Geometric Analysis seminar talk list:
1.
Time:9:00-10:00, Thursday, March 2
Speaker:Junsheng Zhang, UC Berkeley
Title:A note on the supercritical deformed Hermitian-Yang-Mills equation
Abstract:We show that on a compact Kähler manifold all real (1,1)-classes admitting solutions to the supercritical deformed Hermitian-Yang-Mills equation form a both open and closed subset of those which satisfy the numerical condition proposed by Collins-Jacob-Yau. More importantly, we show by an example that it can be a proper subset. This disproves a conjecture made by Collins-Jacob-Yau.
ZOOM ID：974 0641 0454
Password：482702
2.
Time:14:00-15:00, Wednesday, March 15
Speaker:Long Li
Title:ON THE RESIDUAL MONGE-AMPERE MASS OF PLURISUBHARMONIC FUNCTIONS WITH SYMMETRY
Abstract:In this talk, we will study the residual Monge-Ampere mass of a plurisubharmonic function with isolated singularity at the origin in C^2. We proved that the residual mass is zero if its Lelong number is zero at the origin, provided that it is S1-invariant and radially regular. This result answers the zero mass conjecture raised by Guedj and Rashkovskii in this special case.
ZOOM ID：999 7884 2363
Password：295591
3.
Time:14:00-15:00, Wednesday, March 22
Speaker:Jiyuan Han, Westlake University
Title:A study seminar on theta functions
Abstract:In this study seminar, we will talk about the theta functions on elliptic curves with a totally degenerate reduction.
ZOOM ID：977 9416 6368
Password：436688
4.
Time:14:00-15:00, Wednesday, March 29
Speaker:Jakob Hultgren, Umeå University
Title:Real Monge-Ampère equations on reflexive polytopes and the SYZ conjecture
Abstract:The SYZ conjecture on special Lagrangian torus fibrations in Calabi-Yau manifolds close to large complex limits is a central topic in geometry. Recent advances in complex geometry has reduced this conjecture to a comparison principle between solutions to real and non-Archimedean Monge-Ampère equations. In the 'toric setting', i.e. when considering certain families of Calabi-Yau hypersurfaces in toric manifolds, this comparison property can be proved by producing solutions to a real Monge-Ampère equation on the boundary of a reflexive polytope. I will outline a variational approach to such equations, present a necessary and sufficient condition for existence of solutions and discuss implications for the SYZ conjecture. This is based on joint work with M. Jonsson, E. Mazzon and N. McCleerey (arXiv:2208.13697) and joint work with R. Andreasson (arXiv:2303.05276)
ZOOM ID：981 6544 0011
Password：340582
5.
Time:18:00-19:00，Wednesday, April 5
Speaker:Ruadhai Dervan，University of Glasgow
Title:The universal structure of moment maps in complex geometry
Abstract:Much of complex geometry is motivated by linking the existence of solutions to geometric PDEs (producing "canonical metrics") to stability conditions in algebraic geometry. I will answer a more fundamental question: what is the recipe to produce geometric PDEs in complex geometry? The solution to this will use some old tools from symplectic geometry, namely equivariant differential geometry and the theory of moment maps. This is joint work with Michael Hallam.
ZOOM ID：952 802 9961
Password：314159
6.
Time:14:00-15:00, Wednesday, April 12
Speaker:Ved Datar, India Institute of Science
Title:Singular solutions for complex Hessian type equations
Abstract:It is well known that solvability of the complex Monge- Ampere equation on compact Kaehler manifolds is related to the positivity of certain intersection numbers. In fact, this follows from combining Yau’s resolution of the Calabi conjecture, with Demailly and Paun’s generalization of the classical Nakai-Mozhesoin criteria. This correspondence was recently extended to a broad class of complex non-linear PDEs including the J-equation and the deformed Hermitian-Yang-Mills (dHYM) equations by the work of Gao Chen and others. A natural question to ask is whether solutions (necessarily singular) exist in any reasonable sense if the Nakai criteria fails. Results of this nature are ubiquitous in Kaehler geometry - existence of weak Kaehler-Einstein metrics on normal varieties and Hermitian-Einstein metrics on reflexive sheaves to name a couple. Much closer to the present theme, is the work of Boucksom-Eyssidieux-Guedj-Zeriahi on solving the complex Monge-Ampere equation in big classes. In the talk, I will first speak about some joint and ongoing work with Ramesh Mete and Jian Song, that offers a reasonably complete resolution in complex dimension two, at least for the J-equation and the dHYM equations. Next, I will discuss some conjectures on what one can expect in higher dimensions.
ZOOM ID：952 802 9961
Password：314159
7.
Time:14:00-15:00, Wednesday, May 10
Speaker:Goncalo Oliveira，Institute Superior Tecnico
Title:Lagrangian mean curvature flow and the Gibbons-Hawking ansatz
Abstract:In this talk I will report on joint work with Jason Lotay on which we prove versions of the Thomas and Thomas-Yau conjectures regarding the existence of special Lagrangian submanifolds and its relation to the Lagrangian mean flow as a way to find them. If time permits I will also report on some more recent work towards proving some more recent conjectures due to Joyce.
ZOOM ID：952 802 9961
Password：314159
8.
Time:14:00-15:00, Wednesday, May 24
Speaker:Shengxuan Zhou, Peking University
Title:Peak sections and Bergman kernels on complex hyperbolic cusps
Abstract:The asymptotic behavior of Bergman kernel on Kahler manifold has been studied by Tian, Ruan, Zelditch, Catlin, Ma, Marinescu and many others since 1990. The works related to Bergman kernel plays an important role in complex geometry.
In this talk, we will introduce a way to localize the Bergman kernel by using peak sections. As an application, we consider the asymptotic behavior of Bergman kernels on complex hyperbolic cusps.
ZOOM ID：952 802 9961
Password：314159
9.
Time:14:00-15:00, Wednesday, May 31
Speaker:Yaxiong Liu，Tsinghua University
Title:Recent progress on the valuative stability of polarized varieties
Abstract:The valuative criterion has played an essential role in the studying of K-stability of Fano varieties. We will introduce the recent progress on valuative stability of general polarized varieties, including Dervan-Legendre's beta-functional, openness of uniformly valuative stability, and Boucksom-Jonsson's divisorial stability.
ZOOM ID：952 802 9961
Password：314159
10.
Time:20:00-21:00, Wednesday, June 7
Speaker:Samuel Johnston，Cambridge University
Title:Frobenius Structure Conjecture in Intrinsic Mirror Symmetry
Abstract:Given an affine log Calabi-Yau U and a compactification X such that D= X\U is the support of a nef divisor and D is maximally degenerate snc divisor, then the intrinsic mirror algebra of Gross and Siebert admits a trace form which encodes naive counts of rational curves intersecting the boundary finitely many times. By further using results of Keel and Yu, we deduce in the case where U contains a Zariski dense torus that the Gross-Siebert and the Keel-Yu mirror constructions coincide for appropriate choices of compactifications (X,D). In particular, we demonstrate certain log Gromov-Witten invariants are equal to certain counts of non-archimedean analytic disks in the Berkovich analytification of U. If time allows, we will discuss implications for Fano mirror symmetry.
ZOOM ID：952 802 9961
Password：314159
11.
Time:9:00-10:00, Wednesday, September 13
Speaker:Xiaowei Wang, Rutgers University
Title:Moduli space of log Calabi-Yau pairs
Abstract:KSBA stability and K-stability have been successful in constructing moduli spaces of canonically polarized variety and Fano varieties, the case of Calabi-Yau varieties remains to be subtle and less understood. In this talk, we will discuss an approach to this problem in the case of log Calabi-Yau pairs (X,D), where X is a Fano variety and D is an anticanonical Q-divisor, in which we consider all semi-log-canonical degenerations. A challenge of this approach is that the moduli stack can be unbounded. Nevertheless, if we consider log Calabi-Yau pairs as degenerations of P^2 with plane curves, we show that there exists a projective good moduli space despite the unboundedness.
(This is joint work with K. Ascher, D. Bejleri, H. Blum, K. DeVleming, G. Inchiostro, and Y. Liu)
ZOOM ID：952 802 9961
Password：314159
12.
Time:14:00-15:00, Tuesday,September 19
Speaker:Yongqiang Liu, USTC
Title:L^2 type invariants of hyperplane arrangement complement
Abstract:We first give an brief introduction to the topic of hyperplane arrangement. Then we study the L^2 type invariants of hyperplane arrangement complement. In particular, we give concrete formulas for these L^2 type invariants at degree 1 and the connection with combinatoriscs. If tim allows, some similar results for smooth complex quasi-project variety will be presented.
ZOOM ID：952 802 9961
Password：314159
13.
Time:9:00-10:30, September 27 & 28
Speaker:Yucong Jiang, UIUC
Title:An invitation to generalized complex geometry
Abstract:The two lectures will give an introduction to generalized complex and generalized Kähler geometry. The material will include Courant algebroids, Dirac structures, generalized complex structures, the spinor formulation, the generalized Darboux theorem, deformations of GC, generalized metrics, generalized Kähler structures, correspondence between bihermitian structures and GK structures, generalized submanifolds and possibly more.
ZOOM ID：952 802 9961
Password：314159
14.
Time:9:00-10:00, Wednesday, October 11
Speaker:Kai Xu, Duke University
Title:On the existence problem of weak inverse mean curvature flow
Abstract:Inverse mean curvature flow is the flow that evolves a mean-convex hypersurface at the speed of the inverse of its mean curvature. The focus of this talk is a weak version of the inverse mean curvature flow, defined by Huisken and Ilmanen in 2001. We will introduce this weak formulation and explain the behavior of weak solutions. Then we explain how the existence of weak solutions is affected by the geometry of the manifold. Finally, we introduce the speaker's recent theorem, which roughly states that a certain isoperimetric inequality of a manifold implies the existence of weak solutions on it.
ZOOM ID：952 802 9961
Password：314159
15.
Time:10:30-11:30, Wednesday,October 18
Speaker:Man-Chun Lee, CUHK
Title:Ricci Flow and pinched curvature on non-compact manifolds
Abstract:In dimension three, it was proved recently by Deruelle-Schulze-Simon, Lott, Lee-Topping that three-manifolds with non-negative pinched Ricci curvature are compact or flat. In this talk, we will discuss its partial generalisation to higher dimension using the Ricci Flow method. This is based on joint work with P. Topping.
ZOOM ID：952 802 9961
Password：314159
16.
Time:14:00-15:00, Wednesday, November 8
Speaker:Siqi He, CAS AMSS
Title:Desingularization of Branched Immersed Special Lagrangians
Abstract:Special Lagrangian submanifolds, a significant class of calibrated submanifolds within Calabi-Yau manifolds, have been the focus of extensive study. McLean's theorem demonstrates that the space of nearby special Lagrangian submanifolds can be parameterized by harmonic 1-forms. In this presentation, we will explore recent developments in extending McLean's theorem to desingularize branched immersed special Lagrangians. We will delve into the utilization of multi-valued harmonic functions for constructing nearby deformations.
ZOOM ID：952 802 9961
Password：314159
17.
Time:14:00-15:00, Friday, November 10
Speaker:Miaomiao Zhu, Shanghai Jiao Tong University
Title:Energy quantization for geometric PDEs over spaces with varying geometric structures
Abstract:In this talk, we shall firstly give a brief discussion on the compactness of solutions of some geometric PDEs over degenerating Riemann surfaces. Then, inspired by the 2D situations, we explore a scheme for investigating the compactness of solutions of geometric PDEs over 4-manifolds with varying geometric structures. We illustrate this scheme by applying it to two concrete problems, the biharmonic map system and the Yang-Mills system over non-collapsed Einstein 4-manifolds with varying metrics.
ZOOM ID：952 802 9961
Password：314159
18.
Time:14:00-15:00, November 13/20/22
Speaker:Wangjian Jian, CAS
Title:On the finite time solution of the Kahler-Ricci flow
Abstract:In the three talks, we introduce Perelman's proof of the scalar curvature and diameter estimates for the Fano Kahler-Ricci flow. Then we will talk about the new proofs of Perelman's estimates. Then we will show how to extend these new proofs to general finite time Kahler-Ricci flow, and talk about the applications of these new estimates. The talk is based on the recent joint work with Song and Tian.
ZOOM ID：952 802 9961
Password：314159
19.
Time:10:00-11:00, Wednesday, November 22
Speaker:Yuanqi Wang, The University of Kansas
Title:On G2 instantons with 1-dimensional singularities
Abstract:G2-instantons on 7-dimensional manifolds generalize both flat connections in dimension 3, and anti self-dual connections in dimension 4. Donaldson-Segal program expects a certain count of G2-instantons and other objects could yield a topological invariant for 7-manifolds, called the prospective G2--Casson invariant. Related to the compactification/boundary of the moduli space, Walpuski proposed to construct non-trivial singular G2--instantons via gluing. The analytic part of this singular perturbation problem is expected to encounter indicial roots, that are essentially related to the spectrum of a certain Dirac operator on the standard 5-dimensional unit sphere.
In this talk, we report some work on the spectral theory and consequent obstruction theory of G2-instantons with 1-dimensional singularities. This is the preliminary of a joint project with Thomas Walpuski and Henrique Sá Earp.
ZOOM ID：952 802 9961
Password：314159
20.
Time:9:00-12:00, Wednesday, December 6
Speaker:Minghao Liao, Nanjing University
Title:Kahler-Ricci soliton and weighted Abban-Zhuang Estimate
Abstract:Kahler-Ricci soliton is a type of canonical metrics on Fano manifold, which is a generalization of Kahler-Einstein metric and is related to the limiting behaviour of Kahler-Ricci flow. By the Yau-Tian-Donaldson theorem for Kahler-Ricci soliton, one can verify the existence of Kahler-Ricci soliton on Fano variety via checking weighted K-stability condition. So far only examples with large symmetry group can be checked by equivariant method. To handle Fano variety with small automorphism groups, we develop a weighted Abban-Zhuang estimate to give a lower bound of weighted delta invariant. This talk is based on a joint work with Linsheng Wang.
ZOOM ID：952 802 9961
Password：314159
21.
Time:14:00-17:00, Wednesday, December 6
Speaker:Linsheng Wang, Nanjing University
Title:A new example of Fano manifold with Kähler-Ricci soliton
Abstract:In this talk, I will introduce an effective method to show the existence of the Kähler-Ricci soliton on a given Fano manifold. As an application, we show that any Fano threefold X in the family No.2.28 of Mukai-Mori's list (that is, CP^3 with a smooth plane cubic curve C blowup) admits Kähler-Ricci soliton. Furthermore, we show that the weighted K-stability of the Fano manifolds X is equivalent to the GIT-stability of the plane cubic curves C. This is a joint work with Minghao Miao.
ZOOM ID：952 802 9961
Password：314159
22.
Time:14:00-15:00, Tuesday, December 12
Speaker:Feng Hao, Shandong University
Title:Holomorphic 1-forms without Zeros and Smooth Morphism to Abelian varieties
Abstract:A celebrated result of Popa and Schnell shows that any holomorphic 1-form on a smooth complex projective variety of general type admits zeros. More generally, they show that the Kodaira dimension κ(X) of a smooth complex projective variety X satisfies the following inequality κ(X)≤dimX−g where g is the maximal number of pointwise linearly independent holomorphic 1-forms. In this talk I will give a classification of (minimal) varieties satisfying the equality conditions κ(X)=dimX−g. Roughly speaking, they arise as diagonal quotients of the product of an abelian variety with a variety of general type. This is a joint work with Nathan Chen and Benjamin Church.
ZOOM ID：952 802 9961
Password：314159
23.
Time:9:30-10:30, Tuesday, December 26
Speaker:Hong Huang, Beijing Normal University
Title:Topological classification of compact manifolds with positive isotropic curvature
Abstract:The notion of positive isotropic curvature was introduced by Micallef and Moore in 1988. In this talk I'll first briefly survey some of the previous (before 2019) works of various authors on Riemannian manifolds with positive isotropic curvature. Then I'll introduce my recent work on topological classification of compact manifolds of dimension $n\geq 12$ with positive isotropic curvature, which uses Ricci flow with surgery on orbifolds and some techniques from differential topology, and is based on Brendle's curvature pinching estimates.
ZOOM ID：952 802 9961
Password：314159
