Abstract: The heat kernel and Green's function are the minimal fundamental solutions of the heat equation and Laplace equation respectively, which play very important roles in many problems in PDEs and geometric analysis. The dependence of the global behavior of the heat kernel and Green's function on the large-scale geometry of $M$ is an interesting and important problem that has been intensively studied during the past few decades by many mathematicians.

In this talk, on a complete Riemannian manifold $(M,g)$ with $Ric(M)\geq 0$, I will discuss my recent work on the sharp estimates of the heat kernel and Green's function, based on the sharp Li-Yau's Harnack inequality, Cheeger-Yau's heat kernel comparison Theorem, and Bishop-Gromov's volume comparison Theorem on such a manifold. We first prove sharp two-side Gaussian bounds for the heat kernel, then we obtain the rigidity of $R^n$ with respect to the asymptotic lower bound of the heat kernel and the sharp gradient estimates on the logarithmic heat kernel. Secondly, on a complete manifold with $Ric(M)\geq 0$ and Euclidean volume growth, we prove the new pointwise lower and upper bounds for the heat kernel by a natural geometric quantity that is characterized by the decay rate of the Bishop–Gromov quantities. As applications of the two side bounds, we obtain the large-scale behavior (asymptotics) of the heat kernel and Green's function on such a manifold.

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Abstract: Differential equations are a pillar in the field of mathematical analysis, revealing complex behaviors in nature, science, and engineering. However, providing analytical solutions for them often presents significant challenges, especially in real-world applications. This is where computational mathematics comes into play. In this talk, we'll explore the realm where rigorous analysis intersects with the practicality of numerical methods, offering a friendly introduction to the numerical analysis of (partial) differential equations. This presentation is crafted to be accessible to a general audience, with no prior knowledge of computational methods necessary.

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Abstract:What can be viewed as a continuation of a talk given last semester, in this talk I will discuss some improved estimates for the behavior of generic singularities for solutions of the equation $\partial_{\tau} u = \partial_y^2 u +u^3$. Studying the dynamics of nonlinear heat equations such as this one have proven very fruitful in studying the singularities of PDEs arising from geometric flows, including mean curvature flow and Ricci flow.

Because of the cubic nonlinearity, solutions to this equation blow up in finite time. In the previous talk, the generic blowup dynamics was discussed in terms of the rescaled solution $v(y,\tau) = \sqrt{T-t} u(x,t)$, where $y:= x / \sqrt{T-t}$, $\tau:= -\ln(T-t)$, and $T$ is the blowup time of $u$. Ideally, we would like to use the dynamics of this rescaled solution to study the original, however, the spacetime region the previous results were considered in is too small to be of much use. We discuss this issue in more detail, along with some techniques used to expand the region of effective control.

Abstract: Topological insulators are novel materials which are insulators in their bulk and conductors along their boundary. They are characterized by a topological invariant which is a continuous map from the space of Hamiltonians to Z or Z_2. Identifying this invariant with a measurable quantity (such as electric conductivity) is then a macroscopic form of quantization. In this talk I will discuss the problem of defining the ambient topology for this space of Hamiltonians so that the invariant is indeed continuous (and hence locally constant), as well as proving that its path components are bijective to the codomain Z or Z_2, in the regime when Hamiltonians have strong disorder.

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Abstract: CMV matrices are the unitary analogues of one-dimensional discrete Schrodinger operators. Depending on the distribution of their coefficients, random CMV matrices exhibit a transition in their eigenvalue distribution from a Poisson process (no eigenvalue correlation) to “picket fence” (strong eigenvalue repulsion). We investigate this transition from the perspective of the joint entropy of the coefficients of the random CMV matrix.

Abstract: I will describe the limiting behavior of the eigenvalues of minors of large bi-unitarily random matrices and the roots of derivatives of polynomials with independent, random coefficients, by giving a convolution semi-group which relates the two processes together.

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Abstract: It was discovered recently that the joint spectrum of linear operators gives rise to an interesting link between self-similar group representations and complex dynamics. This talk will review this link. In particular, we shall see that, in the case of the infinite dihedral group $D_\infty$, the projective joint spectrum coincides with the Julia set of of a rational map on the projective space ${\mathbb P}^2$ derived from the self-similarity. Such connection also seems to exist for some more complicated groups, such as the lamplighter group and the group of intermediate growth.

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Abstract: We first extend the gradient and Laplacian estimates of R. Hamilton for positive solutions of the heat equation on closed manifolds, to bounded positive solutions on complete non-compact manifolds with $Ric(M)\geq -k$ for constant $k\geq 0$. An application of our results, together with the two side Gaussian bounds of our recent work on the heat kernel, yields sharp estimates on the gradient and Laplacian of the heat kernel for complete manifolds with $Ric(M)\geq -k$, which are sharp with the same leading term in the short time asymptotic for all manifolds.

Abstract: I will present the paper “Random currents and continuity of Ising model’s spontaneous magnetization” by M. Aizenman, H. Duminil-Copin and V. Sidoravicius.

In the paper they considered three dimensional antiferromagnetic Ising model. It is known that at the high temperature, the system is at disorder; at the low temperature, the system exhibits ferromagnetic order, or magnetization. They proved that at the critical temperature, the magnetization is continuous, which was a long standing conjecture.

A crucial technique is the so-called switching lemma. It establishes a bijection between undirected graphs generated by the random current representation. In many important papers this was used, including the ones helping Hugo Duminil-Copin to win a Fields medal in 2022.

However this technique does not work for the other spin models, for example, XY model or most of the quantum models.

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Abstract: I will present some recent results about spectral multipliers for $-\Delta+V$, where V is a scalar potential in an optimal or almost optimal class of potentials. The results are used to prove new estimates for some partial differential equations. All results are in three space dimensions. This is joint work with Gong Chen and, separately, Michael Goldberg.

Abstract: The de Moivre-Laplace theorem says that binomial distributions, when correctly rescaled, resemble normal distributions. This is arguably the simplest non-trivial special case of the central limit theorem. Given the fact that the de Moivre-Laplace theorem can be proved by direct computation, it is natural to ask whether the general version of the central limit theorem follows from it. In this talk, I will briefly go over existing proofs of the (Lindeberg-Lévy) central limit theorem to provide a context, and derive the central limit theorem from the de Moivre-Laplace theorem using relatively elementary arguments. In particular, this proof will avoid the use of characteristic functions and Brownian motions.

Abstract: In this talk I will present the papers “Asymptotically Self-similar Blow-up of Semilinear Heat Equations” by Yoshikazu Giga and Robert V. Kohn, and “Refined Asymptotics for the Blowup of $u_t-\Delta u = u^p$” by Stathis Filippas and Robert V. Kohn. The authors study the blowup dynamics of semilinear heat equation $u_t-\Delta u = u^p, p>1$ in $\mathbb{R}^n$ in both papers, with the first paving the way for the second. This nonlinear heat equation has structure similar to many other nonlinear equations, in particular several which arise in geometric analysis, and so often the techniques used to study the dynamics of spacial blow up of this equation may be used to study the blowup of curvature in geometric flows.

Giga and Kohn show that for blowup solutions u which satisfy a weak blowup growth restriction, a version of u rescaled in time and space approaches its steady state solution asymptotically. Fillipas and Kohn then study the long time behavior of the rescaled solution from a dynamical systems point of view: by projecting the equation satisfied by v onto suitably chosen subspaces, one can show that the long time behavior is dominated by the neutral mode, whose dynamics may be obtained exactly.

Some more recent and current work will be briefly discussed at the end.

Abstract: In this talk I will present a progress we made on the quantum measurement. After a measurement on the quantum system, it will collapse into the observed state. There are two postulates for this, one was made by Von Neumann for the density matrices, the other one was made by Luder for the wave function. Based on a model proposed by Gisin, we prove the equivalence between these two.

This is a joint work with Juerg Froehlich.

This is a new direction for me. I will try to answer the questions.

Abstract: The Kaczmarz algorithm is a classical iterative numerical method for solving large and overdetermined linear systems. It has received increasing attention over the last decade, starting with a proof in 2009 of a convergence rate that applies to general matrices, for a variation of the algorithm known as the randomized Kaczmarz algorithm. This talk will outline extensions of the algorithm to deal with systems perturbed by noise, with applications to machine learning and medical imaging.

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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.

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.)

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., $b_{n}=o(n)$.

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.

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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).

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.

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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.

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.

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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.

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.

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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).

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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.

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.

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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.

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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.

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.

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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.

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.

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.

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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).

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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.

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Abstract: I will present a method to compute various cohomologies of surfaces.

Abstract: I will present a method to compute various cohomologies of surfaces.

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.

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.

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.

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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.

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.

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.

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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.

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.

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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).

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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.

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.

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.

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.

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.

Abstract: In this talk I shall present some new Strichartz-type estimates for wave and Klein–Gordon equations, with a few sample applications.

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.

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.

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 $≥ 0$.

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.

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.

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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 innite 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.

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.

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

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

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.

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

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

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

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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.

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

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

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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.

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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.

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.