seminars:anal

## The Analysis Seminar

The seminar meets Wednesdays in WH-100E at 4:30 p.m. There are refreshments and snacks in WH-102 at 3:15.

The seminar is partly funded as one of Dean's Speaker Series in Harpur College (College of Arts and Sciences) at Binghamton University.

Organizers: Paul Loya, Peng Shao, Xiangjin Xu, and Lu Zhang

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### Spring 2017

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

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

Speaker : Timur Akhunov (Binghamton University)
Topic: Spectrum of Laplacian. Part 1

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)
Topic: Spectrum of Laplacian. Part 2 - Fundamental solutions of the d’Alembertian

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 )
Topic: Spectral heat content for symmetric stable processes for general open sets in $\mathbb{R}^1$

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)
Topic: Spectrum of Laplacian. Part 3 - Laplace-Beltrami Operator and Geodesics

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)
Topic: Spectrum of Laplacian. Part 4 - The Hadamard Parametrix

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.

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

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

Speaker : Xiangjin Xu (Binghamton University)
Topic: Spectrum of Laplacian. Part 5 - the sharp Weyl formula

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).

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

* March 8th, Wednesday (Winter break)

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* March 15th, Wednesday (Snow storm)

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

Speaker : Gang Zhou (Caltech)
Topic: motion of an invading heavy tracer particle in a Bose gas

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)
Topic: Spectrum of Laplacian. Part 6 - Introduction to oscillatory integrals

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)
Topic: Spectrum of Laplacian. Part 7 - Pseudo-differential operators and microlocal analysis

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)

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

Speaker :Mihai Bailesteanu (Central Connecticut State University)
Topic: Harnack inequalities for parabolic equations

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 (3:40-5:00pm)

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

Speaker : Guozhen Lu (University of Connecticut)
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* May 4th,Thursday (4:30-5:30pm) (Dean's Lecture in Analysis)

Speaker : Guozhen Lu (University of Connecticut)
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