# Analysis Seminar

Current contact: Gerardo Mendoza

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

• Monday December 3, 2018 at 14:40, Wachman 617

Joseph Feneuil, Temple University

Let $\Omega$ is a open bounded subset of $\mathbb R^n$ and $\Gamma$ is its boundary. Recent works established a relationship between the geometry of the boundary $\Gamma$ and estimates on the solutions of the Dirichlet problem for the Laplacian in the domain $\Omega$. More precisely, under some conditions of topology, $\Gamma$ is uniformly rectifiable if and only if the harmonic measure is absolutely continuous (in a quantitative way) to the surface measure. This nice criterion is unfortunately limited to the case where $\Gamma$ is of dimension $n-1$, because the condition is necessary to construct the harmonic measure.

I will present in this talk how, together with Guy David and Svitlana Mayboroda, we contructed an analogue of the harmonic measure on $\Gamma$ when $\Gamma \subset \mathbb R^n$ is a set of codimension higher than 1. I will discuss about the properties of our new measure that are similar to the real harmonic measure, and our unsolved problems.

• Monday November 26, 2018 at 14:40, Wachman 617

Sara Leshen, Vanderbilt University

The Uncertainty Principle implies that a function and its Fourier transform cannot both be well-localized. The Balian-Low theorem is a form of the Uncertainty Principle for Riesz bases. In this joint work with A. Powell, we prove a new version of the Balian-Low theorem for Gabor Schauder bases generated by compactly supported functions. Moreover, we show that the classical Balian-Low theorem for Riesz bases does not hold for Schauder bases.

• Monday November 12, 2018 at 14:40, Wachman 617

The six classical Painleve equations found numerous applications in different branches of science. E.g. Painleve II (PII) is related to the celebrated Tracy-Widom distributions of random matrix theory and their universality. Painleve ODEs can be seen as classical one-particle dynamical systems with time-dependent Hamiltonians. The Quantum Painleve equations (QPEs) are linear Fokker-Planck or non-stationary Schroedinger PDEs in two independent variables (time" and space") with spatial operators being quantized Painleve Hamiltonians. QPEs are satisfied by certain eigenvalue probabilities of random matrix beta ensembles (or probabilities of Coulomb gas particle positions restricted to a line). E.g. QPII describes the soft edge limit of beta ensembles while QPIII does so for the hard edge.

We construct classical nonlinear integrable structure associated with QPII, more explicit for even integer beta. The nonlinear PDEs tied with QPII allow one to gain more information about the QPII solutions. The corresponding probability distributions explicitly depend on Hastings-McLeod solution of PII in all known cases and conjecturally for all values of beta.

If time permits, I plan to discuss open problems related to QPEs and their generalizations relevant in various applications including multivariate statistics, Coulomb gases in the plane, stochastic Loewner evolutions (SLEs), quantum Hall effect, black hole physics, string theory and others.

• Monday October 29, 2018 at 14:40, Wachman 617

Narek Hovsepyan, Temple University

Hardy functions over the upper half-plane ($\mathbb{H}_+$) are determined by their values on any curve $\Gamma$ lying in the interior or on the boundary of $\mathbb{H}_+$. Given that such a function $f$ is small on $\Gamma$ (say, is of order $\epsilon$), how does this affect the magnitude of $f$ at the point $z$ away from the curve? When $\Gamma \subset \partial \mathbb{H}_+$, we give a sharp upper bound on $|f(z)|$ of the form $\epsilon^\gamma$, with an explicit exponent $\gamma = \gamma(z) \in (0,1)$ and describe the maximizer function attaining the upper bound. When $\Gamma \subset \mathbb{H}_+$ we give an upper bound in terms of a solution of an integral equation on $\Gamma$. We conjecture that this bound is sharp and behaves like $\epsilon^\gamma$ for some $\gamma = \gamma(z) \in (0,1)$. This is a joint work with Yury Grabovsky.

• Monday October 22, 2018 at 14:40, Wachman 617

Shif Berhanu, Temple University

Abstract: We will discuss recent results on local unique continuation at the boundary for the solutions of a class of elliptic operators in the plane. The results involve a local boundary sign condition either on the solution or the product of the solution with a monomial. The work extends boundary uniqueness theorems for harmonic functions proved by Baouendi and Rothschild.

• Monday October 8, 2018 at 14:40, Wachman 617

Cristian Gutierrez, Temple University

Due to dispersion, light with different wavelengths, or colors, is refracted at different angles.

So when white light is refracted by a single lens, in general, each color comes to a focus at a different distance from the objective. This is called chromatic aberration and plays a vital role in lens design.

A way to correct chromatic aberration is to build lenses that are an arrangement of various single lenses made of different materials.

Our purpose in this talk is to show when is mathematically possible to design a lens made of a single homogeneous material so that it refracts light superposition of two colors into a desired fixed final direction. Two problems are considered: one is when light emanates in a parallel beam and the other is when light emanates from a point source.

The mathematical tools used to solve these problems include fixed point theorems and functional differential equations. This is joint work with A. Sabra.

• Monday October 1, 2018 at 14:40, Wachman 617

Francisco Villarroya, Temple University

In this talk I will introduce some relatively new results that make a T1 Theory for compactness. These results completely characterize those Calderon-Zygmund operators that extend compactly on the appropriate Lebesgue spaces and at the standard endpoint spaces. The presentation will start with a brief introduction to the classical T1 Theory.

• Monday September 17, 2018 at 14:40, Wachman 617

Mihaela Ignatova, Temple University

I will describe results regarding the surface quasi-geostrophic equation (SQG) in bounded domains. The results concern global interior Lipschitz bounds for large data for the critical SQG in bounded domains. In order to obtain these, we establish nonlinear lower bounds and commutator estimates for the Dirichlet fractional Laplacian in bounded domains. As an application, global existence of weak solutions of SQG are obtained. If time permits, I will also discuss an application to an electroconvection model.

• Monday September 10, 2018 at 14:40, Wachman 617

Irina Mitrea, Temple University

In his 1978 ICM plenary address A.P. Calderón has famously advocated the use of boundary layer potentials for much more general elliptic systems than the Laplacian''. One may also attach a Geometric Measure Theoretic component to this directive by insisting on considering the most general geometric setting in which the said boundary layer potentials continue to exhibit a natural behavior.

The present talk is based on joint work with G. Hoepfner, P. Liboni. D. Mitrea and M. Mitrea, and fits into this broad program. Its goal is to discuss key features exhibited by all boundary multi-layer potential operators associated with higher order elliptic systems of partial differential operators in various classes of sets of locally finite perimeter, including uniformly rectifiable domains in ${\mathbb{R}}^n$.

• Monday April 30, 2018 at 14:40, Wachman 617

Jonathan Weitsman, Northeastern University

We review geometric quantization in the symplectic case, and show how the program of formal geometric quantization can be extended to certain classes of Poisson manifolds equipped with appropriate Hamiltonian group actions. These include $b$-symplectic manifolds, where the quantization turns out to be finite dimensional, as well as more singular examples ($b^k$-symplectic manifolds) where the quantization is finite dimensional for odd $k$ and infinite dimensional, with a very simple asymptotic behavior, when $k$ is even.

This is a joint work with Victor Guillemin and Eva Miranda.

• Monday April 16, 2018 at 14:40, Wachman 617

Nestor Guillen, University of Massachusetts at Amherst

It is a well known fact that the Dirichlet-to-Neumman map for an elliptic operator yields an integro-differential operator on the boundary of the domain. As it turns out, one can consider a non-linear analogue of this map to describe free boundary conditions in terms of a non-linear non-local operator satisfying a comparison principle. The end result is that a large class of free boundary problems correspond to a (degenerate) parabolic integro-differential equation on a reference submanifold, making it possible to approach free boundary regularity via non-local methods. Based on joint works with Russell Schwab, Jun Kitagawa, and Hector Chang-Lara.

• Monday March 26, 2018 at 14:40, Wachman 617

Terrence Napier, Lehigh University

Joint work with Mohan Ramachandran on analogues of the Hartogs extension theorem (regarding the extension of a holomorphic function of several complex variables past a compact set) in the setting of Kaehler manifolds will be considered.

• Monday March 19, 2018 at 14:40, Wachman 617

Andy Raich, University of Arkansas

The main goal of this talk is to show that geometric information captured by certain invariant CR tensors provides sufficient information to establish the closed range property for $\bar\partial$ on a domain in $\mathbb{C}^n$. A secondary goal of the talk is to provide a general construction method for establishing when a domain (or its boundary) satisfies weak $Z(q)$.

• Monday March 12, 2018 at 14:40, Wachman 617

José González Llorente, Universidad Autónoma de Barcelona

The Mean Value Property for harmonic functions is at the crossroad of Potential Theory, Geometric Function Theory and Probability. In the last years substantial efforts have been made to build up stochastic models for certain nonlinear PDE's like the $p$-laplacian or the infinity-laplacian and the key is to figure out which are the corresponding (nonlinear) mean value properties. After introducing a "natural" nonlinear mean value property related to the $p$-laplacian we will focus on functions satisfying the so called one-radius mean value property. We will review some classical results in the linear case ($p=2$) and then recent nonlinear versions in the more general context of metric measure spaces.

• Monday March 5, 2018 at 14:30, Wachman 617
No meeting

• Wednesday February 28, 2018 at 17:00, Wachman 617

Nordine Mir, Texas A&M-Qatar (note special day and time)

I will discuss recent joint results with B. Lamel regarding the convergence and divergence of formal holomorphic maps between real-analytic CR submanifolds in complex spaces of possibly different dimension. Our results resolve in particular a long standing open question in the subject and recover all known previous existing ones. We will also discuss the new approach developed in order to understand the convergence/divergence properties of such maps.

• Monday February 19, 2018 at 14:40, Wachman 617

Luis Ragognette, Federal University of São Carlos, Brazil

The theory of hyperfunctions deals with generalized functions that are even more general than distributions. Our goal in this talk is to discuss techniques that allowed us to study microlocal regularity of a hyperfunction with respect to different subspaces of the space of hyperfunctions. In other to do that we used a subclass of the FBI transforms introduced by S. Berhanu and J. Hounie. This is a joint work with Gustavo Hoepfner.

• Monday February 12, 2018 at 14:40, Wachman 617

Gerardo Mendoza, Temple University

Fuchs-type operators and certain generalizations arise on manifolds with conical or more general stratifications. While the elliptic theory of such operators is by now fairly well understood, important aspects of the corresponding theory for complexes are still being developed. In this talk I will describe recent progress (joint work with T. Krainer) in the case of conical singularities on the elucidation of the boundary conditions that can be specified in order to obtain a complex in the Hilbert space category.

• Monday February 5, 2018 at 14:40, Wachman 617

The aim of this talk is to present a characterization of Denjoy-Carleman (local and micro-local) regularity using a general class of FBI transform introduced by S.~Berhanu and J.~Hounie in 2012. As an application we exhibit a microlocal Denjoy-Carleman propagation of regularity theorem, that do not seem possible to prove using the classical FBI transform. This is a joint work with Gustavo Hoepfner.

• Monday January 29, 2018 at 14:40, Wachman 617

Ryan Hynd, University of Pennsylvania

We will consider the dynamics of a finite number of particles that interact pairwise and undergo perfectly inelastic collisions. Such physical systems conserve mass and momentum and satisfy the Euler-Poisson equations. In one spatial dimension, we will show how to derive an extra entropy estimate which allows us to characterize the limit as the number of particles tends to infinity.

• Monday January 22, 2018 at 14:40, Wachman 617

Irina Mitrea, Temple University

One of the classical methods for solving elliptic boundary value problems in a domain $\Omega$ is the method of layer potentials, whose essence resides in reducing the entire problem to solving an integral equation on $\partial\Omega$. In this talk I will discuss spectral properties of the intervening singular integral operators and show how the two-dimensional setting plays a special role in this analysis.