Current contact: Gerardo Mendoza

The seminar takes place Mondays 2:30 - 3:30 pm either in Wachman 617 or virtually via Zoom. Please see the announcment of the specific meeting. For virtual meetings, please write if you wish to join. Click on title for abstract.

• Monday January 25, 2021 at 14:30, Zoom meeting

  Analytic and Gevrey regularity for sums of squares in low dimension

  Antonio Bove, University of Bologna

  We present a couple of results of analytic and Gevrey regularity for sums of squares operators in dimension 2. The reason why we focus on dimension 2 is that we believe it is the only case where Treves conjecture holds. We identify the Poisson strata with some higher multiplicity subvarieties of the characteristic variety.

• Monday February 1, 2021 at 14:30, Zoom meeting

  Two new local $T1$ theorems on non-homogeneous spaces

   

  Francisco Villarroya, Temple University

  We introduce two new $T1$ theorems characterizing all Calder\'on-Zygmund operators $$Tf(x)=\int f(t)K(t,x)d\mu (t)$$ that extend boundedly on $L^{p}(\mathbb R^{n},\mu)$  for $1<p<\infty $ with  $\mu$ a non-doubling measure of power growth.

  We employ a new proof method that, unlike all currently known works on $T1$ theorems  in non-homogeneous spaces,  does not use random grids. The new approach allows the use of a countable family of testing functions, and also testing functions supported on cubes of different dimensions.

   

• Monday February 8, 2021 at 14:30, Zoom meeting

  Global Solutions of the Nernst-Planck-Euler Equations

  Jingyang Shu, Temple University

  The transport and the electrodiffusion of ions in homogeneous Newtonian fluids are classically modeled by the Nernst-Planck-Navier-Stokes (NPNS) equations. When the kinematic viscosity term in the Navier-Stokes equation is neglected, the NPNS system becomes the Nernst-Planck-Euler (NPE) system. In this talk, we consider the initial value problem for the NPE equations with two ionic species in two-dimensional tori. We prove the global existence of weak solutions and the global existence and uniqueness of smooth solutions. We also show that in the vanishing viscosity limits, smooth solutions of the NPNS equations converge to the solutions of the NPE equations. This is joint work with Mihaela Ignatova.

• Monday March 1, 2021 at 14:30, Zoom meeting

  Smoothing effect and Strichartz estimates for some time-degenerate Schroedinger operators

  Serena Federico, Ghent University

  In this talk we will analyze the smoothing effect and the validity of Strichartz estimates for some classes of time-degenerate Schroedinger operators. In the first part of the talk we will investigate the local smoothing effect (both homogeneous and inhomogeneous) for time-degenerate Schr\"odinger operators of the form $$ \mathcal{L}_{\alpha,c}=i\partial_t+t^\alpha\Delta_x+c(t,x)\cdot \nabla_x,\quad \alpha>0,$$where $c(t,x)$ satisfies suitable conditions. Additionally, we will employ the smoothing effect to prove local well-posedness results for the associated nonlinear Cauchy problem. In the second part of the talk we will analyze Strichartz estimates for a class of operators similar to the previous one, that is of the form $$\mathcal{L}_{b}:=i\partial_t+ b'(t)\Delta_x,$$with $b'$ satisfying suitable conditions. An application of these estimates will give a (different) local well-posedness result for a semilinear Cauchy problem associated with $\mathcal{L}_b$.

• Monday March 22, 2021 at 14:30, Zoom meeting

  The Neumann problem for symmetric higher order elliptic differential equations

   

  Ariel Barton, University of Arkansas 

  The second order differential equation $\nabla\cdot A\nabla u=0$ has been studied extensively. It is well known that, if the coefficients $A$ are real-valued, symmetric, and constant along the vertical coordinate (and merely bounded measurable in the horizontal coordinates), then the Dirichlet problem with boundary data in $L^q$ or $\dot W^{1,p}$, and the Neumann problem with boundary data in $L^p$, are well-posed in the half-space, provided $2-\varepsilon<q<\infty$ and $1<p<2+\varepsilon$.

  It is also known that the Neumann problem for the biharmonic operator $\Delta^2$ in a Lipschitz domain in $\mathbb{R}^d$ is well posed for boundary data in $L^p$, $\max(1,p_d-\varepsilon)<p<2+\varepsilon$, where $p_d=\frac{2(d-1)}{d+1}$ depends on the ambient dimension~$d$. 
   

  In this talk we will discuss recent well posedness results for the Neumann problem, in the half-space, for higher-order equations of the form $\nabla^m\cdot A\nabla^m u=0$, where the coefficients $A$ are real symmetric (or complex self-adjoint) and vertically constant.
   

   

   

• Monday March 29, 2021 at 14:30, Zoom meeting

  On the global Gevrey vectors

  Gustavo Hoepfner, Federal University of Sao Carlos
  
  We introduce the notion of global $L^q$ Gevrey vectors and investigate the regularity of such vectors in global and microglobal settings. We characterize the vectors in terms of the FBI transform and prove global and microglobal versions of the Kotake-Narasimhan Theorem. As a consequence we provided a refinement of an earlier result by Hoepfner and Raich relating

  the microglobal wavefront sets of the ultradistributions $u$ and $Pu$ when $P$ is a constant coefficient differential operator. This is a joint work with A. Raich and P. Rampazzo.

   

• Monday April 5, 2021 at 14:30, Zoom meeting

  On Optimal Control Problem related to the Infinity Laplacian

  Henok Mawi, Howard University
  
  The infinity Laplacian equation is given by$$\Delta_{\infty} u := u_{x_i}u_{x_j}u_{x_ix_j} = 0 \quad \text{in } \Omega$$where $\Omega$ is an open bounded subset of $\mathbb R^n.$ This equation is a kind of an Euler-Lagrange equation of  the variational problem of minimizing the functional $$I[v] := \textrm{ess sup} \, |Dv|,$$among all Lipschitz continuous functions $v,$ satisfying a prescribed boundary value on $\partial\Omega.$ The infinity obstacle problem is the minimization problem $$\min \{ I[v]:  v \in W^{1,\infty},\  v\geq \psi \}$$for a given function $\psi \in W^{1, \infty}$ which we refer to as the obstacle.

  In this talk I will discuss an optimal control problem related to the infinity obstacle problem. This is joint work with Cheikh Ndiaye.

   

• Monday April 12, 2021 at 14:30, Zoom meeting

  Global hypoellipticity of sums of squares on compact manifolds

  Gabriel Araujo, University of Sao Paulo, Sao Carlos

  In a recent work with Igor A. Ferra (Federal Univ. of ABC - Brazil) and Luis F. Ragognette (Federal Univ. of Sao Carlos - Brazil) we investigate necessary and sufficient conditions for global hypoellipticity of certain sums of squares of vector fields. Our model is inspired by a rather general one introduced by (Barostichi-Ferra-Petronilho, 2017) when the ambient manifold is a torus; here we extend their results to more general closed manifolds, which requires a new interpretation of their Diophantine conditions (as these are not available in our framework).

• Monday April 19, 2021 at 14:30, Zoom meeting

  On the multiparameter distance set problem

  Yumeng Ou, University of Pennsylvania

  In this talk, we will describe some recent progress on the Falconer distance set problem in the multiparameter setting. The original Falconer conjecture (open in all dimensions) says that a compact set $E$ in $\mathbb{R}^d$ must have a distance set $\{|x-y|: x,y\in E\}$ with positive Lebesgue measure provided that the Hausdorff dimension of $E$ is greater than $d/2$. What if the distance set is replaced by a multiparameter distance set? We will discuss some recent results on this question, which is also related to multiparameter projections of fractal measures. This is joint work with Xiumin Du and Ruixiang Zhang.

• Monday April 26, 2021 at 14:30, Zoom meeting

  The second boundary value problem for a discrete Monge-Ampere equation

   

  Gerard Awanou, University of Illinois, Chicago

  In this work we propose a natural discretization of the second boundary condition for the Monge-Ampere equation of geometric optics and optimal transport. It is the natural generalization of the popular Oliker-Prussner method proposed in 1988. For the discretization of the differential operator, we use a discrete analogue of the subdifferential. Existence, unicity and stability of the solutions to the discrete problem are established. Convergence results to the continuous problem are given.

   

• Monday September 6, 2021 ,

  Labor day (no meeting)

   

• Monday September 20, 2021 at 14:30, Wachman 617

  On some electroconvection models

  Elie Abdo, Temple University

  We consider an electroconvection model describing the evolution of a surface charge density interacting with a 2D fluid. We investigate the model on the two-dimensional torus: we study the existence, uniqueness and regularity of solutions, and we show the existence of a global attractor.

• Monday September 27, 2021 at 14:30, Wachman 617

  Relaxation of functionals with bulk and surface energy terms

  Giovanni Gravina, Temple University

  The minimization of energy functionals has a wide range of applications both in pure and applied disciplines, where the existence of minimizes is routinely proved by means of the so-called "direct method in the Calculus of Variations". This, in turn, relies on showing that the energy under consideration is lower semicontinuous. If this property fails, valuable insight may still be gained by characterizing the lower semicontinuous envelop of the energy, referred to as the relaxed energy.

  Motivated by problems in the van der Waals-Cahn-Hilliard theory of liquid-liquid phase transitions, and by some classical examples due to Modica, in this talk, we will study the lower semicontinuity of energy functionals with bulk and surface terms. Since the presence of corners in the domain can affect the lower semicontinuity of the energies under consideration, we will focus on uncovering how the roughness of the domain enters the relaxation procedure.

• Monday October 4, 2021 at 14:30, Wachman 617

  Rellich type identities and their role in the treatment of Elliptic Boundary Value Problems in Lipschitz domains

  Jeongsu Kyeong, Temple University

  Among other things, integral identities of Rellich type allow one to deduce the $L^{2}(\partial \Omega)$ equivalence of the tangential derivative and the normal derivative of a harmonic function with a square integrable non-tangential maximal function of its gradient in a given Lipschitz domain $\Omega \subset \mathbb{R}^{n}$. In this survey talk, I will establish the integral identities in $\mathbb{R}^{n}$ and I will illustrate the role that the aforementioned equivalence plays in establishing invertibility properties of singular integral operators of layer potential type associated with the Laplacian in Lipschitz domains in $\mathbb{R}^{2}$, through an interplay between PDE, Harmonic Analysis, and Complex Analysis methods.

• Monday October 11, 2021 at 14:30, Zoom meeting

  Heat content and geometric analysis

  Patrick McDonald, New College of Florida

  The heat content associated with a bounded domain in a Riemannian manifold is a function obtained by solving an initial value problem for the heat operator on the domain. Heat content gives rise to a collection of geometric invariants closely related to the Dirichlet spectrum. In this talk I will survey results that compare and contrast the role of heat content invariants to the role of spectral data in geometric analysis. In particular, I will discuss results involving planar polygons and provide explicit examples of where heat content invariants and Dirichlet spectrum behave similarly, and also where they behave differently.

• Monday October 18, 2021 at 14:30, Zoom meeting

  A first-order approach to solvability for singular Schrödinger equations

  Andrew Morris, University of Birmingham, UK

  We will first give a brief overview of the first-order approach to boundary value problems, which factorises second-order divergence-form equations into Cauchy-Riemann systems. The advantage is that the holomorphic functional calculus for such systems can provide semigroup solution operators in tremendous generality, extending classical harmonic measure and layer potential representations. We will then show how recent developments now allow for the incorporation of singular perturbations in the associated quadratic estimates. This allows us to solve Dirichlet and Neumann problems for Schr\"odinger equations with potentials in Sobolev-critical Lebesgue spaces and reverse H\"older spaces. This is joint work with Andrew Turner.

• Monday October 25, 2021 , Wachman 617

  On the Lack of Fredholm Solvability for the $L^p$ Dirichlet Problem for Weakly Elliptic Systems in the Upper Half-Space

  Irina Mitrea, Temple University
  
   

  .The $L^p$ Dirichlet Problem for constant coefficient second-order systems satisfying the Legendre-Hadamard strong ellipticity condition is well posed in the upper half-space. Surprisingly, this result may fail if only weak ellipticity is assumed, and the failure manifests itself at a fundamental level through lack of Fredholm solvability. In this talk I will discuss a couple of pathological cases, sought in the class of weakly elliptic systems that fail to possess a distinguished coefficient tensor. This is joint work with Dorina Mitrea and Marius Mitrea.

   

• Monday November 1, 2021 at 14:30, Zoom meeting

  Wave decay for domains star-shaped with respect to infinity

  Tanya Christiansen, University of Missouri
  
   

  We wish to understand how the geometry of a domain $X\subset {\mathbb R}^d$ affects the decay of solutions to the wave equation on $X$ with Dirichlet boundary conditions.

  The case in which $ \mathcal{O}={\mathbb R}^d\setminus X$ is bounded is a classical obstacle scattering problem.  In the special case when $\mathcal{O}$ is star-shaped, decay of solutions of the wave equation is a  classical result of Morawetz.  We study certain sets $X$ which have ${\mathbb R}^d\setminus X$ unbounded.   These sets $X$ are unbounded in some directions, and bounded in others.  We introduce a notion of "star-shaped with respect to infinity" and show that this condition has implications for the behavior of the resolvent of the Laplacian.  For waveguides which are star-shaped with respect to infinity, this implies some wave decay.

  This talk is based on joint work with K. Datchev.
   

   

• Monday November 8, 2021 at 14:30, Zoom meeting

  Uniqueness of the $C^*$ norm in strict deformation quantization

  Severino Toscano de Rego Melo, University of São Paulo, São Paulo

  Abstract: Rieffel's algebra of pseudodifferential operators, introduced in the context of deformation quantization, will be described from the point of view of somebody who is familiar with pseudodifferential operators. Old results about the characterization of pseudodifferential operators as bounded operators with a smooth orbit under the action of the Heisenberg group will also be explained. Finally I will report on a recent joint paper with Cabral and Forger, in which we prove the uniqueness of the $C^*$ norm on Rieffel's algebra.

• Monday November 15, 2021 at 14:30, Virtual presentation

  Traveling waves close to the Couette flow

  Ángel Castro, Instituto de Ciencias Matemáticas, Madrid

  In this talk we shall study the existence of smooth traveling waves close to the Couette flow for the 2D incompressible Euler equation for an ideal fluid. It is well known that this kind of solutions do not exist arbitrarily close to the Couette flow if the distance is measured in $H^{3/2+}$. In this presentation we will deal with the case $H^{3/2-}$.

• Monday November 22, 2021 ,

  No meeting

   

• Monday November 29, 2021 at 14:30, Zoom meeting

  The Cauchy–Szegő projection and its commutator for domains in $\mathbb C^n$ with minimal smoothness

  Loredana Lanzani, Syracuse University
  
   

  Let $D\subset \mathbb C^n$ be a bounded, strongly pseudoconvex domain whose boundary $bD$ satisfies the minimal regularity condition of class $C^2$. A 2017 result of Lanzani \& Stein states that the Cauchy–Szego projection $\mathscr S_\omega$ defined with respect to any Leray Levi-like measure $\omega$ is bounded in $L^p(bD, \omega)$ for any $1 < p < \infty$. (We point out that for this class of domains, induced Lebesgue measure is Leray Levi-like.) Here we show that $\mathscr S_\omega$ is in fact bounded in $L^p(bD,\Omega_p)$ for any $1 < p < \infty$ and for any $\Omega_p$ in the optimal class of $A_p$ measures, that is $\Omega_p = \psi_p\sigma$  where  $\psi_p$ is a Muckenhoupt $A_p$-weight and $\sigma$ is induced Lebesgue measure. As an application, we characterize boundedness and compactness in $L^p(bD,\Omega_p)$ for any $1 < p < \infty$ and for any $A_p$ measure $\Omega_p$, of the commutator $[b, \mathscr S_\omega]$ for any Leray Levi-like measure $\omega$. We next introduce the notion of holomorphic Hardy spaces for $A_p$ measures, $1 < p < \infty$, and we characterize boundedness and compactness in $L^2(bD,\Omega_2)$ of the commutator $[b, \mathscr S_{\Omega_2}]$ of the Cauchy–Szego projection defined with respect to any $A_2$ measure $\Omega_2$. Earlier closely related results rely upon an asymptotic expansion, and subsequent pointwise estimates of the Cauchy–Szego kernel, but these are unavailable in the settings of minimal regularity of $bD$; at the same time, newer techniques introduced by Lanzani \& Stein to deal with the setting of minimal regularity are not applicable to $A_p$ measures that are not Leray Levi-like. It turns out that the method of extrapolation is an appropriate replacement for the missing tools. 

  This is joint work with Xuan Thinh Duong (Macquarie University), Ji Li (Macquarie University) and Brett Wick (Washington University in St. Louis).

   

• Monday December 6, 2021 at 14:30, Wachman 617

  Dyadic models for fluid equations

  Mimi Dai, University of Illinois-Chicago

  Inspired by the study of dyadic models for the Navier-Stokes equation, we propose some simplified models for the magnetohydrodynamics in order to have a better understanding on various topics. Pathological solutions are constructed, for instance, solution that blows up at finite time and non-unique Leray-Hopf solutions. Challenging questions will be discussed too. Most of the work is joint with Susan Friedlander.

Current contact: Gerardo Mendoza

The seminar takes place Mondays 2:30 - 3:30 pm either in Wachman 617 or virtually via Zoom. Please see the announcment of the specific meeting. For virtual meetings, please write if you wish to join. Click on title for abstract.

• Monday February 3, 2020 at 14:40, Wachman 617

  Rates of convergence to statistical equilibrium: a general approach and applications

  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).

• Monday February 17, 2020 at 14:40, Wachman 617

  Stein spaces with spherical CR boundaries and their hyperbolic metrics

  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 $U$.

• Monday March 9, 2020 at 14:30, Wachman 617

  (Cancelled)

  Thomas Krainer, PennSate-Altoona

  Postponed to Fall semester.

• Monday March 30, 2020 at 14:40, Wachman 617

  TBA

  Eric Stachura, Kennesaw State University

• Monday April 6, 2020 at 14:40, Wachman 617

  (Cancelled) On $p$-ellipticity and connections to solvability of elliptic complex valued PDEs

  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.

• Monday October 12, 2020 at 14:30, Zoom meeting

  On a conjecture of Baouendi and Rothschild regarding unique continuation

  Shif Berhanu, Temple University
  
   

  In $1993$, Baouendi and Rothschild proved the following boundary unique continuation result: Let $B^+$ be a half ball in the upper half space in $\mathbb R^n$, $u$ continuous on $\overline{B^+}$, harmonic in $B^+$, and $u(x',0)\geq 0$ on the flat piece of $\partial B^+$. If $u$ vanishes to infinite order at the origin in the sense that $u(x)=O(|x|^N)$ for all $N$, then $u\equiv 0.$ 

  They conjectured that a similar result holds for more general domains and more general second order elliptic operators. We will present a positive solution of the conjecture for second order elliptic operators with real analytic coefficients with data given on a real analytic hypersurface. Our result will be a special case of a more general theorem for real analytic elliptic differential operators of any order. Our results have applications to unique continuation for CR functions which was the original inspiration for Baouendi and Rothschild.

   

• Monday November 2, 2020 at 14:30, Zoom meeting

  Pseudo-$H$-type nilmanifolds and analysis of associated operators

  Wolfram Bauer, Leibniz University
  
   

  We give a short introduction to subriemannian geometry.  Based on the Popp measure construction for an equiregular distribution an intrinsic sub-Laplacian can be defined. Generalizing the tangent space, 

  nilpotent Lie groups $G$ serve as local models for a subriemannian manifold and themselves are equipped with a left-invariant subriemannian structure. We introduce pseudo-$H$-type groups $G$ which form a class of step-2-nilpotent Lie groups and consider their quotients by a lattice $\Gamma \subset G$ (pseudo-$H$-type nilmanifolds). Based on a well-known expression 

  of the heat kernel of the sub-Laplacian on the compact left-coset space $\Gamma \backslash G$ we can perform an explicit spectral analysis. In a natural way a pseudo-$H$-type group also carries a pseudo-subriemannian structure which from an analytic viewpoint  induces an ultra-hyperbolic operator $\Delta_{\textrm{UH}}$. We aim to discuss the following questions: 

  -- Can we explicitely construct and classify isospectral (in the subriemannian sense) non-homeomorphic nilmanifolds $\Gamma \backslash G$? 
   

  -- Is the operator $\Delta_{\textrm{UH}}$ locally solvable? Can we explicitly construct its inverse in the case of existence?

  The talk is based on the (joint) papers: 
   

  -- W. Bauer, A. Froehly, I. Markina, Fundamental solutions of a class of ultra-hyperbolic operators on pseudo-$H$-type groups}, Adv. Math. 369, (2020), 1-46. 
   

  -- W. Bauer, K. Furutani, C. Iwasaki, A. Laaroussi,  Spectral theory of a class of nilmanifolds attached to Clifford modules}, Math. Z. (2020)
   

   

• Monday November 9, 2020 at 14:30, Zoom meeting

  Weighted Sobolev regularity of Bergman projection on symmetrized bidisk

  Yuan Yuan, Syracuse University
  
   

  The regularity of Bergman projection is one of the classical problems in several complex variables. Some $L^p$ and Sobolev regularities on some domains with nonsmooth boundary (e.g. Hartgos triangle, quotient domains) have been studied intensively recently. The symmetrized bidisk is another interesting model of non-smooth domains. In this talk, I will discuss the regularity of the Bergman projection on the weighted Sobolev space on the symmetrized bidisk. This is a joint work with Chen and Jin. 

   

• Monday November 16, 2020 at 14:30, Zoom meeting

  On the solvability for differential complexes associated to locally integrable structures

  Paulo Cordaro, University of Sao Paulo
  
   

  In this talk I will consider the problem of local (and also global) solvability for the differential complexes associated to a locally integrable structure. I will survey the known results and describe in some detail the case in which the structure is hypocomplex. Some of the recent results were obtained  in collaboration with M.R. Janhke.

   

• Monday November 30, 2020 at 14:30, Zoom meeting

  Degenerate elliptic boundary value problems with non-smooth coefficients

  Elmar Schrohe, Leibniz University
  
   

  Let $X$ be a manifold with boundary and bounded geometry. On $X$ we consider a uniformly strongly elliptic second order operator $A$ that locally is of the form

  $A=-\sum_{j,k} a_{jk} \partial_{x_j}\partial_{x_k}+ \sum_{j} b_j\partial_{x_j} +c. $
   

  $A$ is endowed with a boundary operator $T$ of the form 

  $T=\varphi_0\gamma_0 + \varphi_1\gamma_1,$

  where $\gamma_0$ and $\gamma_1$ denote the evaluation of a function and its exterior normal derivative, respectively, at the boundary, and $\varphi_0$, $\varphi_1$ are non-negative $C^\infty_b$ functions on the boundary with $\varphi_0+\varphi_1\ge c_0>0$. This problem is not elliptic in the sense of Lopatinskij and Shapiro, unless either  $\varphi_1\not=0$ everywhere or $\varphi_1=0$ everywhere. 

  We show that the realization $A_T$ of $A$ in $L^p(\Omega)$ has a bounded $H^\infty$-calculus of arbitrarily small angle whenever the $a_{jk}$ are H\"older continuous and $b_j$ as well as $c$ are $L^\infty$.

  For the proof we first treat the operator with smooth coefficients on $\mathbb R^n_+$. Here we rely on an extension of Boutet de Monvel's calculus to operator-valued symbols of H\"ormander type $(1,\delta)$. We then use   $H^\infty$-perturbation techniques in order to treat the non-smooth case.

  The existence of a bounded $H^\infty$-calculus allows us to apply maximal regularity techniques. We show how a theorem of Cl\'ement and Li can be used to establish the existence of a short time solution to the porous medium equation on $X$ with boundary condition $T$.

  (Joint work with Thorben Krietenstein, Hannover)

   

• Monday December 7, 2020 at 14:30, Zoom meeting

  On Model Operators in Singular Analysis

  Thomas Krainer, PennState Altoona
  
   

  A common theme in PDEs is to exploit invariance properties with respect to scaling of equations. This leads to fundamental solutions, the heat kernel, as well as the notion of principal symbol. Perturbation theory is then used to derive qualitative results for more general equations, where the dominant scaling-invariant pieces are the principal parts on which invertibility assumptions (ellipticity conditions) are placed. While invertibility of the principal symbol of an elliptic operator governs certain qualitative properties of the equation locally, additional conditions are required to determine well-posedness and regularity on spaces with noncompact ends, and especially on manifolds with incomplete geometry such as boundaries and singularities (i.e. one needs to impose boundary conditions). There are operator-valued analogues of the principal symbol of the operator that are associated with the boundaries and singularities that govern the behavior of solutions and the conditions to be placed on them for the equation. These dominant terms again exhibit certain top-order homogeneity properties, i.e., scaling invariance in a suitable sense, and are sometimes referred to as model operators.

  In this talk I will speak about model operators from a purely functional analytic perspective. We will obtain several results, some previously known in special cases, as well as new ones as consequences of generic functional analytic properties.

   

Current contact: Gerardo Mendoza

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

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

  Subelliptic Liouville theorems

  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.

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

  A Tb Theorem for compactness and boundedness of singular integral operators

  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.

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

  Recent results on Generated Jacobian Equations

  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 February 11, 2019 at 14:40, Wachmn 617

  Homogenization of a class of one-dimensional nonconvex viscous Hamilton-Jacobi equations with random potential

  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 25, 2019 at 14:40, Wachman 617

  The $L^p$-boundedness of the Riesz transform on graphs and Riemannian manifolds

  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 March 4, 2019 at 14:40,

   

  Spring break, no meeting

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

  Analytic continuation in an annulus and in a Bernstein ellipse

  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.

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

  A Minkowski problem for nonlinear capacity

  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.

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

  Harmonic measure and quantitative connectivity

  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).

• Monday April 1, 2019 at 14:40, Wachman 617

  Regular finite type conditions for smooth pseudoconvex real hypersurfaces in $\mathbb C^n$ 

  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 $n=3$.

• Monday April 8, 2019 at 14:40, Wachman 617

  The Atiyah-Patodi-Singer index on manifolds with non-compact boundary

  Maxim Braverman, Northeastern University

  We study the index of the APS boundary value problem for a strongly Callias-type operator $D$ on a complete Riemannian manifold $M$. We use this index to define the relative eta-invariant of two strongly Callias-type operators $A$ and $A'$, which are equal outside of a compact set. Even though in our situation the $\eta$-invariants of $A$ and $A'$ are not defined, the relative $\eta$-invariant behaves as if it were the difference of the $\eta$-invariants of $A$ and $A'$. We also define the spectral flow of a family of such operators and use it compute the variation of the relative $\eta$-invariant. (Joint work with Pengshuai Shi.)

• Monday April 15, 2019 at 14:40, Wachman 617

  Two inverse problems for hyperbolic PDE in three space dimensions

   

  Zachary Bailey, Temple University

   We consider two inverse problems for hyperbolic PDE in three space dimensions. The two problems are associated with a single hyperbolic PDE with a zero order coefficient and the goal is the recovery of this coefficient from two different types of "backscattering data" - backscattering data coming from a fixed offset distribution of sources and receivers on the boundary or backscattering data coming from a single incoming spherical wave. For these problems we prove a stability result provided the difference of the two coefficients is horizontally or angularly controlled respectively. Our work adapts the techniques used by Eemeli Bl&cira;sten, Rakesh and Gunther Uhlmann to solve problems similar to theirs.

   

• Monday April 22, 2019 at 14:40, Wachman 617

  Analysis of a new eigenvalue problem related to scattering by a crack

  Samuel Cogar, University of Delaware

  In this talk I will introduce a new modified transmission eigenvalue problem for scattering by a partially coated crack. Rather than study this problem in isolation, I will present a generalized Robin eigenvalue problem depending on a bounded linear operator that encodes the information for a given scattering medium. Results obtained in this general setting will then be applied to the case of scattering by a partially coated crack, including a new proof that finitely many eigenvalues exist when the surface impedance of the crack is sufficiently small. I will conclude with some numerical examples that both verify the theoretical results and demonstrate the sensitivit

• Monday September 9, 2019 at 14:40, Wachman 617

  A local Tb Theorem for compact singular integral operators with non-homogeneous measures

  Francisco Villarroya, Temple University

  We introduce a new local $Tb$ Theorem for Calder\'on-Zygmund operators \begin{equation*} Tf(x)=\int f(t)K(t,x)d\mu (t) \end{equation*} that extend compactly on $L^{p}(\mathbb R^{n},\mu)$ for $1< p<\infty$ and $\mu $ in a class of non-homogeneous measures. In the main result, compactness is deduced from the following two hypotheses:

  $\bullet$ appropriate decay estimates satisfied by either the operator kernel or the operator measure, and

  $\bullet$ the action of the operator over families of testing functions $(b_{Q})_{Q\in \mathcal D}$ supported on dyadic cubes, which in general may not be accretive.

  As an application we describe the measures $\mu $ such that the Cauchy integral defines a compact operator.

• Monday September 16, 2019 at 14:40, Wachman 617

  Non-vanishing of $L$-functions of Hilbert modular forms in the critical strip

  Wissam Raji, American University of Beirut

  Modular forms are analytic functions defined on the upper half-plane with a specific transformation law under elements of the full modular group $\mathrm{SL}_2(\mathbb{Z})$. In this talk, we give different motivations to the theory and then give an explicit introduction about the main definitions in the theory of modular forms. Interesting series called $L$-series, constructed using the Fourier coefficients of modular forms have important connections to elliptic curves. We show that, on average, the $L$-functions of cuspidal Hilbert modular forms (a generalization of classical modular forms) with sufficiently large weight $k$ do not vanish on the line segments $ \Im(s)=t_0, \ \Re(s) \in (\frac{k-1}{2},\frac{k}{2}-\epsilon)\cup (\frac{k}{2}+\epsilon,\frac{k+1}{2})$.

• Monday September 23, 2019 ,

  No meeting (Grosswald Lectures)

   

• Friday October 18, 2019 at 11:00, Wachman 617 (note special date and time)

  Equivalence of Cauchy-Riemann manifolds and multisummability theory

  Laurent Stolovitch, Université Nice Sophia Antipolis

  We prove that if two real-analytic hypersurfaces in $\mathbb C^2$ are equivalent formally, then they are also $C^\infty$ CR-equivalent at the respective point. As a corollary, we prove that all formal equivalences between real-algebraic Levi-nonflat hypersurfaces in $\mathbb C^2$ are algebraic (in particular are convergent). The result is obtained by using the recent CR - DS technique, connecting degenerate CR-manifolds and Dynamical Systems, and employing subsequently the multisummability theory of divergent power series used in the Dynamical Systems theory. This is a joint work with I. Kossovskiy and B. Lamel.

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

  Comparison Theorems on H-type Foliations, an Invitation to sub-Riemannian Geometry

  Gianmarco Molino, University of Connecticut

  Sub-Riemannian geometry is a generalization of Riemannian geometry to spaces that have a notion of distance, but have restrictions on the valid directions of motion. These arise in a natural way in remarkably many settings.

  This talk will include a review of Riemannian geometry and an introduction to sub-Riemannian geometry; we'll then introduce the notion of H-type foliations; these are a family of sub-Riemannian manifolds that generalize both the K-contact structures arising in contact geometry and the H-type group structures. Our main focus will be recent results giving uniform comparison theorems for the Hessian and Laplacian on a family of Riemannian metrics converging to sub-Riemannian ones. From this we can conclude a sharp sub-Riemannian Bonnet-Myers type theorem.

• Monday November 18, 2019 at 13:30, Wachman 617 (note special time)

  Elastic binodal in Calculus of Variations: a blackboard talk at the graduate student level

  Yury Grabovsky, Temple University

  The Cauchy-Born principle in physics says that macroscopic affine deformations cause microscopic affine deformations. Mathematically this principle can be formulated in the language of Calculus of Variations: $y(x)=F_{0}x$ is the minimizer of the (energy) functional \[ E[y]=\int_{\Omega}W(\nabla y(x))dx\qquad(\Omega\subset\mathbf{R}^{d}-\mbox{a Lipschitz domain}) \] among all Lipschitz functions $y:\Omega\to\mathbf{R}^{m}$, such that $y(x)=F_{0}x$ on $\partial\Omega$. In this form the Cauchy-Born principle can be viewed as a version of Jensen's inequality for convex functions: \[ \frac{1}{|\Omega|}\int_{\Omega}W(\nabla y(x))dx\ge W(F_{0})= W\left(\frac{1}{|\Omega|}\int_{\Omega}\nabla y(x)dx\right), \] since $y(x)=F_{0}x$ on $\partial\Omega$. When the above inequality holds, we say that $W(F)$ is quasiconvex at $F_{0}\in\mathbf{R}^{m\times d}$. The boundary of the set of points of quasiconvexity is called the elastic binodal. When quasiconvexity fails, the gradients of minimizers of $E[y]$ can become discontinuous or even cease to exist, while minimizing sequences develop fine scale oscillations that people call the microstructure.

  In this talk I will discuss my joint work with Lev Truskinovsky, aiming to understand when and why such spontaneous discontinuities and microstructures form. This lecture is geared towards graduate students and is meant to be widely accessible.

• Monday December 2, 2019 at 02:40, Wachman 617

  Sharp estimates of the spherical heat kernel.

  Tomasz Z. Szarek, Rutgers University

  The classical spherical heat kernel is an important object in analysis, probability and physics, among other fields. It is the integral kernel of the spherical heat semigroup and thus provides solutions to the heat equation based on the Laplace-Beltrami operator on the sphere. It is also a transition probability density of the spherical Brownian motion. In this talk we prove sharp two-sided global estimates for the heat kernel associated with a Euclidean sphere of arbitrary dimension. If time permits, we will present a generalization of this result to the compact rank-one symmetric spaces. The talk is based on joint papers with Adam Nowak and Peter Sjögren

Current contact: Gerardo Mendoza

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

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

  Spectral properties for layer potentials associated with second and higher order elliptic PDE in rough domains in two dimensions

  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.

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

  Sticky particles and the Euler-Poisson equations

  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 February 5, 2018 at 14:40, Wachman 617

  A class of FBI transforms and their use in Denjoy-Carleman regularities

  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.

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

  Elliptic complexes of Fuchs-type operators

  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 19, 2018 at 14:40, Wachman 617

  Microlocal analysis in the context of hyperfunctions

  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.

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

  Convergence and divergence of formal CR mappings

  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 March 5, 2018 at 14:30, Wachman 617

  No meeting

   

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

  Nonlinear one-radius mean value properties in metric measure spaces

  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 19, 2018 at 14:40, Wachman 617

  CR Invariants and Solvability of the d-bar equation

  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 26, 2018 at 14:40, Wachman 617

  The Bochner--Hartogs dichotomy

  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 April 16, 2018 at 14:40, Wachman 617

  Free boundary problems as integro-differential equations (cancelled)

  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 April 30, 2018 at 14:40, Wachman 617

  On the geometric quantization of (some) Poisson manifolds

  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 September 10, 2018 at 14:40, Wachman 617

  Multi-layer potentials for higher order systems in non smooth domains

  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 September 17, 2018 at 14:40, Wachman 617

  SQG in bounded domains

  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 October 1, 2018 at 14:40, Wachman 617

  Compact T1 Theory

  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 October 8, 2018 at 14:40, Wachman 617

  On the existence of dichromatic single element lenses

  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 22, 2018 at 14:40, Wachman 617

  Unique continuation at the boundary for some elliptic operators

  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 29, 2018 at 14:40, Wachman 617

  Extrapolation of $H^2$ functions in the upper half-plane.

  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 November 12, 2018 at 14:40, Wachman 617

  Quantum Painleve II equation (QPII) and random matrix beta ensembles

  Igor Rumanov, University of Colorado-Boulder

  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 November 26, 2018 at 14:40, Wachman 617

  A Balian-Low type theorem for Gabor Schauder bases

  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 December 3, 2018 at 14:40, Wachman 617

  A harmonic measure for sets of higher codimensions

  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.

Current contact: Gerardo Mendoza

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

• Monday January 23, 2017 at 14:40, Wachman 617

  Quantitative homogenization and regularity theory

  Scott Armstrong, NYU

   

  I will describe recent developments in the quantitative homogenization of linear elliptic equations in divergence form, emphasizing some ideas arising from the calculus of variations and the role played by a new elliptic regularity theory for equations with random coefficients.

• Monday January 30, 2017 at 14:40, Wachman 617

  The Dirichlet problem for elliptic systems with data in Köthe function spaces

  Irina Mitrea

   

  In this talk I will discuss well-posdness results for the Dirichlet problem for second-order, homogeneous, elliptic systems, with constant complex coefficients, in the upper half space, with boundary data from Lebesgue spaces, variable exponent Lebesgue spaces, Lorentz spaces, Zygmund spaces, as well as their weighted versions. A key tool in this analysis is establishing boundedness of the Hardy-Littlewood maximal operator on appropriate Köthe function spaces. This is joint work with Dorina Mitrea, Marius Mitrea and Jose Maria Martell.

• Monday February 20, 2017 at 14:40, Wachman 617

  Differentiability and rectifiability on metric planes

  Guy David, Courant Institute, New York University

  Since the work of Cheeger, many non-smooth metric measure spaces are now known to support a differentiable structure for Lipschitz functions. The talk will discuss this structure on metric measure spaces with quantitative topological control: specifically, spaces whose blowups are topological planes. We show that any differentiable structure on such a space is at most 2-dimensional, and furthermore that if it is 2-dimensional the space is 2-rectifiable. This is partial progress on a question of Kleiner and Schioppa, and is joint work with Bruce Kleiner.

• Monday February 27, 2017 at 14:40, Wachman 617

  Higher order elliptic equations

  Federico Tournier, University of La Plata and IAM, Argentina

  We look at the local problem in free space and in half space of an elliptic operator with Hölder coefficients.

• Monday March 20, 2017 at 14:40, Wachman 617

  Degenerate diffusions on manifolds with corners

  Charles Epstein, University of Pennsylvania

• Monday April 17, 2017 at 14:40, Wachman 617

  Singular perturbation limits of fractional Allen-Cahn

  Yannick Sire, John Hopkins

  I will report on recent work with V. Millot and K. Wang on the singular limit for a fractional Allen-Cahn equation leading to stationary nonlocal minimal surfaces. I will introduce these latter concepts and will prove the convergence result, based on a deep Geometric Theory argument from Marstrand.

• Monday May 1, 2017 at 14:40, Wachman 617

  Sobolev regularity estimates for solutions to spectral fractional elliptic equations

  Tadele Mengesha, University of Tennessee, Knoxville

  Global Calderon-Zygmund type estimates are obtained for solutions to fractional elliptic problems over a smooth domain. Our approach is based on the 'extension problem' where the fractional elliptic operator is realized as a Dirichlet-to-Neumann map to a degenerate elliptic PDE in one more dimension. This approach allows the possibility of deriving estimates for solutions to the fractional elliptic equation from that of a corresponding degenerate elliptic equation. We will confirm this first by obtaining weighted estimates for the gradient of solutions to a class of linear degenerate/singular elliptic problems. The class consists of those with coefficient matrix that is symmetric, nonnegative definite, and both its smallest and largest eigenvalues are proportion to a particular weight that belongs to a Muckenhoupt class. The weighted estimates are obtained under a smallness condition on the mean oscillation of the coefficients with a weight. This is a joint work with T. Phan.

• Monday September 11, 2017 at 14:40, Wachman 617

  Boundary Value Problems for the Biharmonic Equation in Rough Domains

  Irina Mitrea, Temple University.

  This talk is focused on Singular Integral Operator methods for boundary value problems for the Bi-Laplacian in irregular domains in the Euclidean Space and is largely motivated by the study of the classical free-plate problem arising in the Kirchhoff-Love theory of thin plates. This is based on joint work with Gustavo Hoepfner, Paulo Liboni and Marius Mitrea.

   

• Monday September 18, 2017 at 14:40, Wachman 617

  Recent progress on Landis' conjecture

  Blair Davey, City College of New York

  In the late 1960s, E.M. Landis made the following conjecture: If $u$ and $V$ are bounded functions, and $u$ is a solution to $\Delta u = V u$ in $\mathbb{R}^n$ that decays like $|u(x)| \le c \exp(- C |x|^{1+})$, then $u$ must be identically zero. In 1992, V. Z. Meshkov disproved this conjecture by constructing bounded functions $u, V: \mathbb{R}^2 \to \mathbb{C}$ that solve $\Delta u = V u$ in $\mathbb{R}^2$ and satisfy $|u(x)| \le c \exp(- C |x|^{4/3})$. The result of Meshkov was accompanied by qualitative unique continuation estimates for solutions in $\mathbb{R}^n$. In 2005, J. Bourgain and C. Kenig quantified Meshkov's unique continuation estimates. These results, and the generalizations that followed, have led to a fairly complete understanding of the complex-valued setting. However, there are reasons to believe that Landis' conjecture may be true in the real-valued setting. We will discuss recent progress towards resolving the real-valued version of Landis' conjecture in the plane.

   

• Monday September 25, 2017 at 14:40, Wachman 617

  Ultradifferential operators in the study of Gevrey solvability and regularity

  Luis Fernando Ragognette, Federal University of São Carlos, Brazil

  The goal of this talk is to present results on infinite order differential operators and its applications to local solvability of a differential complex associated to a locally integrable structure in a Gevrey environment.

  One of the reasons why infinite order differential operators are important in this setting is a structural theorem that says that every ultradistribution of order $s$ can be locally represented by an infinite order differential operator applied to a Gevrey function of order $s$, this new kind of representation is crucial in several applications that we are going to discuss.

   

• Monday October 2, 2017 at 14:40, Wachman 617

  Applications of Mizohata type vector fields to solutions of first order nonlinear PDE's

  Gustavo Hoepfner, Federal University of São Carlos, Brazil

  The goal of this talk is twofold. Firstly, we will recall the notion of quasi $\ell$ Mizohata vector fields first introduced by L. Nunes and R. dos Santos Filho (UFSCar) and show that it can be written normal form which is very special when we are in $\mathbb R^2$.

  Secondly, we will establish a connection of these quasi $\ell$ Mizohata vector fields with recent results by Z. Adwan and S. Berhanu on the solutions of first order nonlinear PDE's and extend them to the classes of ultradifferentiable functions.

  This is a joint work with R. Medrado from Universidade Federal do Cear\'a.

   

• Monday October 9, 2017 at 14:40, Wachman 617

  Applications of Mizohata type vector fields to solutions of first order nonlinear PDE's, Part II

  Gustavo Hoepfner, Federal University of São Carlos, Brazil

  This is a continuation of last week's talk. The goal of this talk is twofold. Firstly, we will recall the notion of quasi $\ell$ Mizohata vector fields first introduced by L. Nunes and R. dos Santos Filho (UFSCar) and show that it can be written normal form which is very special when we are in $\mathbb R^2$.

  Secondly, we will establish a connection of these quasi $\ell$ Mizohata vector fields with recent results by Z. Adwan and S. Berhanu on the solutions of first order nonlinear PDE's and extend them to the classes of ultradifferentiable functions.

  This is a joint work with R. Medrado from Universidade Federal do Cear\'a.

   

• Monday October 16, 2017 at 14:40, Wachman 527

  Regularity results for degenerate elliptic and parabolic equations in non-divergence form

  Farhan Abedin, Temple University

  I will present some new results concerning regularity of solutions to certain degenerate elliptic and parabolic equations in non-divergence form. In the first half of the talk, I will discuss recent work on Harnack’s inequality for operators structured on Heisenberg vector fields, with coefficients that are uniformly positive definite, continuous, and symplectic. This is joint work with Cristian Gutierrez (Temple) and Giulio Tralli (University of Rome). In the second half, I will outline work in progress with Giulio Tralli on parabolic equations of Kolmogorov type.

• Monday October 30, 2017 at 14:40, Wachman 617

  The families index formula on stratified spaces

  Pierre Albin, University of Illinois, Urbana-Champaign

  Stratified spaces arise naturally even when studying smooth objects, e.g., as algebraic varieties, orbit spaces of smooth group actions, and many moduli spaces. There has recently been a lot of activity developing analysis on these spaces and studying topological invariants such as the signature. I will report on joint work with Jesse Gell-Redman in which we study families of Dirac-type operators on stratified spaces and establish a formula for the Chern character of their index bundle.

   

• Monday November 6, 2017 at 14:40, Wachman 617

  The singular free boundary in the Signorini problem

  Mariana Smit Vega Garcia, University of Washington

  In this talk I will overview the Signorini problem for a divergence form elliptic operator with Lipschitz coefficients, and I will describe a few methods used to tackle two fundamental questions: what is the optimal regularity of the solution, and what can be said about the singular free boundary in the case of zero thin obstacle. The proofs are based on Weiss and Monneau type monotonicity formulas. This is joint work with Nicola Garofalo and Arshak Petrosyan.

• Monday November 13, 2017 at 14:40, Wachman 617

  Sufficient conditions for existence of Least Gradient solutions

  Ahmad Sabra, University of Warsaw

  In this talk we study the following variational problem: \begin{equation} \inf\Big\{\int_\Omega |Du|:u\in \mathrm{BV}(\Omega),\ u|_{\partial\Omega}=f\Big\}, \end{equation} with $\Omega$ a Lipschitz bounded domain and $f\in L^1(\partial\Omega)$. Solutions to (1) are called least gradient functions and do not always exist for every boundary data $f$. It is well-known that level sets of $\textrm{LG}$ functions are minimal surfaces. Using this geometrical property, we construct solutions to (1) for convex domains $\Omega$ and $f\in \mathrm{BV}(\partial\Omega)$ satisfying some monotonicity properties. We also establish a connection between solutions to (1) and variational problems that appear in Free Material Design applications.

• Monday December 4, 2017 at 14:40, Wachman 617

  On unique continuation at the boundary for harmonic functions and solutions of the Helmholtz equation

  Shif Berhanu, Temple University

  We will discuss results on local unique continuation at the boundary for holomorphic functions of one variable and for the solutions of the Helmholtz equation $L_cu=\Delta u+cu=0,\, c\in \mathbb R$ in an open set of the half space $\mathbb R^n_+$ generalizing the theorems proved by Baouendi and Rothschild for harmonic functions. The results involve a local boundary sign condition on the product of the solution and a monomial. Applications to unique continuation for CR mappings will also be discussed.

Current contact: Gerardo Mendoza

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

• Monday February 1, 2016 at 14:40, Wachman 617

  Spectral instability of selfadjoint extensions

  Gerardo Mendoza, Temple University

• Monday February 8, 2016 at 14:40, Wachman 617

  Fourier Integral Operators: an overview

  Gerardo Mendoza, Temple University

• Monday February 15, 2016 at 14:40, Wachman 617

  Fourier Integral Operators: an overview, Part II

  Gerardo Mendoza, Temple University

• Monday February 29, 2016 at 14:40, Wachman 617

  Spring break

  No meeting

• Monday March 7, 2016 at 14:40, Wachman 617

  Constrained shape analysis through flows of diffeomorphisms

  Sylvain Arguillère, Johns Hopkins University

• Monday March 14, 2016 at 14:40, Wachman 617

  Local solvability of a class of degenerate second order operators with smooth and non smooth coefficients

  Serena Federico, University of Bologna

• Monday March 21, 2016 at 14:40, Wachman 617

  Jump Formulas for Tempered Distributions

  Hussein Awala, Temple University

• Monday March 28, 2016 at 14:40, Wachman 617

  A new generation of Calderon-Zygmund theory for singular integrals on Riemannian manifolds

  Marius Mitrea, University of Missouri

• Monday April 4, 2016 at 14:40, Wachman 617

  Nonreflexive representations of Jordan multialgebras in the theory of exact relations for effective tensors of composite materials with an application to Calculus of Variations

  Yury Grabovsky, Temple University

• Monday April 11, 2016 at 14:40, Wachman 617

  Higher dimensional scattering theory and integral representation formulas

  Dorina Mitrea, University of Missouri

• Monday April 18, 2016 at 14:40, Wachman 617

  Strongly Correlated Topological Insulators

  Peter S. Riseborough, Temple University

• Monday April 25, 2016 at 14:40, Wachman 617

  On the numerical solution of the far field refractor problem

  Cristian Gutiérrez, Temple University.

• Monday October 3, 2016 at 14:40, Wachman 617

  Vector valued estimates via the helicoidal method

  Camil Muscalu, Cornell University

   

  The plan is to describe a new method of proving (multiple) vector valued inequalities in harmonic analysis.It is extremely robust, yet conceptually simple, and allowed us to give positive answers to a number of open questions that have been circulating for some time. Joint work with Cristina Benea.

   

• Monday October 17, 2016 at 14:40, Wachman 617

  A Sharp Higher-Order Integration by Parts Formula with Non-Tangential Traces

  Irina Mitrea, Temple University

   

  In this talk I will discuss an optimal higher-order integration by parts formula with non-tangential traces in non-smooth domains and sketch its proof. This is joint work with Gustavo Hoepfner, Paulo Liboni, Dorina Mitrea, and Marius Mitrea.

   

• Monday October 24, 2016 at 14:40, Wachman 617

  Optimal regularity for Bellman equation in two dimensions

  Ovidiu Savin, Columbia University

   

  I will discuss about the optimal regularity in a two-phase free boundary problem involving different elliptic operators and its connection with the Bellman equation. The proofs are based on some old ideas of Bernstein concerning elliptic equations in two dimensions. This is a joint work with L. Caffarelli and D. De Silva.

   

• Monday November 14, 2016 at 14:40, Wachman 617

  Existence of Propagators for Coulomb-Like Potentials in Density Functional Theory

  Eric Stachura

   

  Density Functional Theory (DFT) is one of the most widely used methods for electronic structure calculations in materials science. It was realized that for N ≥ 103 electrons, it is impractical to find the N particle wave function for this system. One of the gems of DFT is the Hohenberg-Kohn Theorem, which says that the ground state electron density alone provides all properties of a given static system. When the system is allowed to evolve in time, the corresponding time dependent theory (TDDFT) was initiated by E. Runge and E. K. U Gross in the early 1980’s, and is one of the most popular theories for computing electronic excitation spectra. Runge and Gross proved a time dependent analog of the Hohenberg- Kohn Theorem, which is the starting point for our work. While attempting to develop a new proof of the Runge-Gross Theorem, there came a need to solve a Schr ̈odinger equation with time dependent Hamiltonian in R3N . By smoothing out the classical Coulomb potential, we show existence of unitary propagators for a general time dependent Schr ̈odinger equation where we allow the atomic nuclei to move along classical trajectories. By appealing to a classical 1973 result of Barry Simon, we can also understand the spectrum of the underlying time dependent Hamiltonian. This is joint work with Maxim Gilula (MSU) and is inspired by work of John Perdew (Temple). Preliminary report.

   

• Monday December 5, 2016 at 14:40, Wachman 617

  Mellin Transform Techniques for the Mixed Problem in Two Dimensions

  Hussein Awala, Temple University

   

  In this talk I will discuss the boundary value problem with mixed Dirichlet and Neumann boundary conditions for the Laplacian and the Lame system in infinite sectors in two dimensions. Using a potential theory approach the problem is reduced to inverting a singular integral operator (SIO) naturally associated with the problem on appropriate function spaces. Mellin transform techniques are then employed in the study of the spectrum of the aforementioned SIO.

   

Current contact: Gerardo Mendoza

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

• Monday January 26, 2015 at 14:40, Wachman 617

  Szegő Projections and Kerzman-Stein Formulas II

  Irina Mitrea, Temple University.

• Monday February 2, 2015 at 14:40, Wachman 617

  Geometric analysis on singular and non-compact spaces. 

  Jesse Gell-Redman, Johns Hopkins University.

• Monday February 9, 2015 at 14:40, Wachman 617

  Inverse iteration for p-ground states

  Ryan Hynd, University of Pennsylvania.

• Monday February 16, 2015 at 14:40, Wachman 617

  Fredholm Theory for Higher Order Elliptic Boundary Value Problems in Non-Smooth Domains 

  Irina Mitrea, Temple University

• Monday February 23, 2015 at 14:40, Wachman 617

  High frequency estimates for the Helmholtz equation with applications to boundary integral equations

  Jared Wunsch, Northwestern University.

• Monday March 9, 2015 at 14:40, Wachman 617

  Variational models for prestrained plates with Monge-Ampere constraint.

  Marta Lewicka, University of Pittsburgh

• Monday March 16, 2015 at 14:40, Wachman 617

  Can you hear the metric on a sphere?

  Emily Dryden, Bucknell University.

• Monday March 23, 2015 at 14:40, Wachman 617

  On the geometry and topology of zero sets of Schrödinger eigenfunctions

  Yaiza Canzani, Harvard University and IAS

• Monday March 30, 2015 at 14:40, Wachman 617

  Wiener-type tests for evolution equations

  Giulio Tralli, University of Bologna

• Monday April 6, 2015 at 14:40, Wachman 617

  On the reflection principle for CR mapping between CR manifolds. 

  Ming Xiao, Rutgers University

• Monday April 13, 2015 at 14:40, Wachman 617

  Inverting double layers on Lebesgue spaces on the boundary of Lipschitz domains

  Hussein Awala, Temple University

• Monday April 20, 2015 at 14:40, Wachman 617

  Higher order Poisson Kernels and Lp polyharmonic boundary value problems in Lipschitz domains

  Zhihua Du, Jinan University

• Monday September 14, 2015 at 14:40, Wachman 617

  Some new results on Dolbeault cohomology groups

  Christine Laurent-Thiébaut, Institut Fourier, Université de Grenoble I

• Monday September 21, 2015 at 14:40, Wachman 617

  Radiation Conditions and Integral Representations in Higher Dimensional Scattering

  Irina Mitrea, Temple University

• Monday October 5, 2015 at 14:40, Wachman 617

  Sharp trace-Sobolev inequalities of order 4

  Antonio Aché, Princeton University

• Monday October 12, 2015 at 14:40, Wachman 617

  Positivity of the complex Laplacian

  Siqi Fu, Rutgers University

• Monday October 19, 2015 at 14:40, Wachman 617

  Smooth measures in infinite dimensions

  Tai Melcher, University of Virginia

• Monday October 26, 2015 at 14:40, Wachman 617

  Oscillatory integrals satisfying Varchenko's condition

  Maxim Gilula, University of Pennsylvania

• Monday November 2, 2015 at 14:40, Wachman 617

  Fluid limits and queueing policies

  Christian Gromoll, University of Virginia

• Monday November 9, 2015 at 14:40, Wachman 617

  Harmonic analysis on compact manifolds and metric-measure spaces

  Isaac Pesenson, Temple University

• Monday November 16, 2015 at 14:40, Wachman 617

  On the Hopf lemma in the plane and its applications

  Shif Berhanu, Temple University

• Monday November 30, 2015 at 14:40, Wachman 617

  Weak solutions to some refraction problems in metamaterials

  Eric Stachura, Temple University

• Monday December 7, 2015 at 14:40, Wachman 617

  Spectral Properties of the Hardy Kernel Operators and Application to Second Order Elliptic Boundary Value Problems

  Hussein Awala, Temple University

Current contact: Gerardo Mendoza

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

• Monday January 27, 2014 at 14:40, Wachman 617

  Regularity for solutions of a class of fully nonlinear PDEs from complex geometry 

  Annamaria Montanari, Universita di Bologna

   

• Monday February 24, 2014 at 14:40, Wachman 617

  Inverse Spectral Problems for Transmission Eigenvalues 

  David Colton, University of Delaware

   

• Monday March 10, 2014 at 14:40, Wachman 617

  Boundary Value Problems for Elliptic Wedge Operators of First Order 

  Thomas Krainer, Penn State Altoona

   

• Monday March 17, 2014 at 14:40, Wachman 617

  Hearing the shapes of graphs and related sampling problems 

  Michael Robinson, American University

• Monday March 24, 2014 at 14:40, Wachman 617

  The reflector Problem and the Inverse Square Law

  Ahmad Sabra, Temple University

   

• Monday March 31, 2014 at 14:40, Wachman 617

  The mixed problem in Lipschitz domains 

  Russell Brown, University of Kentucky

• Monday April 7, 2014 at 14:40, Wachman 617

  Reverse entropy power inequalities

  Mokshay Madiman, University of Delaware

• Monday April 14, 2014 at 14:40, Wachman 617

  Intermittency and the Stochastic Heat Equation 

  Daniel Conus, Lehigh University

• Monday April 21, 2014 at 14:40, Wachman 617

  How frequently can Sobolev maps distort Hausdorff dimension? 

  Zoltan Balogh, University of Bern, Switzerland

• Monday April 28, 2014 at 11:00, Barton 108

  Approximation of Polyatomic FPU Lattices by KdV Equations 

  Special seminar:

  J. Douglas Wright, Drexel University

   

• Monday April 28, 2014 at 14:40, Wachman 617

  On the regularity of CR mappings between CR manifolds 

  Shiferaw Berhanu, Temple University

• Monday May 5, 2014 at 14:40, Wachman 617

  Asymptotic spectral properties of phase space localization operators of Gabor type 

  Krzysztof Nowak, Drexel University

   

• Monday September 15, 2014 at 14:40, Wachman 617

  Eigenspace decomposition of the kernel of R-invariant operators

  Gerardo Mendoza, Temple University.

• Monday September 22, 2014 at 14:40, Wachman 617

  The Witten deformation on singular spaces

  Ursula Ludwig, Département de Mathématiques d'Orsay.

• Monday September 29, 2014 at 14:40, Wachman 617

  Existence of solution of an Imaging Problem in Geometrical Optics

  Ahmad Sabra, Temple University.

• Monday October 20, 2014 at 14:40, Wachman 617

  Szegő Projections and Kerzman-Stein Formulas

  Irina Mitrea, Temple University

• Monday November 3, 2014 at 14:40, Wachman 617

  Pressureless Euler-Poisson systems

  Truyen Nguyen, Department of Mathematics, The University of Akron.

• Monday November 17, 2014 at 14:40, Wachman 617

  Aberrant CR Structures

  Howard Jacobowitz, Department of Mathematics, Rurgers University.

• Monday December 1, 2014 at 14:40, Wachman 617

  Failure of the Weierstrass Preparation Theorem in Quasi-analytic Denjoy-Carleman Rings

  Andreea Nicoara, University of Pennsylvania

Current contact: Gerardo Mendoza

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

• Monday September 9, 2013 at 14:40, Wachman 617

  Homogenization of elliptic PDEs in Fourier space 

  Yury Grabovsky, Temple University

   

• Monday September 16, 2013 at 14:40, Wachman 617

  Analytical issues in quasi-static cohesive fracture evolution 

  Christopher J. Larsen, Worcester Polytechnic Institute

   

• Monday September 30, 2013 at 14:40, Wachman 617

  Existence of global symmetry-breaking solutions for a phase-field model of lipid-bilayer vesicles 

  Timothy J. Healey Cornell University

   

• Monday October 7, 2013 at 14:40, Wachman 617

  Asymptotic Expansions for Transmission Eigenvalues for Media with Small Inhomogeneities 

  Shari Moskow, Drexel University

   

• Monday October 14, 2013 at 14:40, Wachman 617

  The mixed problem for the Lame system in two dimensions 

  Katharine Ott, University of Kentucky

   

• Monday October 21, 2013 at 14:40, Wachman 617

  Universality for the Stochastic Airy Operator 

  Brian Rider, Temple University

   

• Monday October 28, 2013 at 14:40, Wachman 617

  New monotonicity formulas and the optimal regularity in the thin obstacle problem with Lipschitz coefficients 

  Mariana Smit Vega Garcia, Purdue University

   

• Monday November 4, 2013 at 14:40, Wachman 617

  PDEs in Lipschitz Domains and Lipschitz Manifolds

  Kevin Brewster, University of Missouri, Columbia

   

• Monday November 11, 2013 at 14:40, Wachman 617

  Scalable frames 

  Kasso A. Okoudjou, University of Maryland

   

• Monday November 18, 2013 at 14:40, Wachman 617

  Density functional theory and Donsker-Varadhan 

  Omar Hijab, Temple University

   

• Monday December 2, 2013 at 14:40, Wachman 617

  Korn and Korn-like inequalities for perfect cylindrical shells 

  Davit Harutyunyan, University of Utah

   

Current contacts: Vasily Dolgushev, Jaclyn Lang, Ed Letzter or Martin Lorenz.

The Seminar usually takes place on Mondays at 1:30 PM in Room 617 on the sixth floor of Wachman Hall. Click on title for abstract.

• Monday January 22, 2024 at 13:30, Wachman 527

  Algebra Seminar. Organizational Meeting

  This is the organizational meeting of the Algebra Seminar. 

   

• Monday January 29, 2024 at 13:30, Wachman 527

  The Herbrand—Ribet Theorem

• Monday February 5, 2024 at 13:30, Wachman 527

  Toward a local Langlands correspondence in families

  Gilbert Moss, University of Maine
  
  Let $G$ be a connected reductive algebraic group, such as $GL_n$, and let $F$ be a nonarchimedean local field, such as the p-adic numbers $\mathbb{Q}_p$. The local Langlands program describes a connection, which has been established in many cases, between irreducible smooth representations of $G(F)$ and Langlands parameters, which are described in terms of the absolute Galois group of $F$. The local Langlands correspondence "in families" is concerned with an aspect of the local Langlands program that seeks to upgrade this connection beyond irreducible representations to a smoothly varying morphism between natural moduli spaces of $G(F)$ representations and Langlands parameters. We will describe a precise conjecture in this direction and summarize past work establishing the conjecture for $GL_n(F)$, as well as ongoing work toward establishing it for classical groups. 

• Monday February 12, 2024 at 13:30, Wachman 527

  Galois theory for infinite algebraic extensions

  Vasily Dolgushev, Temple University
  
  This is an overview of the series of talks on Galois theory for infinite algebraic extensions. I will introduce the set-up and formulate the main theorem of the Galois theory for infinite algebraic extensions (the theorem is due to W. Krull). I will formulate the Nikolov-Segal theorem on finitely generated profinite groups and talk about examples of non-open subgroups of finite index in the absolute Galois group of rational numbers. If time permits, I will also formulate the Shafarevich conjecture.  
  
  
   

• Monday February 19, 2024 at 13:30, Wachman 527

  Introduction to topological groups

  Chathumini Kondasinghe, Temple University
  
  This is a brief introduction to topological groups. I will define topological groups, give several examples and prove selected statements. This is a part of the series on talks on Galois theory for infinite algebraic extensions.
  
   

• Monday February 26, 2024 at 09:30, Wachman Hall 108

  Zeroes of Period Polynomials of Cusp Forms

  Wissam Raji, American University of Beruit
  
  We consider the period polynomials $r_f (z)$ associated with cusp forms $f$ of weight $k$ on all of $SL_2(\mathbb{Z})$, which are generating functions for the critical L-values of the modular L-function associated to f. In 2014, El-Guindy and Raji proved that if $f$ is an eigenform, then $r_f (z)$ satisfies a “Riemann hypothesis” in the sense that all its zeros lie on the natural boundary of its functional equation. We show that this phenomenon is not restricted to eigenforms, and we provide large natural infinite families of cusp forms whose period polynomials almost always satisfy the Riemann hypothesis. For example, we show that for weights $k ≥ 120$, linear combinations of eigenforms with positive coefficients always have unimodular period polynomials.
  tba

• Monday March 11, 2024 at 13:30, Wachman 307

  Number fields with small discriminants

  Frauke Bleher, University of Iowa

  The discriminant d_F of a number field F is a basic invariant of F.  The smaller d_F is relative to [F:Q], the more elements there are in the ring of integers O_F of F that have a given bounded size.  This is relevant, for example, to cryptography using elements of O_F.  In 2007, two cryptographers (Peikert and Rosen) asked whether one could give an explicit construction of an infinite family of number fields F having d_F^{1/[F:Q]} bounded by a constant times [F:Q]^d for some d < 1.  By an explicit construction we mean an algorithm requiring time bounded by a polynomial in log([F:Q]) for producing a set of polynomials whose roots generate F.  In this talk I will describe work with Ted Chinburg showing how this can be done for any d > 0.  The proof uses the group theory of profinite 2-groups as well as recent results in analytic number theory.

   

• Monday March 18, 2024 at 13:30, Wachman 527

  Introduction to profinite groups

  Sean O'Donnell, Temple University
  
  We will start the talk with a review of limits of functors and present selected examples for categories of groups, topological spaces and topological groups. We will also discuss natural transformations and upgrade the limit assignment to a functor. Motivated by Galois Theory, we will present several properties of limits from downward directed posets and their co-initial sub-posets. We will define the profinite completion of a group G as the topological group. If time permits, we will conclude the talk with a practical description of the profinite completion of the ring of integers.
  
  
   

• Monday April 8, 2024 at 13:30, Wachman 527

  Non-open subgroups of finite index

  Aniruddha Sudarshan, Temple University

  In this talk, we will show the existence of a non-open subgroup of finite index of the absolute Galois group of the rationals. If time permits, we will also talk about the Nikolov-Segal theorem. Among other things, this theorem implies that every finite index subgroup of a topologically finitely generated profinite group G is open in G. 

   

• Monday April 15, 2024 at 13:30, Wachman 527

  Orbit problems and the mod p properties of Markoff numbers

  William Chen, Rutgers University

  Markoff numbers are positive integers which encode how resistant certain irrational numbers are to being approximated by rationals. In 1913, Frobenius asked for a description of all congruence conditions satisfied by Markoff numbers modulo primes p. In 1991 and 2016, Baragar, Bourgain, Gamburd, and Sarnak conjectured a refinement of Frobenius’s question, which amounts to showing that the Markoff equation x^2 + y^2 + z^2 - xyz = 0 satisfies “strong approximation”; that is to say: they conjecture that its integral points surject onto its mod p points for every prime p. In this talk we will show how to prove this conjecture for all but finitely many primes p, thus reducing the conjecture to a finite computation. A key step is to understand this problem in the context of describing the orbits of certain group actions. Primarily, we will consider the action of the mapping class group of a topological surface S on (a) the set of G-covers of S, where G is a finite group, and (b) on the character variety of local systems on S. Questions of this type have been related to many classical problems, from proving that the moduli space of curves of a given genus is connected, to Grothendieck’s ambitious plan to understand the structure of the absolute Galois group of the rationals by studying its action on "dessins d’enfant". We will explain some of this history and why such problems can be surprisingly difficult.

   

• Monday August 26, 2024 at 13:30, Wachman 617

  Algebra Seminar: organizational meeting 

   

• Monday September 9, 2024 at 13:30, Wachman 617

  Galois theory for infinite algebraic extensions

  Vasily Dolgushev, Temple University

  We will show that the Galois group of an infinite (algebraic) Galois extension is naturally a profinite group and give several examples. We will formulate the main theorem of the Galois theory for infinite algebraic extensions (the theorem is due to W. Krull) and show that, in this set-up, Galois groups have non-closed subgroups.  

   

• Monday September 16, 2024 at 13:30, Wachman 617

  Local Fields, Inertia Groups, And Frobenius Elements

  Xiaoyu Huang, Temple University

  We will begin by discussing local fields and valuations, followed by an exploration of inertia groups and Frobenius elements. Next, we will cover unramified extensions and then connect these concepts to the global field setting.

   

• Monday September 23, 2024 at 13:30, Wachman 617

  The Chebotarev Density Theorem

  Aniruddha Sudarshan, Temple University

  Dirichlet's prime number theorem states that there exists infinitely many primes in a given arithmetic progression. Chebotarev's theorem is a vast generalization of this classical result. We state Chebotarev's result and talk about a few of its applications.  We will mainly focus on the density result of Frobenius elements in the absolute Galois group of a number field. 

   

• Monday September 30, 2024 at 13:20, Wachman 617

  Cyclotomic Characters and Compatible Systems of p-adic Galois Representations

  Stephen Liu, Temple University

  The p-adic cyclotomic character is an important example of one-dimensional Galois representations. We will introduce the construction and some properties of cyclotomic characters, such as ramification and certain compatibility across different primes. Motivated by these properties, we will try to define compatible systems of  p-adic Galois representations. 

   

• Monday October 7, 2024 at 13:20, Wachman 617

  Galois representations on Elliptic curves

  Erik Wallace, Temple University

  This talk will give a brief introduction to Galois Representations on Elliptic curves, and a survey of some of the main results. We will draw particular attention to connections with previous talks, such as on the inertia group, on the Chebotarev density theorem. Specific examples will be included, as well as some sage code.

   

• Monday October 14, 2024 at 13:20, Wachman 617

  Complex Galois representations

  Chathumini Kondasinghe, Temple University

  In this talk we will show that every complex Galois representation factors through the Galois group of a finite extension of Q. Then we will use this result to show that every Galois representation over C can be seen as a compatible system of Galois representations. 

   

• Monday October 21, 2024 at 13:20, Wachman 617

  Class Field Theory and 1-Dimensional Galois Representations

  Sean O'Donnell, Temple University

  This talk will give a brief summary of some results of class field theory and their relevance to the study of 1-dimensional Galois representations of local and global fields. Results covered will include the existence of the local and global Artin homomorphisms, the Kronecker-Weber Theorem, and a brief overview of the connection between local and global class field theory.

   

• Thursday October 31, 2024 at 15:30, Wachman 617

  An invitation to the BSD conjecture

  Ashay Burungale, University of Texas Austin
  
  The talk plans to present an introduction to the BSD conjecture predicting a mysterious link between rational points on an elliptic curve defined over rational numbers, and analytic properties of the associated Hasse-Weil L-function. Some recent progress will be discussed.

• Monday November 4, 2024 at 13:20, Wachman 617

  Automorphic functions and harmonic analysis on groups

  Ross Griebenow, Temple University

  We introduce the theory of automorphic functions for a discrete cofi- nite group Γ ≤ PSL2(R), and the spectral theory of the Laplace operator on H. Combining these ideas, we develop the “Selberg trace formula” which relates the spectral decomposition of square-integrable functions on Γ\H with the geometry of Γ\H. The trace formula allows us to under- stand the properties of a certain zeta function which is analogous to the Riemann zeta-function, and characterize the number of closed geodesics on Γ\H with bounded length.

• Friday November 8, 2024 , University of Pennsylvania

  Zeta functions and symplectic duality

  Yiannis Sakellaridis, Johns Hopkins University
  
  Abstract:  The Riemann zeta function was introduced by Euler, but carries Riemann's name because he was the one who extended it to a meromorphic function on the entire complex plane, and discovered its importance for the distribution of primes. It admits a vast class of generalizations, called L-functions, but, as in Riemann's case, one usually cannot prove anything about them without relying on seemingly unrelated integral representations. 
  
  In joint work with David Ben-Zvi and Akshay Venkatesh, we elucidate the origin of such integral representations, showing that they are manifestations of a duality between nice Hamiltonian spaces for a pair $(G,\check G)$ of ``Langlands dual'' groups. Over the geometric cousins of number fields -- algebraic curves and Riemann surfaces -- such dualities had been anticipated and constructed in many cases by Gaiotto and others, motivated by mathematical physics. 
  
  The first talk will be a gentle and example-oriented introduction to problems in the ``relative'' Langlands program, introducing automorphic L-functions, and various ways of generalizing Riemann's integral representation. We will also talk about the idea of quantization, and why it might be an appropriate framework for studying such constructions. In the second talk, I will introduce a conjectural duality between nice (``hyperspherical'') Hamiltonian spaces, and how it gives rise to a hierarchy of conjectures, both function- and sheaf-theoretic, refining the Langlands correspondence.

• Friday November 8, 2024 , University of Pennsylvania

  Mac Lane valuations and algebraic geometry

  Andrew Obus, CUNY Baruch
  
  Abstract:  Almost 90 years ago, Mac Lane discovered the technique of "inductive valuations", which allows one to write down valuations on a rational function field over a discretely valued field in a particularly explicit way.  The first talk will be a hands-on introduction to the theory, requiring no background beyond the definition of a discrete valuation (which we will recall).  At the end of the talk, we will fast forward 80 years or so and discuss the relationship between Mac Lane valuations and models of the projective line.

   

  In the second talk give various examples of how the relationship between Mac Lane valuations and models of the projective line can be used to resolve singularities and find regular models of arithmetic surfaces.  In particular, we will overview how Mac Lane valuations can be used to give explicit minimal regular normal crossings models of superelliptic curves (joint with Padmavathi Srinivasan), and also how they can used to understand “stabilization indices” of curves with potentially multiplicative reduction (joint with Daniele Turchetti).

   

• Monday November 11, 2024 at 13:30, Wachman 617

  Sum of Two Squares and Four Squares: A Proof by Modular Forms

  Anthony Pasles, Temple University

  We solve the Sum of Two Squares and Sum of Four Squares Problems using modular forms. In particular, we develop explicit formulas for the number of representations of a positive integer as both a sum of two squares of integers and as a sum of four squares of integers. We introduce the Jacobi theta series (theta), and remark that understanding the Fourier expansions of powers of this function will provide solutions to the problems in question. Then we verify theta satisfies certain functional equations in order to illustrate that theta squared is a modular form of weight 1 and level Gamma (a specific congruence subgroup). We show this space of modular forms has dimension 1, and then use an Eisenstein series living in the same space to determine the Fourier coefficients of theta squared and solve the sum of two squares problem. We employ a similar method with modular forms of weight 2 and level Gamma to solve the sum of four squares problem.

   

• Monday November 18, 2024 at 13:30, Wachman 617

  Introduction to Atkin-Lehner Theory

  Violet Nguyen, Temple University

  In the 1930s, E. Hecke introduced linear operators to study the space of modular forms of level N, which are now fundamental to the theory. He showed that there is a basis of normalized eigenforms with respect to the commuting operators T_n for n coprime to N, and each eigenspace is one-dimensional. An extension of these results would be done by A.O.L. Atkin and J. Lehner in 1970 for the congruence subgroups Gamma_0(N) and all positive integers n. This talk gives a detailed overview of Atkin-Lehner theory for Gamma_1(N), starting from our knowledge of the Hecke operators T_n on modular forms of level Gamma(1) and coprime n.