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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.
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.
Igor Rumanov, University of Colorado-Boulder
TBA
Joseph Feneuil, Temple University
TBA
Shif Berhanu, Temple University
TBA
Cristian Gutierrez, Temple University
TBA
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.
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.
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 ${\mathbb{R}}^n$.
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.
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.
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.
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 $Z(q)$.
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.
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.
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.
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.
Renan Medrado, Universidade Federal do Ceará, Brazil
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.
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.
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.