his classic textbook as follows: "The following calculations are often technically difficult but eventually yield extremely powerful and useful assertions concerning the smoothness of weak solutions. As always, the heart of each computation is the invocation of ellipticity: the point is to derive analytic estimates from the structural, algebraic assumption of ellipticity."
In this talk we present a recently discovered structural condition, called $p$-ellipticity, which generalizes classical ellipticity and plays a fundamental role in many seemingly mutually unrelated aspects of the $L^p$ theory of elliptic complex valued PDE. So far, $p$-ellipticity has proven to be the key condition for:
(i) convexity of power functions (Bellman functions) (ii) dimension-free bilinear embeddings, (iii) $L^p$-contractivity and boundedness of semigroups $(P_t^A)_{t>0}$ associated with elliptic operators, (iv) holomorphic functional calculus, (v) multilinear analysis, (vi) regularity theory of elliptic PDE with complex coefficients.
During the talk I will describe my contribution to this development in particularly to (vi). It is of note that the $p$-ellipticity was co-discovered independently by Carbonaro and Dragicevic on one side (from the perspective of (i) and (ii)), and Pipher and myself on the other.
Eric Stachura, Kennesaw State University
Thomas Krainer, PennSate-Altoona
Postponed to Fall semester.
Xiaojun Huang, Rutgers University
Let $\Omega$ be a Stein space (of complex dimension at least two) with possibly isolated singularities and a connected compact strongly pseudoconvex smooth boundary $M = \partial \Omega$. Let $(f,D)$ be a non-constant CR mapping, where $D$ is an open connected subset of $M$. Suppose that $(f,D)$ admits a CR continuation along any curve in $M$ and for each CR mapping element $(g,D^*)$ with $D^*\subset M$ obtained by continuing $(f,D)$ along a curve in $M$, it holds that $\|g\|\leq C$ for a certain fixed constant $C$. Then $(f,D)$ admits a holomorphic continuation along any curve $\gamma$ with $\gamma(0) \in D$ and $\gamma(t) \in \mathrm{Reg}(\Omega)$ for $t \in (0, 1]$. Moreover, for any holomorphic mapping element $(h,U)$ with $U \subset \mathrm{Reg}(\Omega)$ obtained from continuation of $(f,D)$, we have $\|h\| < C$ on $U$.
Cecilia Freire Mondaini, Drexel University
This talk focuses on the study of convergence/mixing rates for stochastic dynamical systems towards statistical equilibrium. Our approach uses the weak Harris theorem combined with a generalized coupling technique to obtain such rates for infinite-dimensional stochastic systems in a suitable Wasserstein distance. In particular, we show two scenarios where this approach is applied in the context of stochastic fluid flows. First, to show that Markov kernels constructed from a suitable numerical discretization of the 2D stochastic Navier-Stokes equations converge towards the invariant measure of the continuous system. This depends crucially on a spectral gap result for the discrete Markov kernel that is independent of the level of discretization. Second, to approximate the posterior measure obtained via a Bayesian approach to inverse PDE problems, particularly when applied to advection-diffusion type PDEs. In this latter case, the Markov transition kernel is constructed with an exact preconditioned Hamiltonian Monte Carlo algorithm in infinite dimensions. A rigorous proof of mixing rates for such algorithm was an open problem until quite recently. Our approach provides an alternative and flexible methodology to establish mixing rates for other Markov Chain Monte Carlo algorithms. This is a joint work with Nathan Glatt-Holtz (Tulane U).