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2019Äê1ÔÂ14ÈÕ-1ÔÂ19ÈÕ

Part I: Jan.14-17, "Tianyuan Advanced Seminar on Geometry and Analysis"

Introduction£º There will be three mini-courses during this winter school.Graduate students and
advanced undergraduate students are very welcome. The speakers and course  titles/abstracts are
as follows.

Course Title£º Asymptotic Analysis for Yamabe Equations
Speaker£º Prof.Han, Qing (University of Notre Dame)
Abstract£º The Yamabe equation is an important class of equations in the geometric analysis and
nonlinear elliptic  equations. It arises  from the conformal  geometry and  describes the scalar
curvatures of the conformal metrics. Many  famous mathematicians  made significant contributions
in this field. In this course, we studyasymptotic behaviors of solutions of the Yamabe  equation
with isolated singularities and discuss someimportant results by  Caffarelli, Schoen, and Spruck,
and  by Korevaar,  Mazzeo, Pacard, and  Schoen.  We  will also present  some recent results that
solutions  near isolated  singularities  are well approximated by  series that decay at discreet
orders.

Course Title£º Extreme eigenvalue problems on Riemannian manifolds
Speaker£º Prof.Lu, Zhiqin (University of California at Irvine)
Abstract£º We shall first discuss basic properties of eigenvalues of compact Riemannian manifolds.
Then we will study the Riemannian manifolds or metric  spaces whose k-th eigenvalue reach maximum
within certain metric classes, from the classic results of Hersch to more recent developments.

Course Title£º Frequency and Estimates of Nodal Set of Harmonic Function
Speaker£º Prof.Zhang, Zhenlei (Capital Normal University)
Abstract£º It  has  been known for  decades  that the  frequency  relates closely to the nodal set
estimate. In this course, we  first  recall the  basic  properties  of the  frequency of  harmonic
functions, then we introduce Logunov's work on lower bound of nodal set estimate.

Winter School Schedule£º

 Time Speaker Title Jan.14 08:30£­10:20 Han, Qing Asymptotic Analysis for Yamabe Equations 10:40£­12:00 Lu, Zhiqin Extreme eigenvalue problems on Riemannian manifolds 14:10£­15:30 Han, Qing Asymptotic Analysis for Yamabe Equations 16:00£­17:20 Zhang, Zhenlei Frequency and Estimates of Nodal Set of Harmonic Function Jan.15 08:30£­10:20 Han, Qing Asymptotic Analysis for Yamabe Equations 10:40£­12:00 Lu, Zhiqin Extreme eigenvalue problems on Riemannian manifolds 14:10£­15:30 Han, Qing Asymptotic Analysis for Yamabe Equations 16:00£­17:20 Zhang, Zhenlei Frequency and Estimates of Nodal Set of Harmonic Function Jan.16 08:30£­10:20 Han, Qing Asymptotic Analysis for Yamabe Equations 10:40£­12:00 Lu, Zhiqin Extreme eigenvalue problems on Riemannian manifolds 16:20£­17:40 Zhang, Zhenlei Frequency and Estimates of Nodal Set of Harmonic Function 19:10£­20:30 Zhang, Zhenlei Frequency and Estimates of Nodal Set of Harmonic Function Jan.17 08:30£­10:20 Lu, Zhiqin Extreme eigenvalue problems on Riemannian manifolds 10:40£­12:00 Lu, Zhiqin Extreme eigenvalue problems on Riemannian manifolds

Room: Teaching Building No. 5, Eastern District, USTC
Jan.14, 5206
Jan.15, 5207
Jan.16, 5207
Jan.17, 5206

Part II: Jan.17-19, "Tianyuan Forum on Geometry and Analysis"

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Conference Schedule£º

 & Room Time Speaker Title Chair Jan.17   5206 14:00£­14:05 Àî¼ÎÓí ¿ªÄ»ÖÂ´Ç Âé Ï£ ÄÏ 14:05£­14:55 Àî¼ÎÓí Compactness and singularity related to harmonic maps Break 15:30£­16:20 Âí  Ô¾ Global solutions to wave-Klein-Gordon system in 2+1 dimension Îâ ÔÆ »Ô 16:30£­17:20 Ð¤  ½¨ Positivity in the inverse $\sigma_k$ equations Jan.18   5206 09:00£­09:50 ÅíË«½× Location and uniqueness of concentration solutions »Æ   Õý Break 10:20£­11:10 ·½×ÓºÀ Self-Similar and Self-Conformal Solutions to Inverse Curvature Flows 11:20£­12:10 ÀîÎÄ¿¡ Steklov eigenvalues and free boundary minimal surfaces Lunch 14:00£­14:50 Íõ  ±ø Heat kernel on Ricci shrinkers Áõ ÊÀ Æ½ Group photo taken, Break 15:40£­16:30 Öì  ÃÈ Li-Yau gradient bounds under integral curvature conditions 16:40£­17:30 Àî  Æ½ Nonnegative Hermitian vector bundles and Chern numbers Banquet Jan.19   2104 09:00£­09:50 Â½Ö¾ÇÚ Analysis of the Laplacian on the moduli space of polarized Calabi-Yau manifolds ¸ð ½¨ È« Break 10:20£­11:10 Íõ  ·¼ Obata's rigidity theorem on manifolds with boundary 11:20£­12:10 ËÎ  Áˆ Singularity of Yang-Mills-Higgs fields on surfaces Lunch 14:00£­14:50 ½ð  Áú Semiclassical measures on compact hyperbolic surfaces. ³Â Ñ§ ³¤ Break 15:20£­16:10 ÂéÐ¡ÄÏ Generalized Bergman kernels on symplectic manifolds and applications.

Room£ºTeaching Building No.5, Eastern District, USTC

Teaching Building No.2, Eastern District, USTC

Annex: Summary of the Conference Report

Title: Location and uniqueness of concentration solutions

AbstractThis talk is concerned with the following nonlinear Schr\"odinger equation $$-\varepsilon^2\Delta u+ V(x)u=|u|^{p-2}u,\,\,\,u\in H^1(R^N),$$where $\varepsilon>0$ is a small parameter, $N\geq 1$, $2<p<2^*$.For a class of  $V(x)$ which possesses non-isolated critical points, we obtain the necessary condition, existence and local uniqueness of the positive single peak solution with concentrating at this kind of points. Here the main difficulty is the degeneracy and inhomogeneity of $V(x)$ at the  concentrating point.

Title: Global solutions to wave-Klein-Gordon system in 2+1 dimension

Abstract: In this talk we will present some results on the global existence of regular solutions to wave-Klein-Gordon system in 2+1 dimension. Compared with the 3 + 1 dimensional case, this problem is more delicate due to the slow decay of the solutions to the free linear wave/Klein-Gordon equation in dimension 2+1. We will refine some techniques developed in 3 + 1 case, including the normal form transform, the semi-hyperbolic frame and the conformal energy estimate on hyperboloids. Some new observations will be applied in order to obtain more precise estimates.

Title: Positivity in the inverse $\sigma_k$ equations

Abstract: We discuss a conjecture by Lejmi-Szekelyhidi, which is a numerical criterion on the solvability of inverse $\sigma_k$ equations over a compact K\"ahler manifold. We show how it is related to the positivity theory in complex algebraic and analytic geometry. In particular, we show how a weaker version of their conjecture can be verified by obtaining the desired positivity results for cohomology classes of bidegree (n-1, n-1) when k=n-1 or when the manifold is a 3-fold.

Title: Analysis of the Laplacian on the moduli space of polarized Calabi-Yau manifolds

Abstract: In this talk, we first discuss some recently results on the extension of the Laplacian on non-complete manifolds. We then use our results into the analysis on the geometry of Calabi-Yau moduli spaces. This is joint with Hang Xu.

Title: Self-Similar and Self-Conformal Solutions to Inverse Curvature Flows

Abstract: In this talk, the speaker will examine a large class of curvature flows by degree -1 homogeneous functions of principal curvatures in Euclidean spaces. This class of curvature flows includes the well-known inverse mean curvature flow and many others in the current literature. Self-similar (and more generally self-conformal) solutions to these curvature flows are solutions that evolves homothetically (conformally resp.) without changing their shapes (conformal class resp.). We will talk about the uniqueness, rigidity, and constructions problems of both compact and non-compact self-similar (self-conformal) solutions to these flows.

Title: Steklov eigenvalues and free boundary minimal surfaces

Abstract: Steklov eigenvalues are the spectrum of the Dirichlet-to-Neumann operator on compact manifolds with boundary. These are geometric invariants which satisfy an additional conformal invariance on two dimensional surfaces. In a seminal work of Fraser and Schoen, they discovered an intriguing relationship between an extremal Steklov eigenvalue problem on surfaces with boundary and free boundary minimal surfaces in the Euclidean unit ball. In particular, they proved an upper bound on the normalized first Steklov eigenvalue only in terms of the topological type of the surface. In this talk, we will discuss some known results and open questions about the Steklov eigenvalues of free boundary minimal surfaces in the unit ball.

Title: Heat kernel on Ricci shrinkers.

Abstract: We develop heat kernel estimates on Ricci shrinkers. Using these estimates, we improve many classical theorems for closed Ricci flows, including no-local-collapsing, pseudo-locality theorem and curvature tensor maximum principle. As applications, we show that any Ricci shrinker with \lambda_2>0 must be a quotient of sphere.  This is a joint work with Yu Li.

Titile: Li-Yau gradient bounds under integral curvature conditions

Abstract: Li-Yau type gradient bounds has been widely used in geometric analysis, and become a powerful tool in deriving geometric and topological properties of differential manifolds. Since the celebrated work of P. Li and S.-T. Yau, numerous efforts have been made in improving the Li-Yau bound on manifolds with Ricci curvature bounded from below.  In this talk, we will present our recent works on Li-Yau type gradient bounds for positive solutions of the heat equation on complete manifolds with certain integral curvature bounds, namely,  |Ric_| in L^p for p>n/2 or certain Kato type of norm of |Ric_| being bounded together with a Gaussian upper bound of the heat kernel. These assumptions allow the lower bound of the Ricci curvature to tend to negative infinity, which is weaker than the assumptions in the known results. These are joint works with Qi S. Zhang.

Title: Nonnegative Hermitian vector bundles and Chern numbers

Abstract: In this talk we review a notion of nonnegativity for Hermitian holomorphic vector bundles introduced by Bott and Chern and its later development. Then we discuss our recent work around it, stressing the relations between analytic positivity and topological positivity.

Title: Compactness and singularity related to harmonic maps

Abstract: In the talk I will review our recent works on compactness and regularity related to harmonic maps.

Title: Obata's rigidity theorem on manifolds with boundary

Abstract: We study the Obata equation with Robin boundary condition $\frac{\partial f}{\partial \nu}+af=0$ on manifolds with boundary Dirichlet boundary and Neumann boundary conditions were previously studied by Reilly and Escobar, respectively. Unlike their resutls, the sign of $a$ plays a role here. The new discovery shows besides spherical domains, there are other manifolds for both $a>0$ and $a<0$. We also considered the non-vanishing Neumann condition $\frac{\partial f}{\partial \nu}=1$ and give a full discussion of the manifolds. This is joint work with Mijia Lai and Xuezhang Chen.

Title: Singularity of Yang-Mills-Higgs fields on surfaces

Abstract: The Yang-Mills-Higgs fields are critical points of the famous Yang-Mills-Higgs functional which plays a fundamental role in quantum field theories. It is known that the isolated singularities of two dimensional YMH fieds are in general not removable. In this talk, we will give a sharp estimate of the singular YMH fields, namely, we establish a precise relation between the order of singularity and the limiting holonomy around that point. This is a recent joint work with Bo Chen.

Title: Semiclassical measures on compact hyperbolic surfaces.

Abstract: We study the behavior of the eigenfunctions of the Laplacian operator on a compact hyperbolic surface in the semiclassical limit. The quantum ergodicity theorem by Shnirelman, Zelditch and Colin de Verdiere implies that there is a density-one sequence of eigenfunctions converging weakly to the Liouville measure on the unit cosphere bundle. The quantum unique ergodicity conjecture by Rudnick and Sarnak states that the Liouville measure is the only semiclassical measure. The conjecture is only known in arithmetic cases by the work of Lindenstrauss and Soundararjan. We prove that semiclassical measures on compact hyperbolic surfaces always have full support on the unit cosphere bundle. The key ingredient is the fractal uncertainty principle of Bourgain-Dyatlov, which states that no function can localize close to fractal sets both in position and frequency. This is joint work with Semyon Dyatlov.

Title: Generalized Bergman kernels on symplectic manifolds and applications.

Abstract: A suitable notion of  ¡°holomorphic section¡± of a prequantum line bundle on a compact symplectic manifold is the eigensections of low energy of the Bochner Laplacian acting on high $p$-tensor powers of the prequantum line bundle. We explain the asymptotic expansion of the  corresponding kernel of the orthogonal projection as the power p tends to infinity. This implies the compact symplectic manifold  can be embedded in the corresponding  projective space. With extra effort, we show the Fubini-Study metrics induced by these embeddings converge at speed rate $1/p^{2}$ to the symplectic form. We explain also its implication on Berezin-Toeplitz quantizations.

Organizers£º

¡¡¡¡¡¡¡¡Bobo Hua     (Fudan University)
¡¡¡¡¡¡¡¡Mijia Lai    (Shanghai Jiao Tong University
¡¡¡¡¡¡¡¡Yi Li        (Southeast University)
¡¡¡¡¡¡¡¡Zuoqin Wang  (University of Science and Technology of China)
¡¡¡¡¡¡¡¡Guoyi Xu     (Tsinghua University)

Supported by£ºNSFC£¬Tianyuan Funding