Problem of the Week
BUGCAT
Zassenhaus Conference
Hilton Memorial Lecture
BingAWM
Math Club
* January 22nd, Wednesday (4:00-5:00pm)
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* January 29th, Wednesday (4:00-5:00pm)
Speaker : organizational meeting
Topic: organizational meeting
Abstract: organizational meeting
* February 5th, Wednesday (4:00-5:00pm)
Speaker : David Renfrew (Binghamton)
Topic: The circular law
Abstract: I will discuss the eigenvalues of random matrices with i.i.d. entries and show they converge to the uniform measure on the unit disk.
* February 12th, Wednesday (3:30-4:30pm)
Speaker : Guillaume Dubach (CUNY Baruch)
Topic: Words and surfaces
Abstract: Words of random matrices with i.i.d. complex Gaussian entries (a.k.a. complex Ginibre matrices) can be studied using a topological expansion formula, or genus expansion. This results in a generalization of well-known properties of products of i.i.d. complex Ginibre matrices on the one hand, and powers of one Ginibre matrix on the other hand. For instance, the empirical distribution of singular values of any word is shown to converge to a Fuss-Catalan distribution whose parameter only depends on the length of the word. (Joint work with Yuval Peled.)
* February 19th , Wednesday (3:40-4:40pm)
Speaker: Indrajit Jana (Temple University)
Topic: CLT for non-Hermitian random band matrices with variance profiles.
Abstract: We show that the fluctuations of the linear eigenvalue statistics of a non-Hermitian random band matrix of bandwidth $b_{n}$ with a continuous variance profile converges to a Gaussian distribution. We obtain an explicit formula for the variance of the limiting Gaussian distribution, which depends on the test function, and as well as the growth rate of the bandwidth $b_{n}$. In particular, if the band matrix is a full matrix i.e., $b_{n}=n$, the formula is consistent with Rider, and Silverstein (2006). We also compute an explicit formula for the limiting variance even if the bandwidth $b_{n}$ grows at a slower rate compared to $n$ i.e., $b_{n}=o(n)$.
* February 26th , Wednesday (4:00-5:00pm)
Speaker: Liming Sun (John Hopkins)
Topic: Some convexity theorems of translating solitons in the mean curvature flow
Abstract:I will be talking about the translating solitons (translators) in the mean curvature flow. Convexity theorems of translators play fundamental roles in the classification of them. Spruck and Xiao proved any two dimensional mean convex translator is actually convex. Spruck and I proved a similar convex theorem for higher dimensional translators, namely the 2-convex translating solitons are actually convex. Our theorem implies 2-convex translating solitons have to be the bowl soliton. Our second theorem regards the solutions of the Dirichlet problem for translators in a bounded convex domain . We proved the solutions will be convex under appropriate conditions. This theorem implies the existence of n-2 family of locally strictly convex translators in higher dimension. In the end, we will show that our method could be used to establish a convexity theorem for constant mean curvature graph equation.
* March 4th, Wednesday (4:00-5:00pm) (Winter Break)
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* March 11th, Wednesday (4:00-5:00pm)
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* March 18th, Wednesday (4:00-5:00pm)
Speaker: Gang Zhou
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* March 25th, Wednesday (4:00-5:00pm)
Speaker: Xiangjin Xu
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* April 1st, Wednesday (4:00-5:00pm)
Speaker: Paul Barber
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* April 8th, Wednesday (4:00-5:00pm) (Spring break)
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* April 15th, Wednesday (4:00-5:00pm)
Speaker: Xiangjin's visitor(?)
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* April 22nd, Wednesday (4:00-5:00pm)
Speaker: Gang's visitor
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* April 29th, Wednesday (4:00-5:00pm)
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* May 6th, Wednesday (4:00-5:00pm)
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W
* August 28th, Wednesday (3:30-4:30pm)
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Topic: organizational meeting
* September 4th, Wednesday (3:30-5:00pm)(No talk due Monday schedule)
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* September 5th, Thursday (WH 309, 2:30-4:30pm)(Special time and location)
Speaker : David Cervantes-Nava (Binghamton University)
Topic: Admissions to Candidacy Exam
Abstract: TBD
* September 11th, Wednesday (3:30-4:55pm)
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* September 18th, Wednesday (4:00-5:00pm)
Speaker : Xiangjin Xu (Binghamton University)
Topic: Characterization of Carleson measures on compact manifolds with boundary
Abstract: On the subspaces of $L^2(M)$ generated by eigenfunctions of eigenvalues less than $L(>1)$ associated to the Dirichlet (Neumann) Laplace–Beltrami operator on a compact Riemannian manifold $(M,g)$ with boundary, we discuss some positive and negative results on the characterization of the Carleson measures and the Logvinenko–Sereda sets for Dirichlet (or Neumann) Laplacian on $M$, which generalized the corresponding results of J. Ortega-Cerda and B. Pridhnani on a compact boundaryless manifold (Forum Math.25 (2013), DOI 10.1515/FORM.2011.110).
* September 25th , Wednesday (3:30-4:55pm)
Speaker: Gang Zhou (Binghamton University)
Topic: The dynamics of effective equation of polaron
Abstract: Polaron theory is a model of an electron in a crystal lattice. It is studied in the framework of nonequilibrium statistic mechanics, and it has a lot of applications. In the recent year, jointly with Rupert Frank, we studied the quantum and classical models and obtained different results. Still there are open problems. In this talk I present the results for the dynamics of classical model.
* October 2nd, Wednesday (3:30-4:55pm)
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* October 9th, Wednesday (3:30-4:55pm)(Holiday, Yom Kippur)
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* October 17th, Thursday (1:00-2:00pm, WH 309) Note the special time and location
Speaker: Yuan Yuan, Syracuse University
Topic: Bergman projection on pseudoconvex domains
Abstract: Bergman projection plays important roles in function theory and d-bar Neumann problem on pseudoconvex domains. After giving a brief introduction to the general theory, I will focus on the boundedness of the Bergman projection in L^p spaces. This talk is based on joint work with Chen and Krantz.
* October 23rd, Wednesday (4:00-5:00pm)
Speaker: Adam Weisblatt (Binghamton University)
Topic: The wraparound universe
Abstract: Cosmologists have been trying to determine the shape of the universe. Although most of the evidence says the universe is flat, it need not imply the universe looks like $R^3$. In this talk we discuss the most plausible candidates for the shape of the universe and how to go about detecting such models. Much of the studies into the shape of the the universe has been topological. I will present some new results on how to do analysis on them.
* October 30th, Wednesday (3:30-4:55pm)
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* November 6th, Wednesday (4:00-5:00pm)
Speaker: Alexis Drouot, Columbia University
Topic: Transport at interfaces of topological insulators
Abstract: In this talk, I consider a PDE modeling interface effects between insulators: a Schrodinger equation with periodic asymptotics (the bulk), away from a strip (the interface). I will state the bulk-edge correspondence. This theorem predicts that the interface between two topologically distinct insulators always conducts energy. I will illustrate it in the context of graphene; explain applications to robust waveguides; and provide dynamical interpretations.
* November 13th, Wednesday (4:00-5:00pm)
Speaker: Steven Gindi (Binghamton University)
Topic: Long Time Limits of Generalized Ricci Flow
Abstract: We derive modified Perelman-type monotonicity formulas for solutions to the generalized Ricci flow equation with symmetry on principal bundles. This leads to rigidity and classification results for nonsingular solutions.
* November 20th, Wednesday (3:30-4:55pm)
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* November 27th, Wednesday (3:30-4:55pm)(Thanksgiving break)
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* December 6th, Friday (2:00-3:00pm) (Special date and time)
Speaker: Cheng Zhang (University of Rochester)
Topic: Eigenfunction estimates of the fractional Laplacian on a bounded domain
Abstract: We will introduce the eigenvalue problem of the Dirichlet fractional Laplacian on a bounded domain in $R^n$. We obtained new interior $L^p$ estimates for the eigenfunctions by using latest results on sharp resolvent estimates, heat kernels, and commutator estimates. This is a joint work with Xiaoqi Huang and Yannick Sire (arXiv:1907.08107).
* January 23rd, Wednesday (3:30-4:30pm)
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Topic: organizational meeting
* January 30th, Wednesday (3:30-5:00pm)(No talk due Monday schedule)
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* February 6th, Wednesday (3:30-4:55pm)
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* February 13th, Wednesday (3:30-4:55pm)
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* February 20th , Wednesday (3:30-4:55pm)
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* February 27th , Wednesday (3:30-4:55pm)
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* March 6th, Wednesday (3:30-4:55pm)
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* March 13th, Wednesday (3:30-4:55pm)
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* March 20th, Wednesday (3:30-4:55pm)
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* March 27th, Wednesday (3:30-4:55pm)
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* April 3rd, Wednesday (3:30-4:55pm)
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* April 10th, Wednesday (3:30-4:55pm)
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* April 17th, Wednesday (3:30-4:55pm)
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* April 24th, Wednesday (3:30-4:55pm)
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* May 1st, Wednesday (3:30-4:55pm)
Speaker: Lu Wang (University of Wisconsin & IAS)
Topic: Mean Curvature Flow Expanders of Low Entropy
Abstract: Colding and Minicozzi introduced a notion of entropy of a hypersurface, which is defined by the supremum over all Gaussian integrals with varying centers and scales. In this talk, we will discuss the properties of self-expanding solutions of mean curvature flow with small entropy. This is joint work with Jacob Bernstein.
* May 8th, Wednesday (3:30-4:55pm)
Speaker:John Ma (Rutgers University)
Topic: Uniqueness Theorem for non-compact Mean Curvature Flow with possibly unbounded curvatures
Abstract: We discuss uniqueness for mean curvature flow of non-compact manifolds. We use an energy argument to prove a uniqueness theorem for mean curvature flow with possibly unbounded curvatures. These generalize the results in Chen and Yin (CAG, 07). This is a joint work with Man-Chun Lee.
* May 15th, Wednesday (3:30-4:55pm)
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* August 29th, Wednesday (3:30-4:30pm)
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Topic: organizational meeting
* September 5th, Wednesday (3:30-5:00pm)(No talk due Monday schedule)
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Topic: No talk
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* September 12th, Wednesday (3:30-4:55pm)
Speaker : Gang Zhou (Binghamton University)
Topic: A description of generic singularities formed by mean curvature flow
Abstract: In this talk I will present the progresses my collaborators, including Michael Sigal and Dan Knopf, and I made in the past few years. We developed a new way of studying mean curvature flow, and I am trying to use it to understand the evolution of hypersurfaces under mean curvature flow.
* September 19th, Wednesday (3:30-4:55pm)(Holiday, Yom Kippur)
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* September 26th , Wednesday (3:30-4:55pm)
Speaker: Adam Weisblatt (Binghamton University)
Topic: The heat equation on planar diagrams.
Abstract: The heat kernel on a surface helps to describe its geometry. However, solving the heat equation explicitly and extracting the geometric information can be difficulty. In this talk, I will offer a new approach to the heat equation using planar diagrams. The heat kernel constructed will not be the authentic heat kernel for the surface, but we will show how it captures geometry.
* October 3rd, Wednesday (3:30-4:55pm)
Speaker: Brian Allen (West Point)
Topic: Stability Questions and Convergence of Riemannian Manifolds
Abstract: We will start by surveying the stability of the scalar torus rigidity theorem, a result about the impact of geometry on topology, and the stability of the positive mass theorem, an important theorem in mathematical relativity. Since stability requires a notion of closeness this will lead us naturally to consider various notions of distance between and convergence of Riemannian manifolds. We will end by discussing theorems and important examples which aim at contrasting these notions of convergence which have been, and will continue to be, applied to stability problems.
* October 10th, Wednesday (3:30-4:55pm)
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* October 17th, Wednesday (3:30-4:55pm)
Speaker: Shengwen Wang (Binghamton University)
Topic: Mean curvature flow with surgery and applications
Abstract: I will first review about mean curvature flow with surgery for 2-convex hypersurfaces. Then I will report on joint work with Mramor for mean curvature flow with surgery for low entropy mean-convex hypersurfaces and an application to the classification of self-shrinkers. I will also discuss what elements we still lack to do surgery for generic mean curvature flow.
* October 24th, Wednesday (3:30-4:55pm)
Speaker: Lu Zhang (Binghamton University)
Topic: Some useful methods for Fourier multipliers
Abstract: I will give a introduction of some methods that have been recently used to study the Lp bounds for the multi-parameter Fourier multipliers, which include one method that was applied in my recent work.
* October 31st, Wednesday (3:30-4:55pm)
Speaker: Xiangjin Xu (Binghamton University )
Topic: New heat kernel estimates on manifolds with negative Ricci curvature
Abstract: In this talk, we first introduce some new sharp Li–Yau type gradient estimates, both in local and global version, for the positive solution $u(x,t)$ of the heat equations $$\partial_t u-\Delta u=0$$ on a complete manifold with $Ric(M)\geq -k$. As applications, some new parabolic Harnack inequalities, both in local and global version, are derived. Based on the new parabolic Harnack inequalities, some new sharp Gaussian type lower bound and upper bound of the heat kernel on a complete manifold with $Ric(M)\geq -k$ are proved, which are new even for manifold $M$ with nonnegative Ricci curvature, $Ric(M)\geq 0$. An upper bound of $\mu_1 (M) \geq 0$, the greatest lower bound of the $L^2$-spectrum of the Laplacian on a complete noncompact manifold $M$, is achieved. At the end, we discuss some open questions related to the sharp Li–Yau type estimates.
* November 7th, Wednesday (3:30-4:55pm)
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* November 14th, Wednesday (3:30-4:55pm)
Speaker: Phil Sosoe, Cornell University
Topic: Applications of CLTs and homogenization for Dyson Brownian Motion
to Random Matrix Theory
Abstract: I will explain how two recent technical developments in Random Matrix Theory allow for a precise description of the fluctuations of single eigenvalues in the spectrum of large symmetric matrices. No prior knowledge of random matrix theory will be assumed.
(Based on joint work with B Landon and HT Yau).
* November 21st, Wednesday (3:30-4:55pm)(Thanksgiving break)
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* November 28th, Wednesday (3:30-4:55pm)
Speaker: Martin Fraas, Virginia Tech
Topic: Perturbation Theory of Quantum Trajectories
Abstract: Quantum trajectories are certain Markov processes on a complex projective space. They describe the evolution of a quantum system subject to a repeated indirect measurement. For a given set of matrices $A$ and a unit vector $x$, a probability of a sequence of matrices $V_1, V_2, \dots , V_n$, $Vj \in A$ is proportional to $||V_n \dots V_1x||^2$. The Markov process is given by $x_n \sim V_n \dots V_1 x$. In this talk, I will review the basic properties of this process, in particular, conditions that guarantee the uniqueness of the stationary measure. Then I will discuss how the measure and the process change if the underlying set of matrices A changes.
* December 5th, Wednesday (3:30-4:55pm)
Speaker: Kunal Sharma (Binghamton University)
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* January 17th, Wednesday (3:30-4:30pm)
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Topic: organizational meeting
* January 24th, Wednesday (3:30-5:00pm)
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Topic: No talk
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* January 31st, Wednesday (3:30-4:30pm)
Speaker : Adam Weisblatt (Binghamton University)
Topic: Computation of Cohomology
Abstract: I will present a method to compute various cohomologies of
surfaces.
* February 7th, Wednesday (3:30-4:30pm)(Cancelled due weather)
Speaker : Adam Weisblatt (Binghamton University)
Topic: Computation of Cohomology (continue)
Abstract: I will present a method to compute various cohomologies of
surfaces.
* February 14th , Wednesday (3:30-4:30pm)(Cancelled)
Speaker: Kunal Sharma (Binghamton University)
Topic: Some remarks on Calderon-Seeley projector
Abstract: We will show how Calderon-Seeley projector comes up in study of boundary values problems for elliptic operators on a compact manifold with boundary. Its properties and applications to address Fredholmness of the operator will be discussed.
* February 21st, Wednesday (3:30-4:30pm)
Speaker: Binbin Huang (Binghamton University)
Topic: Some Geometric Constructions on Manifolds with Corners
Abstract: Manifolds with corners are of little new interest for pure topologists - they are just the manifolds with boundaries. For differential geometers, there are a few intriguing phenomena to study. On the other hand, they are (at least philosophically) unavoidable for analysts who study linear differential operators. In this talk, we will look at some fundamental notions in the theory of manifolds with corners. Some geometric constructions closely related to linear differential operators will be discussed, paving the way to the study of various (pseudo-)differential calculi.
* February 28th, Wednesday (3:30-4:30pm)
Speaker: Binbin Huang (Binghamton University)
Topic: Some Geometric Constructions on Manifolds with Corners (Continue)
Abstract: Manifolds with corners are of little new interest for pure topologists - they are just the manifolds with boundaries. For differential geometers, there are a few intriguing phenomena to study. On the other hand, they are (at least philosophically) unavoidable for analysts who study linear differential operators. In this talk, we will look at some fundamental notions in the theory of manifolds with corners. Some geometric constructions closely related to linear differential operators will be discussed, paving the way to the study of various (pseudo-)differential calculi.
* March 7th, Wednesday (3:30-4:30pm)(Winter break)
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* March 14th, Wednesday (3:30-4:30pm)
Speaker: Timur Akhunov (Binghamton University)
Topic: Changing dispersion for KdV
Abstract: Dispersive partial equations describe evolution of waves, whose speed of propagation depends on wave frequency. The uncertainty principle of quantum mechanics is intimately tied to the dispersion in the Schrodinger equation. The Korteweg-de Vries (KdV) equation was originally derived in 1890s to explain surface waves in a shallow fluid is among the most studied nonlinear dispersive PDE. Dispersion has since then found a way to connect with harmonic analysis, number theory and algebraic geometry. In a series of papers (the last in collaboration with David Ambrose and Doug Wright from Drexel) we have independently rediscovered and adapted techniques from thin-film equations to the context of KdV.
* March 21th, Wednesday (3:30-4:30pm)
Speaker: Shengwen Wang (John Hopkins University)
Topic: Hausdorff stability of round spheres under small-entropy perturbation
Abstract: The Colding-Minicozzi entropy functional is defined on the space of all hypersurfaces and it measures the complexity of a hypersurface. It is monotonic non-increasing along mean curvature flow and the entropy minimizer among all closed hypersurfaces are round spheres. In this talk I will present a Hausdorff stability result of round spheres under small entropy perturbation.
* March 28th, Wednesday (3:30-4:30pm)
Speaker: Binbin Huang (Binghamton University)
Topic: On an extension of the b-calculus
Abstract: The b-calculus developed by R. Melrose, is among the first materializations of his program of “microlocalizing boundary fibration structures”. Along with other closed related calculi, it provides a convenient framework to study geometric-analytic problems on manifolds with certain singular structures. Due to its nice mapping properties on (b-)Sobolev spaces, techniques from functional analysis can be applied, which makes it a natural choice for the study of index theory. With a more geometric approach initiated by P. Loya, we developed a theory that extends the classical b-calculus. It is obtained by replacing the boundary decay condition by a more modest one. In this talk, we will begin with a brief review of the b-calculus, then we will give a detailed description of our calculus, and study its Fredholm problem.
* April 4th, Wednesday (3:30-4:30pm)(Spring break)
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* April 11th, Wednesday (3:30-4:30pm)
Speaker: Kunal Sharma (Binghamton University)
Topic: Some remarks on Calderon-Seeley projector
Abstract: We will show how Calderon-Seeley projector comes up in study of boundary values problems for elliptic operators on a compact manifold with boundary. Its properties and applications to address Fredholmness of the operator will be discussed.
* April 18th, Wednesday (3:30-4:30pm)
Speaker:Benjamin Harrop-Griffiths (NYU)
Topic: Degenerate dispersive equations
Abstract: We discuss recent work on some quasilinear toy models for the phenomenon of degenerate dispersion, where the dispersion relation may degenerate at a point in physical space. In particular, we present a proof of the existence of solutions using a novel change of variables reminiscent of the classical hodograph transformation. This is joint work with Pierre Germain and Jeremy L. Marzuola.
* April 24th, Tuesday (2:50-4:10pm at WH 309) (Special date, time and location)
Speaker: Binbin Huang (Binghamton University)
Topic: Thesis Defense
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* April 25th, Wednesday (3:30-4:30pm)
Speaker: Marius Lemm (Institute for advanced studies, Princeton)
Topic: On the averaged Green's function for an elliptic equation with random coefficients
Abstract: We consider an elliptic operator on the discrete d-dimensional lattice whose coefficient matrix is a small i.i.d. perturbation of the identity. Recently, Jean Bourgain introduced novel techniques from harmonic analysis to prove the convergence of the Feshbach-Schur perturbation series for the averaged Green's function of this model. Our main contribution is a refinement of Bourgain's approach which yields a conjecturally nearly optimal decay estimate. As an application, we derive estimates on higher derivatives of the averaged Green's function which go beyond the second derivatives considered by Delmotte-Deuschel and related works. This is joint work with Jongchon Kim (IAS).
* May 2nd, Wednesday (3:30-5:00pm)
Speaker: Adam Weisblatt (Binghamton University)
Topic: Thesis Defense
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* August 23th, Wednesday (3:30-4:30pm)
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Topic: organizational meeting
* August 30th, Wednesday (3:30-5:00pm)
Speaker : Adam Weisblatt (Binghamton University)
Topic: The dirichlet problem on manifolds with corners.
Abstract: It is well known that the dirichlet problem in R^2 has a solution
when the boundary of the region is smooth. We will use geometric
techniques to construct an integral operator which gives the solution of
the dirichlet problem when the boundary has corners.
* September 6th, Wednesday (3:30-4:30pm)
Speaker : Adam Weisblatt (Binghamton University)
Topic: The dirichlet problem on manifolds with corners.(continued)
Abstract: It is well known that the dirichlet problem in R^2 has a solution
when the boundary of the region is smooth. We will use geometric
techniques to construct an integral operator which gives the solution of
the dirichlet problem when the boundary has corners.
* September 13th, Wednesday (3:30-4:30pm)
Speaker : Gang Zhou (Binghamton University)
Topic: Derivation of an effective evolution equation for a strongly coupled polaron
Abstract: Polaron theory is a model of an electron in a crystal lattice. It is in the framework of nonequilibrium statistic mechanics, which becomes important in recent year because people can conduct better experiments. There are two different mathematical models for polaron: H. Frohlich proposed a quantum model in 1937; L. Landau and S. I. Pekar proposed a system of nonlinear PDEs in 1948. In a joint work of Rupert Frank, we proved that these two models are equivalent to certain orders.
* September 20th, Wednesday (3:30-4:30pm) (Rosh Hashanah)
Speaker: Adam Weisblatt (Binghamton University)
Topic: The dirichlet problem on manifolds with corners.(continued)
Abstract: It is well known that the dirichlet problem in R^2 has a solution
when the boundary of the region is smooth. We will use geometric
techniques to construct an integral operator which gives the solution of
the dirichlet problem when the boundary has corners.
* September 27th, Wednesday (3:30-4:30pm)
Speaker: Gang Zhou (Binghamton University)
Topic: On the evolution of surfaces under mean curvature flow
Abstract: In this talk I will present the progresses made in the past few years on the evolution of surfaces under mean curvature flow. Our contributions were to prove the uniqueness of limit cylinder at the time of blowup and to unify different approaches by different parties, and to address some open problems, especially in four dimensional manifolds. These were made possible by applying different techniques learned from theoretical physics and mathematical physics. Joint works of Dan Knopf and Michael Sigal.
* October 4th, Wednesday (3:30-4:30pm)
Speaker: Marius Beceanu (Albany University -SUNY)
Topic: Strichartz estimates for the wave and Klein–Gordon equations
Abstract: In this talk I shall present some new Strichartz-type estimates for wave and Klein–Gordon equations, with a few sample applications.
* October 11th, Wednesday (3:30-4:30pm)
Speaker: Lu Zhang (Binghamton University)
Topic: Multi-parameter singular Radon transforms
Abstract: I will give a brief introduction to a type of the multi-parameter singular Randon transforms. Such type of operators was originally studied by Christ, Nagel, Stein and Wainger. The theory was
extended to the cases involving product kernels and general multi-parameter setting by B. Street and Stein.
* October 18th, Wednesday (3:30-4:30pm)
Speaker: Philippe Sosoe (Cornell University)
Topic: A sharp quasi-invariance result for the quartic NLS equation with Gaussian initial data
Abstract: I will discuss a recent result, with T. Oh and N. Tzvetkov, proving that the distribution of the solution of a dispersive equation on the circle with random initial data according to some Gaussian measure remains regular at positive times. This result is optimal in two senses which will be clarified in the talk.
* October 25th, Wednesday (3:30-4:30pm)
Speaker: Yakun Xi (University of Rochester)
Topic: Geodesic period integrals of eigenfunctions on Riemannian surfaces.
Abstract: We use the Gauss-Bonnet theorem and the triangle comparison theorems of Rauch and Toponogov to show that on compact Riemannian surfaces of negative curvature, period integrals of eigenfunctions over geodesics go to zero at the rate of $O((\log \lambda)^{(-1/2)})$ if $\lambda$ are their frequencies. No such result is possible in the constant curvature case if the curvature is $≥ 0$.
* November 1st, Wednesday (3:30-4:30pm)
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Topic: the Analysis Caucus Meeting for next year's teaching
Abstract: To discuss and make recommendations on what analysis courses, numbered above Math 330 for both graduate and undergraduate, should be offered in 2018-19, and who should (or would like to) teach them.
* November 8th, Wednesday (3:30-4:30pm)
Speaker: Zhenfu Wang (University of Pennsylvania)
Topic: Quantitative estimates of propagation of chaos for stochastic particle systems
Abstract: We derive quantitative estimates proving the propagation of chaos for large stochastic systems of interacting particles. We obtain explicit bounds on the relative entropy between the joint law of the particles and the tensorized law at the limit. We have to develop for this new laws of large numbers at the exponential scale. But our result only requires very weak regularity on the interaction kernel in the negative Sobolev space $W^{-1, \infty}$ , thus including the Biot-Savart law and the point vortices dynamics for the 2d incompressible Navier-Stokes.
* November 15th, Wednesday (3:30-4:30pm)
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* November 22rd, Wednesday (Thanksgiving)
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* November 29th, Wednesday (3:30-4:30pm)
Speaker: Rongwei Yang (SUNY Albany)
Topic: Projective Spectrum and Finitely Generated Groups
Abstract: For a tuple $A= (A_1; A_2;\ldots.. A_n)$ of elements in a unital Banach algebra $B$, its projective spectrum $P(A)$ is the collection of $z\in \mathbb{C}^n$ such that the multiparameter pencil $A(z) =z_1 A_1+z_2 A_2+\ldots+z_n A_n$ is not invertible. If $(\rho;H)$ is a unitary representation of a finitely generated group $G=< g_1;g_2;;g_{nj}>$ and $Ai=(g_i);i= 1,2,\ldots,n;$ then $P(A)$ reflects the structure of $G$ as well as the property of $\rho$. In this talk we will see how projective spectrum characterizes amenability, Haagerup's property and Kazhdan's property (T) of the groups. Projective spectrum can be computed explicitly for some groups. We will have an in-depth look at the case of the innite dihedral group $D1$, and will indicate a connection with group of intermediate growth.
A big part of this talk is joint work with R. Grigorchuk.
* December 6th, Wednesday (3:30-4:30pm)(Cancelled)
Speaker: Kunal Sharma (Binghamton University)
Topic: Some remarks on Calderon-Seeley projector
Abstract: We will show how Calderon-Seeley projector comes up in study of boundary
values problems for elliptic operators on a compact manifold with boundary. Its properties and applications to address Fredholmness of the operator will be discussed.
* January 18th, Wednesday (3:30-4:30pm)
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Topic: organizational meeting
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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 (No seminar talk)
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* May 3rd, Wednesday (3:40-5:00pm)
Speaker : Guozhen Lu (University of Connecticut)
Topic: Hardy-Adams inequalities on hyperbolic spaces and Hardy-Sobolev-Maz'ya inequalities on half spaces
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)
Topic: Sharp geometric and functional inequalities and applications to geometry and PDEs
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.
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.
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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.
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.