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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.
Samuel Cogar, University of Newark
TBA
Zachary Bailey, Temple University
TBA
Maxim Braverman, Northeastern University
TBA
Wanke Yin, Wuhan University and Rutgers University
Let $M$ be a smooth real hypersurface in $\mathbb C^n$ with $n\geq 2$. For any $p\in M$ and any integer $s\in [1,n-1]$, Bloom in 1981 defined the following three kinds of integral invariants: invariant $a^{(s)}(M,p)$ defined in terms of contact order by complex submanifolds, invariant $t^{(s)}(M,p)$ defined by the iterated Lie bracket of vector fields and invariant $c^{(s)}(M,p)$ defined through the degeneracy of the trace of the Levi form. When $M$ is pseudoconvex, Bloom conjectured that these three invariants are equal. Bloom and Graham gave a complete solution of the conjecture for $s=n-1$. Bloom showed that the conjecture is true for $a^{(1)}(M,p)=c^{(1)}(M,p)$ when $n=3$. In this talk, I will present a recent joint work with Xiaojun Huang, in which we gave a solution of the conjecture for $s=n-2$. In particular, this gave a complete solution of the Bloom conjecture for $n=3$.
Jose Maria Martell, ICMAT, Madrid, Spain
In nice environments, such as Lipschitz or chord-arc domains, it is well-known that the Dirichlet problem for the Laplacian with data in Lebesgue spaces $L^p$ is solvable for some finite $p$. This property is equivalent to the fact that the associated harmonic measure is absolutely continuous, in a quantitative way, with respect to the surface measure on the boundary. In this talk we will study under what circumstances the harmonic measure for a rough domain is a well-behaved object. We will also present some results for the converse, in which case good properties for the domain and its boundary can be proved by knowing that the harmonic measure satisfies a quantitative absolute continuity property with respect to the surface measure. We will describe the two main features appearing in this context: one related to the regularity of the boundary, expressed via its uniform rectifiablity, and another one related to the connectivity of the domain, written in terms of some quantitative connectivity towards the boundary using non-tangential paths. The results that we will present are higher dimensional scale-invariant extensions of the F. and M. Riesz theorem and its converse. That classical result says that, in the complex plane, the harmonic measure is absolutely continuous with respect to the arc-length measure for simply connected domains (a strong connectivity condition) with rectifiable boundary (a regularity condition).
Murat Akman, University of Connecticut
The classical Minkowski problem consists in finding a convex polyhedron from data consisting of normals to their faces and their surface areas. In the smooth case, the corresponding problem for convex bodies is to find the convex body given the Gauss curvature of its boundary, as a function of the unit normal. The proof consists of three parts: existence, uniqueness and regularity.
In this talk, we study a Minkowski problem for certain measure, called $p$-capacitary surface area measure, associated to a compact convex set with nonempty interior and its $p$-harmonic capacitary function. We will discuss existence, uniqueness, and regularity of this problem under this setting and see connections with the Brunn-Minkowski inequality and Monge-Ampere equation.
Narek Hovsepyan, Temple University
Analytic continuation problems are notoriously ill-posed without additional regularizing constraints, even though every analytic function has a rigidity property of unique continuation from every curve inside the domain of analyticity. In fact, well known theorems, guarantee that every continuous function can be uniformly approximated by analytic functions (polynomials or rational functions, for example). We consider several analytic continuation problems with typical global boundedness constraints. All such problems exhibit a power law precision deterioration as one moves away from the source of data. In this talk we demonstrate the effectiveness of our general Hilbert space-based approach for determining these exponents. The method identifies the ``worst case'' function as a solution of a linear equation with a compact operator. In special geometries, such as the circular annulus this equation can be solved explicitly. The obtained solution is then used to determine the power law exponent for the analytic continuation from an interval between the foci of a Bernstein ellipse to the entire ellipse. In those cases where such exponents have been determined in prior work our results reproduce them faithfully.
This is joint work with Yury Grabovsky.
Spring break, no meeting
Joseph Feneuil, Temple University
The Riesz transform $\nabla \Delta^{-1/2}$ on $\mathbb R^n$ is bounded on $L^p$ for all $p\in (1,+\infty)$. This well known fact can quickly be proved by using the Fourier transform. Strichartz asked then whether this property is transmitted to Riemannian manifold, more exactly, what are the geometric conditions needed on our manifold to get the boundedness of the Riesz transform.
We shall present (part of) the literature on the topic, including the results of the speaker (together with Li Chen, Thierry Coulhon, and Emmanuel Russ) on fractal-like spaces. We shall also talk about the case of graphs, that can be seen as discrete version of Riemannian manifolds, which will allow us to give concrete examples of application of our work.
If time permits, we will provide equivalent statements for an assumption frequently met when working on graphs (which implies $L^2$-analyticity of the Markov operator). In particular, we will see a way to weaken this assumption to $L^2$-analyticity.
Atilla Yilmaz, Temple University
I will present joint work with Elena Kosygina and Ofer Zeitouni in which we prove the homogenization of a class of one-dimensional viscous Hamilton-Jacobi equations with random Hamiltonians that are nonconvex in the gradient variable. Due to the special form of the Hamiltonians, the solutions of these PDEs with linear initial conditions have representations involving exponential expectations of controlled Brownian motion in a random potential. The effective Hamiltonian is the asymptotic rate of growth of these exponential expectations as time goes to infinity and is explicit in terms of the tilted free energy of (uncontrolled) Brownian motion in a random potential. The proof involves large deviations, construction of correctors which lead to exponential martingales, and identification of asymptotically optimal policies.
Nestor Guillen, University of Massachusetts, Amherst
A large number of problems involve mappings with a prescribed Jacobian, from optimal transport mappings to problems of lenses and antenna design in geometric optics. Many of these problems originate from what is now known as a "generating function", e.g. the cost function in optimal transport, in which case the equation is known as Generated Jacobian Equation. This class of equations has been proposed by Trudinger, and it covers not only optimal transport problems, but also near-field problems in optics. In this talk I will discuss work with Jun Kitagawa were we prove Holder continuity for the gradient of weak solutions to GJE, under natural assumptions. The results are in the spirit of, and extend, Caffarelli's theory for the real Monge-Ampere equation. The key observation is that a quasiconvexity property of the underlying generating function (related to MTW tensor) guarantees the validity of an estimate akin to Aleksandrov's estimate for convex functions.
Francisco Villarroya, Temple University
I will introduce a Tb Theorem that characterizes all Calderón-Zygmund operators that extend compactly on $L^p(\mathbb R^n)$ by means of testing functions as general as possible. In the classical theory of boundedness, the testing functions satisfy a non-degeneracy property called accretivity, which essentially implies the existence of a positive lower bound for the absolute value of the averages of the testing functions over all dyadic cubes. However, in the the setting of compact operators, due to their better properties, the hypothesis of accretivity can be relaxed to a large extend. As a by-product, the results also describe those Calderon-Zygmund operators whose boundedness can be checked with non-accretive testing functions.
Alessia Elisabetta Kogoj, University of Urbino "Carlo Bo"
Several Liouville-type theorems are presented, related to evolution equations on Lie Groups and to their stationary counterpart. Our results apply in particular to the heat operator on Carnot groups, to linearized Kolmogorov operators and to operators of Fokker-Planck-type like the Mumford operator. An application to the uniqueness for the Cauchy problem is also shown.
These results are based on joint publications with A. Bonfiglioli, E. Lanconelli, Y. Pinchover and S. Polidoro.