**Problem of the Week**

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**Hilton Memorial Lecture**

seminars:anal:2014_2015

* **January 18th, Wednesday ** (3:30-4:30pm)

Speaker* ***:
**

* **January 25th, Wednesday ** (3:30-5:00pm)

Speaker* ***: Timur Akhunov (Binghamton University)
**

*Abstract*: Spectrum of Laplacian reveals properties of heat, sound, light and atomic properties. Addressing some of these questions motivated Fourier in the 18th to develop harmonic analysis that decomposes signals into distinct frequencies. Fast forward to the 21st century - how does the distribution of frequencies or spectrum is influenced by the curved geometry of space (or space-time). In the series of expository lectures over the course of the semester, several members of the analysis faculty will address these questions. The first lecture will begin with the overview of the Laplace and wave equation in the Euclidean space. It should be broadly accessible.

The lectures are based on the book: **Hangzhou Lectures on Eigenfunctions of the Laplacian, Christopher D. Sogge, (Annals of Mathematics Studies-188), Princeton University Press. 2014**

* **February 1st, Wednesday ** (3:30-4:30pm)

Speaker* ***: Timur Akhunov (Binghamton University)
**

*Abstract*: Spectrum of Laplacian reveals properties of heat, sound, light and atomic properties. Addressing some of these questions motivated Fourier in the 18th to develop harmonic analysis that decomposes signals into distinct frequencies. Fast forward to the 21st century - how does the distribution of frequencies or spectrum is influenced by the curved geometry of space (or space-time). In the series of expository lectures over the course of the semester, several members of the analysis faculty will address these questions. This lecture will overview the fundamental solutions of the wave equation in the Euclidean space. It should be broadly accessible.

The lectures are based on the book: **Hangzhou Lectures on Eigenfunctions of the Laplacian, Christopher D. Sogge, (Annals of Mathematics Studies-188), Princeton University Press. 2014**

* **February 8th, Wednesday ** (3:40-4:40pm)

Speaker* ***: Hyunchul Park (SUNY - New Paltz )
**

*Abstract*: In this talk, we study asymptotic behavior of spectral heat content with respect to symmetric stable processes for arbitrary open sets with finite Lebesgue measure in a real line. Spectral heat content can be interpreted as fractional heat particles that remain in the open sets after short time $t > 0$. We are mainly interested in the relationship between the heat content and the geometry of the domain. Three different behaviors appear depending on the stability indices $\alpha$ of the stable processes and in each case different geometric objects of the domain are discovered in the asymptotic expansion of the corresponding heat content expansion. This is a joint work with R. Song and T. Grzywny.

* **February 15th, Wednesday ** (3:40-5:00pm)

Speaker* ***: Lu Zhang (Binghamton University)
**

*Abstract*: The Laplace operator on Euclidean space can be generalized to Laplace-Beltrami operator on compact manifolds, which is defined as the divergence of the gradient. We will do a brief review of some properties of the Lapace-Beltrami operator such as the related elliptic regularity
estimates. Moreover, we will see for any point in the domain, by choosing proper local coordinate system vanishing at this point, rays through the origin will be geodesics for the metric involved in the Laplace-Beltrami operator.

The lectures are based on the book: **Hangzhou Lectures on Eigenfunctions of the Laplacian, Christopher D. Sogge, (Annals of Mathematics Studies-188), Princeton University Press. 2014**

* **February 22nd, Wednesday ** (3:40-5:00pm)

Speaker* ***: Lu Zhang (Binghamton University)
**

*Abstract*: To study the fundamental solution of the wave operator, We will introduce the Hadamard parametrix, in which the error term can be made arbitrarily smooth. Such construction gives the singularities of the fundamental solution with any desired precision. Also, we will see the use of geodesic normal coordinates in the establishment of a uniqueness theorem for the Cauchy problem.

* **March 1st, Wednesday ** (3:40-5:00pm)

Speaker* ***: Xiangjin Xu (Binghamton University)
**

*Abstract*: This talk is mainly devoted to the proof of the sharp Weyl formula of the spectrum of Laplacian on compact boundaryless Riemannian manifolds.The proof presented uses the Hadamard parametrix. If time allows, we will discuss that no improvements of the sharp Weyl formula are possible for the standard sphere, and one can make significant improvements for bounds for the remainder term in the Weyl law for manifolds with nonpositive curvature (especially for flat n-torus).

* **March 8th, Wednesday ** (Winter break)

Speaker* ***:
**

*Abstract*:

* **March 15th, Wednesday ** (Snow storm)

Speaker* ***:
**

*Abstract*:

* **March 22nd, Wednesday ** (3:40-5:00pm)

Speaker* ***: Gang Zhou (Caltech)
**

*Abstract*: I will present recent results on a non-relativistic Hamiltonian model of quantum friction, about the motion of an invading heavy tracer particle in a Bose gas exhibiting Bose Einstein condensate. We prove the following observations: if the initial speed of the tracer particle is lower than the speed of sound in the Bose gas, then in large time the particle will travel ballistically; if the initial speed is higher than the speed of sound, the it will converge to the speed of sound. In both regimes the system will converge to some inertial states. Joint works with Juerg Froehlich, Michael Sigal, Avy Soffer, Daneil Egli and Arick Shao.

* **March 29th, Wednesday ** (3:40-5:00pm)

Speaker* ***: Adam Weisblatt (Binghamton University)
**

*Abstract*: We will define what it means to be an oscillatory integral and
investigate it's stationary phase properties.

The lectures will partially base on the book: Hangzhou Lectures on Eigenfunctions of the Laplacian, Christopher D. Sogge, (Annals of Mathematics Studies-188), Princeton University Press. 2014

* **April 5th, Wednesday ** (3:40-5:00pm)

Speaker* ***: Lu Zhang (Binghamton University)
**

*Abstract*: We will do a brief introduction to Pseudo-differential operators on Riemannian manifold, as well as some related microlocal analysis. By taking advantage of their properties, one can prove the propagation of singularities for the half wave equation, which involves the square root of Laplace Beltrami, and also a special case of the Egorov's theorem.

The lectures will base on the book: Hangzhou Lectures on Eigenfunctions of the Laplacian, Christopher D. Sogge, (Annals of Mathematics Studies-188), Princeton University Press. 2014

* **April 12th, Wednesday ** (Spring break)

Speaker* ***:
**

*Abstract*:

* **April 19th, Wednesday ** (3:40-5:00pm)

Speaker* ***:Mihai Bailesteanu (Central Connecticut State University)
**

*Abstract*: We discuss an aglorithm to produce Harnack inequalities for various parabolic equations. As an application, we obtain a Harnack inequality for the curve shortening flow and one for the parabolic Allen Cahn equation on a closed n-dimensional manifold.

* **April 26th, Wednesday ** (No seminar talk)

Speaker* ***:
**

*Abstract*:

* **May 3rd, Wednesday ** (3:40-5:00pm)

Speaker* ***: Guozhen Lu (University of Connecticut)
**

*Abstract*: We establish sharp Hardy-Adams inequalities on hyperbolic spaces and
Hardy-Sobolev-Maz'ya inequalities with high order derivatives on half spaces. The Hardy-Sobolev-Maz'ya inequalities follow from sharpened Sobolev inequalities for Paneitz operators on hyperbolic spaces.

* **May 4th,Thursday ** (4:30-5:30pm) (Dean's Lecture in Analysis)

Speaker* ***: Guozhen Lu (University of Connecticut)
**

*Abstract*: Sharp geometric and functional inequalities play an important role in applications to geometry and PDEs. In this talk, we will discuss some important geometric inequalities such as Sobolev inequalities, Hardy inequalities, Hardy-Sobolev inequalities Trudinger-Moser and Adams inequalities, Gagliardo-Nirenberg inequalities and Caffarelli-Kohn-Nirenberg inequalities, etc. We will also brief talk about their applications in geometry and nonlinear PDEs. Some recent results will also be reported.

This talk is intended to be for the general audience.

**September 7th**(3:30-4:30pm)

Speaker**: Lu Zhang (Binghamton University)**

*Topic***: Equivalence of Critical and Subcritical Sharp Trudinger-Moser Inequalities.**

*Abstract*: Trudinger-Moser inequalities describe the limiting case of the Sobolev embeddings. There are two types of such optimal inequalities: critical and subcritical inequalities, both with the best constants. Surprisingly, we are able to show these two types of inequalities are actually equivalent. Moreover, we can provide a precise relationship between their supremums.

**September 14th**(4:40-5:40pm)

Speaker**: Timur M Akhunov (Binghamton University)**

*Topic***: On hypoellipticity of degenerate elliptic operators**

*Abstract*: Solutions of the laplace equation are always smooth in the interior of the domain. This property, called hypoellipticity, is inherited by the solutions of the uniformly elliptic operators. However, if the elliptic operator is degenerate in some directions, would solutions still be smooth? Ellipticity is such a powerful effect, that degeneracy may not be enough to create singular solutions. The type of degeneracy matters and we investigate a large class of indefinitely degenerate operators.

**September 19th, Monday**(4:40-5:40pm)

Speaker**: Lu Zhang (Binghamton University)**

*Topic***: Trudinger-Moser Inequalities with Exact Growth**

*Abstract*: Original Trudinger-Moser inequality on the bounded domain with sharp constant fails on the whole plane. In this case a subcritial inequality holds, or the full Sobolev norm instead of the seminorm is needed to attain a critical inequality. In fact, we can establish a version of critical inequality under the restriction of the seminorm only, where instead we should add a polynomial decay into the inequality.

**September 28th, Wednesday**(4:40-5:30pm)

Speaker**: Timur M Akhunov (Binghamton University)**

*Topic***: When is it possible to have wellposedness of the fully non-linear KdV equation without resorting to weighted spaces?**

*Abstract*: The Korteweg-de Vries equation is a famous model for the propagation of long waves in a shallow canal. In generalization of this model with stronger nonlinear effects a competition between dispersion and anti-diffusion is possible. Solutions to these equations can fail to depend continuously on data unless data has extra decay. In this talk, joint work with David Ambrose and Doug Wright, we investigate a wide class of equations, where this extra decay is not needed.

**October 5th, Wednesday**(4:40-5:30pm)

Speaker**: Adam Weisblatt (Binghamton University)**

*Topic***: Pricing in financial mathematics**

*Abstract*: We will discuss the philosophy and analysis required to price financial derivatives.

**October 12th (Yom Kippur)**

Speaker**:**

*Topic***:**

*Abstract*:

**October 19th, Wednesday**(4:40-5:30pm)

Speaker**: Danyu Zhang (Binghamton University)**

*Topic***: Introduction to Riemannian Geometry**

*Abstract*: I am going to introduce Riemannian metric, connections, geodesics, different curvatures and Jacobi fields, with examples, based on Do Carmo's book Remannian Geometry.

**October 26th, Wednesday**(4:40-5:30pm)

Speaker**: David Renfrew (Binghamton University)**

*Topic***: Eigenvalues of large non-Hermitian random matrices with a variance profile.**

*Abstract*: The eigenvalues of non-Hermitian random matrices with independent, identically distributed entries are governed by the circular law. We consider the eigenvalues of random matrices with independent entries but remove the assumption of identical distributions, allowing entries to have different variances. We describe the eigenvalue density of such matrices under certain assumptions on the graph theoretic properties on the connectivity of the variance profile.

**November 2nd**

Speaker**:**

*Topic***:**

*Abstract*:

**November 9th**

Speaker**: Chenyun Luo (Johns Hopkins University)**

*Topic***: On the motion of a slightly compressible liquid**

*Abstract*: I would like to go over some recent results on the compressible Euler equations with free boundary. We first provide a new apriori energy estimates which are uniform in the sound speed, which leads to the convergence to the solutions of the incompressible Euler equations.This is a joint work with Hans Lindblad.

On the other hand, the energy estimates can be generalized to the compressible water wave problem, i.e., the domain that occupied by the fluid is assumed to be unbounded. Our method requires the detailed analysis of the geometry of the moving boundary.

**November 16th**

Speaker**: Mathew Wolak (Binghamton University)**

*Topic***: Invariant differential operators for the classical Cartan Motion Groups**

*Abstract*: Lie group contraction is a process that ``flattens out'' a Lie group, similar to the process by which a sphere becomes a plane as the radius tends to infinity. The Cartan motion groups are special contractions of semisimple Lie groups. I will present generators for the algebra of bi-invariant differential operators for the Cartan motion groups.

**November 23rd (Thanksgiving)**

Speaker**:**

*Topic***:**

*Abstract*:

**November 30th, Wednesday**(4:40-5:30pm)

Speaker**: Binbin Huang (Binghamton University)**

*Topic***: Introduction to Spectral Geometry via Heat Trace Asymptotic Expansion**

*Abstract*: The study of spectral geometry concerns concerns relationships between geometric structures of manifolds and spectra of canonically defined differential operators, e.g., Laplace–Beltrami operator on a closed Riemannian manifold. It's also closely related to the heat kernel approach for Atiyah-Singer index theorem. The heat trace and its Asymptotic expansion provide an elegant way in this study. We will go over this method from scratch, beginning with the definition of trace-class operators and culminating in a proof of the celebrated Weyl's law.

**December 2nd, Friday**(4:40-5:40pm) (Colloquium)

Speaker**: Ling Xiao (Rutgers)**

*Topic***: Translating Solitons in Euclidean Space.**

*Abstract*:
Mean curvature flow may be regarded as a geometric version of the heat equation. However, in contrast to the classical heat equation, mean curvature flow is described by a quasilinear evolution system of partial
differential equations, and in general the solution only exists on a finite time interval. Therefore, it's very typical that the flow develops singularities.

Translating solitons arise as parabolic rescaling of type II singularities. In this talk, we shall outline a program on the classification of translating solitons. We shall also report on some recent progress we have made in the joint work with Joel Spruck.

**December 5th, Monday**(4:40-5:40pm) (Colloquium)

Speaker**: Gang Zhou (Caltech)**

*Topic***: On Singularity Formation Under Mean Curvature Flow**

*Abstract*:
In this talk I present our recent works, jointly with D.Knopf and I.M.Sigal, on singularity formation under mean curvature flow. By very different techniques, we proved the uniqueness of collapsing cylinder for a generic class of initial surfaces. In the talk some key new elements will be discussed. A few problems, which might be tackled by our techniques, will be formulated.

**December 7th, Wednesday**(4:40-5:40pm) (Colloquium)

Speaker**: Chen Le (University of Kansas)**

*Topic***: Stochastic heat equation: intermittency and densities.**

*Abstract*:
Stochastic heat equation (SHE) with multiplicative noise is an important model. When the diffusion coefficient is linear, this model is also called the parabolic Anderson model, the solution of which traditionally gives the Hopf-Cole solution to the famous KPZ equation. Obtaining various fine properties of its solution will certainly deepen our understanding of these important models. In this talk, I will highlight several interesting properties of SHE and then focus on the probability densities of the solution.

In a recent joint work with Y. Hu and D. Nualart, we establish a necessary and sufficient condition for the existence and regularity of the density of the solution to SHE with measure-valued initial conditions. Under a mild cone condition for the diffusion coefficient, we establish the smooth joint density at multiple points. The tool we use is Malliavin calculus. The main ingredient is to prove that the solutions to a related stochastic partial differential equation have negative moments of all
orders.

**March 9**

Speaker**: Adam Weisblatt (Binghamton University)**

*Topic***: Constructing heat kenels**

*Abstract*: We will carefully examine the properties of heat kernels in euclidean space. Then we construct the most natural manifold where the kernels should exist and be studied.

**April 13**

Speaker**: Binbin Huang (Binghamton University)**

*Topic***: An elementary introduction to spectral sequences and applications in differential geometry (Part 1)**

*Abstract*: The technique of spectral sequences was applied to study the isomorphism between De Rham cohomology and Cech cohomology. This could be thought as supplementary material to Dr. Loya's class this semester on the geometry and analysis on manifolds. Definitions and proof details would be shown.

**April 20**

Speaker**: Lu Zhang (Wayne State University)**

*Topic***: $L^p$ estimates for some pseudo-differential operators.**

*Time***: 3:30pm-4:30pm**

*Abstract*: We study the H\”older's type $L^p$ estimates for a class of pseudo-differential operators in one and bi-parameter setting. Such operators include some trilinear pseudo-differential differential operators with symbols as products of two H\”omander class $BS^0_{1,0}$ functions defined on lower dimensions, and also a bi-parameter bilinear Calder\'on-Vaillancourt theorem, where the symbols are taken from the bi-parameter H\”omander class $BBS^m_{0,0}$.

**April 20**

Speaker**: Pearce Washabaugh (Boulder)**

*Topic***: Model Fluid Mechanics Equations and Universal Teichmüller Spaces**

*Time***: 4:40pm-5:40pm**

*Abstract*: One of the ways of approaching the problems of 3D fluid mechanics is to study simpler lower dimensional model equations that capture some of the key analytic properties of the full 3D situation. The Wunsch equation, a special case of a generalization of the Constantin-Lax-Majda equation, is one such one dimensional model. It, along with the Euler-Weil-Petersson equation, arise as geodesic equations on the Universal Teichmüller Curve and Universal Teichmüller Space respectively. In this talk, I will discuss new results on blowup and global existence for these equations, numerical simulations applying conformal welding to their solutions, and how the Surface Quasi-Geostrophic equation, a two dimensional model for the 3D Euler equation, is a possible higher dimensional version of this picture. This is joint work with Stephen Preston.

**April 27**

Speaker**: Kyle Thompson (Toronto)**

*Topic***: Superconducting Interfaces**

*Abstract*: In this talk we will look for solutions to a two-component system of nonlinear wave equations with the properties that one component has an interface and the other is exponentially small except near the interface of the first component. The second component can be identified with a superconducting current confined to an interface. In order to find solutions of this nature, we will carry out a formal analysis which will suggest that for suitable initial data, the energy of solutions concentrate about a codimension one timelike surface whose dynamics are coupled in a highly nonlinear way to the phase of the superconducting current. We will finish by discussing a recent result confirming the predictions of this formal analysis for solutions with an equivariant symmetry in two dimensions.

**May 4**

Speaker**: Binbin Huang (Binghamton University)**

*Topic***: An elementary introduction to spectral sequences and applications in differential geometry (Part 2)**

*Abstract*: The technique of spectral sequences was applied to study the isomorphism between De Rham cohomology and Cech cohomology. This could be thought as supplementary material to Dr. Loya's class this semester on the geometry and analysis on manifolds. Definitions and proof details would be shown.

**October 7**

Speaker**: Changwei Zhou (Binghamton University)**

*Topic***: Hochschild homology for polynomial algebra in R^n**

*Abstract*: In this talk we will attempt to compute the Hochschild homology of polynomial algebra in R^n using Richard Melrose's elementary approach. The audience is welcome to offer critical, honest opinion whenever they felt it is needed. The speaker welcomes the proof to be debated to foster a laid back atmosphere helping a refined understanding of the subject.

**October 14**

Speaker**: Kunal Sharma (Binghamton University)**

*Topic***: Homology of pseudo differential symbols**

*Abstract*: We will consider the algebra of classical pseudo-differential operators on a compact closed manifold and will compute its homology.

**October 22, 2:50-3:50pm (Special Date, Joint with Geometry and Topology Seminar)**

Speaker**: Jiuyi Zhu (Johns Hopkings University)**

*Topic***: Doubling estimates, vanishing order and nodal sets of Steklov eigenfunctions**

*Abstract*: Recently the study of Steklov eigenfunctions has been attracting much attention. We investigate the qualitative and quantitative properties of Steklov eigenfunctions. We obtain the sharp doubling estimates for Steklov eigenfunctions on the boundary and interior of the manifold using Carleman inequalities. As an application, optimal vanishing order is derived, which describes quantitative behavior of strong unique continuation property. We can ask Yau's type conjecture for the Hausdorff measure of nodal sets of Steklov eigenfunctions. We derive the lower bounds for interior and boundary nodal sets. In two dimensions, we are able to obtain the upper bounds for singular sets and nodal sets. Part of work is joint with Chris Sogge and X. Wang.

**October 28**

Speaker**: Adam Weisblatt (Binghamton University)**

*Topic***: Heat kernel on a manifold**

*Abstract*: We will explicitly construct the heat kernel on a closed manifold using semiclassical pseudodifferential operators.

**November 4**

Speaker**: Adam Weisblatt (Binghamton University)**

*Topic***: Heat kernel on a manifold (continued)**

*Abstract*: We will continue constructing the heat kernel on a closed manifold using semiclassical pseudodifferential operators.

**November 18**

Speaker**: Adam Weisblatt (Binghamton University)**

*Topic***: Heat kernel on a manifold (continued)**

*Abstract*: We continue the construction of the heat kernel on a closed manifold.

**December 2**

Speaker**: Adam Weisblatt (Binghamton University)**

*Topic***: Heat kernel on a manifold (continued)**

*Abstract*: We will extend the heat kernel from Euclidean spaces to closed manifolds.

**February 26 (unusual day & time: Thursday, 4:30pm)**

Speaker**: Niels Martin Moeller (Princeton University)**

*Title***: Gluing of Geometric PDEs - Obstructions vs. Constructions for Minimal Surfaces & Mean Curvature Flow Solitons**

*Abstract*: For geometric nonlinear PDEs, where no easy superposition principle holds, examples of (global, geometrically/topologically interesting) solutions can be hard to come about. In certain situations, for example for 2-surfaces satisfying an equation of mean curvature type, one can generally “fuse” two or more such surfaces satisfying the PDE, as long as certain global obstructions are respected - at the cost (or benefit) of increasing the genus significantly. The key to success in such a gluing procedure is to understand the obstructions from a more local perspective, and to allow sufficiently large geometric deformations to take place. In the talk I will introduce some of the basic ideas and techniques (and pictures) in the gluing of minimal 2-surfaces in a 3-manifold. Then I will explain two recent applications, one to the study of solitons with genus in the singularity theory for mean curvature flow (rigorous construction of Ilmanen's conjectured “planosphere” self-shrinkers), and another to the non-compactness of moduli spaces of finite total curvature minimal surfaces (a problem posed by Ros & Hoffman-Meeks). Some of this work is joint w/ Steve Kleene and/or Nicos Kapouleas.

**November 5**

Speaker**: Yuanzhen Shao (Vanderbilt University)**

*Topic***: Continuous maximal regularity on manifolds with singularities and applications to geometric flows**

*Abstract*: In this talk, we study continuous maximal regularity theory for a class of degenerate or singular differential operators on manifolds with singularities. Based on this theory, we show local existence and uniqueness of solutions for several nonlinear geometric flows and diffusion equations on non-compact, or even incomplete, manifolds, including the Yamabe flow and parabolic p-Laplacian equations. In addition, we also establish regularity properties of solutions by means of a technique consisting of continuous maximal regularity theory, a parameter-dependent diffeomorphism and the implicit function theorem.

**December 3**

Speaker**: Douglas J. Wright (Drexel University)**

*Topic***: Approximation of Polyatomic FPU Lattices by KdV Equations**

*Abstract*: Famously, the Korteweg-de Vries equation serves as a model for the propagation of long waves in Fermi-Pasta-Ulam (FPU) lattices. If one allows the material coefficients in the FPU lattice to vary periodically, the “classical” derivation and justification of the KdV equation go awry. By borrowing ideas from homogenization theory, we can derive and justify an appropriate KdV limit for this problem. This work is joint with Shari Moskow, Jeremy Gaison and Qimin Zhang.

seminars/anal/2014_2015.txt · Last modified: 2017/08/21 16:24 by xxu

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