# Analysis Seminar

Current contact: Gerardo Mendoza

The seminar takes place Mondays 2:40 - 3:30 pm in Wachman 617. Click on title for abstract.

• Tuesday April 2, 2019 at 14:40, Wachman 617
TBA

Wanke Yin, Wuhan University and Rutgers University

TBA

• Tuesday March 26, 2019 at 15:00, Wachman 617 (note special day and time)
TBA

Marius Mitrea, University of Missouri

TBA

• Monday March 25, 2019 at 14:40, Wachman 617
TBA

Jose Maria Martell, ICMAT, Madrid, Spain

TBA

• Monday March 18, 2019 at 14:40, Wachman 617
TBA

Murat Akman, University of Connecticut

TBA

• Monday March 11, 2019 at 14:40, Wachman 617
TBA

Narek Hovsepyan, Temple University

TBA

• Monday March 4, 2019 at 14:40,

Spring break, no meeting

• Monday February 25, 2019 at 14:40, Wachman 617

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.

• Monday February 11, 2019 at 14:40, Wachmn 617

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.

• Monday February 4, 2019 at 14:40, Wachman 617

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.

• Monday January 28, 2019 at 14:40, Wachman 617

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.

• Tuesday January 22, 2019 at 14:00, Wachman 617 (note special day and time)

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.