seminars:anal

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

—-

* **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. In time allows, we will discuss that no improvements of the sharp Weyl formula are possible for the standard sphere and for manifolds with nonpositive curvature (especially for flat n-torus) one can make significant improvements for bounds for the remainder term in the Weyl law.

* **March 8th, Wednesday ** (3:40-5:00pm)

Speaker* ***:
**

*Abstract*:

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

Speaker* ***:
**

*Abstract*:

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

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

*Abstract*:

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

Speaker* ***:
**

*Abstract*:

* **April 5th, 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 12th, Wednesday ** (Spring break)

Speaker* ***:
**

*Abstract*:

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

Speaker* ***:
**

*Abstract*:

* **April 26th, Wednesday ** (3:40-5:00pm)

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

*Abstract*:

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

Speaker* ***:
**

*Abstract*:

seminars/anal.txt · Last modified: 2017/02/27 19:20 by xxu

Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Noncommercial-Share Alike 3.0 Unported