### Geometric Analysis Seminar

The seminar will be a hybrid of in-person presentation and zoom meeting.

Organizers:
Ben McReynolds,
Sai-Kee Yeung.

Date, Time and Place:
Mondays, 3:30pm (EST), Math 731, Purdue University

Schedule:

Aug. 30: Alexandre Eremenko, Purdue University.
Title: Metrics of positive curvature with conic singularities
Abstract:It is a survey of recent work on the following problem. How to describe the space of Riemannian metrics of constant positive curvature with finitely many conic singularities with prescribed angles on a compact surface of given genus? I will discuss what is known, and how this problem is related with other areas of mathematics, besides geometry.

Sep. 13: Laszlo Lempert, Purdue University.
Title: Bergman kernels in holomorphic vector bundles
Abstract:Consider a holomorphic vector bundle $E\to X$ over a compact complex manifold, and the space $\mathcal O(E)$ of its holomorphic sections. Somewhat traditionally, any positive definite inner product on $\mathcal O(E)$ induces a Bergman--type kernel function, a holomorphic section of a certain vector bundle over $X\times\overline X$. Here $\overline X$ is $X$ with the opposite complex structure. In this talk I will argue that it is more natural to associate kernel functions with inner products on the dual $\mathcal O(E)^*$, not necessarily positive definite ones, and I will describe various properties of this association.

Sep. 20: Alexandre Eremenko, Purdue University.
Title: Metrics of constant positive curvature with 4 conic singularities on the sphere.
Abstract:It will be shown that the forgetful map from the space of equivalence classes of metrics indicated in the title to the moduli space of 4-punctured spheres is finite.

Oct. 4: Donu Arapura, Purdue University.
Title: Euler characteristics of aspherical K\"ahler manifolds.
Abstract: This is joint work with Botong Wang. I want to start by recalling the Hopf-Singer conjecture on the sign of the Euler characteristic of a compact aspherical manifold. In the Kaehler setting, we have a stronger conjecture involving Euler characteristics of perverse sheaves. (I won't assume everyone knows what those are, so I'll sumarize the needed facts.) Our results are that the stronger conjecture holds when X is compact Kaehler with nonpositive curvature, or X is aspherical projective with a faithful rigid local system.

Oct. 18: Sai Kee Yeung, Purdue University.
Title: Torelli image of the moduli space of curves in Siegel moduli varieties.
Abstract: The classical Abel-Jacobi map induces the Torelli map from a moduli space of curves of genus $g$ into the corresponding Siegel modular variety. The mapping is in general highly curved but difficult to describe. A conjecture of Oort states that the image does not contain any totally geodesic complex subvarieties for $g$ sufficiently large, which in turn implies a conjecture of Coleman that the number of complex multiplication points on the moduli space is finite. We will explain some approaches to the problems involved and explain new results obtained, in particular the case that the supposed totally geodesic complex subvarieties are complex ball quotients of dimension at least $2$.

Oct. 25: Sam Nariman, Purdue University.
Title: Thurston's fragmentation and c-principles.
Abstract: I will discuss a remarkable generalization of Mather's theorem by Thurston that relates the identity component of diffeomorphism groups to the classifying space of Haefliger structures aka singular foliations". The homotopy type of this classifying space plays a fundamental role in foliation theory. However, it is notoriously difficult to determine its homotopy groups. Mather-Thurston theory relates the homology of diffeomorphism groups to these homotopy groups. Hence, this h-principle type theorem has been used as the main tool to get at the homotopy groups of Haefliger spaces. We talk about generalizing Thurston's method to prove analogue of MT for other subgroups of diffeomorphism groups that was conjectured to hold in particular contactomorphisms and foliation preserving diffeomorphisms. Most h-principle methods use the local statement about M=R^n to prove a statement about compact manifolds. But Thurston's method is intrinsically compactly supported method and it is suitable when the local statement for M=R^n is hard to prove. As we also shall explain Thurston's point of view on this local to global" method implies non abelian Poincare duality.

Nov. 1: Lei Ni, University of California, San Diego.
Title: Quadratic curvatures and Kaehler C-spaces.
Abstract: Quadratic bisectional curvature has been around since 1960s. Its positivity was proposed to characterize the simply-connected homogenous Kaehler spaces by Wu-Yau-Zheng in 2009. Recently there are new notions of quadratic curvatures were proposed as perhaps the `right' candidates. I shall explain what they are, the connection with the so-called generalized Hartshorne conjecture and some progresses.

Nov. 8: R\'emi Reboulet, Institut Fourier, Grenoble.
Title:Non-Archimedean plurisubharmonic geodesics and complex limits. .
Abstract:We study the space of finite-energy plurisubharmonic metrics on the Berkovich analytification of an ample line bundle on a variety over a non-Archimedean field. We discuss the construction of plurisubharmonic geodesics in this space, in parallel with classical results in the complex setting. We then see how specific examples of such geodesics can be interpreted as limits of complex plurisubharmonic geodesics on degenerations of complex manifolds. Zoom meeting: https://purdue-edu.zoom.us/j/95279418547?pwd=T1pza2RlMnV0YUhuWlIreEFrWUFOdz09 > Meeting ID: 952 7941 8547> Passcode: 449355>

Nov. 29: Lvzhou Chen, University of Texas, Austin.
Title:The Kervaire conjecture and the minimal complexity of surfaces .
Abstract:We use topological methods to solve special cases of a fundamental problem in group theory, the Kervaire conjecture, which has connection to various problems in topology. The conjecture asserts that, for any nontrivial group G and any element w in the free product G*Z, the quotient (G*Z)/<> is still nontrivial. We interpret this as a problem of estimating the minimal complexity (in terms of Euler characteristic) of surface maps to certain spaces. This gives a conceptually simple proof of Klyachko's theorem that confirms the Kervaire conjecture for any G torsion-free. I will also explain new results obtained using this approach.

Dec. 6 (postponed): David Ben McReynolds
Title:Spectra and arithmeticity .
Abstract:I will discuss how knowing whether a manifold is arithmetic or not can be useful in solving problems related to the eigenvalue/length spectrum of the manifold.

Jan. 24: Laszlo Lempert, Purdue University.
Title: To the geometry of spaces of plurisubharmonic functions on Kahler manifolds.
Abstract: Consider a compact Kahler manifold (X,\omega) and the space of \omega--plurisubharmonic functions on X. In the past 30 years various subspaces of this space have been studied by endowing them with various metrics. The central object of the talk is similar: a quantity to measure some sort of distance between two \omega--plurisubharmonic functions. This quantity is not a number, but a decreasing function on a certain interval, from which all previously studied metrics, and their properties, can be derived.

Jan. 31: Changyou Wang, Purdue University.
Title: Heat flow of 1/2 harmonic maps into manifolds.
Abstract: There has seen growing interests toward nonlocal nonlinear PDEs and their applications. In this talk, I will describe the gradient flow of fractional Dirichlet energy functional for mappings between manifolds. I will present a recent result concerning the heat flow of 1/2 harmonic maps from $R^n$ to a compact Riemannian manifold $N$. The main technique involves lifting the problem into the free boundary problem of a local PDE but in one spatial dimension higher. As consequences, we establish the existence of a global weak solution of 1/2-harmonic map flow, that is smooth away from a closed set with finite parabolic $(n+1)$-dimensional Hausdorff measure.

Feb. 7: Chi Li, Rutgers University.
Title:On the algebraic uniqueness of Kaehler-Ricci flow limits on Fano manifolds. .
AbstractLet X be a Fano manifold. The Hamilton-Tian conjecture, which has been studied by Perelman, Tian-Zhang, Chen-Wang, Bamler and others, states that the normalized Kaehler-Ricci flow on X converges in the Gromov-Hausdorff topology to a possibly singular Kaehler-Ricci soliton as the time goes to infinity. Chen-Sun-Wang further conjectured that the limit space does not depend on the initial Kaehler metric but depends only on the algebraic structure of X. I will discuss a joint work with Jiyuan Han, which confirms this uniqueness conjecture. The proof is based on the study of an optimization problem for real valuations on the functional field of X. Our result has applications in identifying limits in concrete examples.
Zoom meeting: https://purdue-edu.zoom.us/j/98781403290?pwd=RTAwRExWcTd0K2JpUDNzNlJwSlFiZz09 > Meeting ID: 987 8140 3290> Passcode:523599

Feb. 21: Mei-Chi Shaw, University of Notre Dame.
Title:The Cauchy-Riemann Equations on Hartogs triangles .
AbstractThe Hartogs triangle in C^2 serves as an important example in several complex variables. In this talk, we discuss the extendability and the Cauchy-Riemann equations on Hartogs triangles. We also discuss some recent progress for the Cauchy-Riemann equations on Hartogs triangles in complex projective space of dimension two.
Zoom meeting: https://purdue-edu.zoom.us/j/98781403290?pwd=RTAwRExWcTd0K2JpUDNzNlJwSlFiZz09 > Meeting ID: 987 8140 3290> Passcode:523599

Feb. 28: Che-Hung Huang, Purdue University.
Title: Plurisubharmonicity of the Dirichlet energy.
Abstract: Harmonic mappings of Riemannian manifolds are characterized as critical points of the Dirichlet energy. In this talk, I will discuss a new result on subharmonicity of the Dirichlet energy and highlight its connection with moduli theory.

Mar. 7: David Ben McReynolds, Purdue University
Title:Spectra and arithmeticity .
Abstract:I will discuss how knowing whether a manifold is arithmetic or not can be useful in solving problems related to the eigenvalue/length spectrum of the manifold.

Mar. 21 (Postponed) : Justin E. Katz, Purdue University
Title:You can hear the shape of certain Shimura varieties .
AbstractIt has been known since the 60's that one cannot hear the shape of a drum, in general. That being said, one should expect that special-enough drums produce special-enough sounds to determine their source. In this talk, I will outline my proof that some very special drums (principal congruence covers of certain shimura curves) are determined by the way they sound.

Apr. 4: Sai-Kee Yeung, Purdue University.
Title: Semi-continuity of cohomology in a real analytic setting.
Abstract: Consider a real analytic family of self-adjoint elliptic operators on a real analytic Hermitian vector bundle on a real analytic manifold. We would like to explain that the dimension of the kernel of the operators is semi-continuous in the Zariski topology of the parameter space. An application is on the study of geometry of a family of almost complex manifolds. It also gives rise to some sort of Direct Image Theorem in real analytic setting.

Apr. 11, 8:30 pm (Unusual time): Pak-Tung Ho, Sogang University, Korea
Title: Chern-Yamabe problem .
AbstractI will explain what the Chern-Yamabe problem is, and talk about the Chern-Yamabe flow which is a geometric flow approach to solve the Chern-Yamabe problem. I will also mention other results related to the Chern-Yamabe problem.
Zoom meeting: https://purdue-edu.zoom.us/j/98781403290?pwd=RTAwRExWcTd0K2JpUDNzNlJwSlFiZz09 > Meeting ID: 987 8140 3290> Passcode:523599

Apr. 18 : Justin E. Katz, Purdue University
Title:You can hear the shape of certain Shimura varieties .
AbstractIt has been known since the 60's that one cannot hear the shape of a drum, in general. That being said, one should expect that special-enough drums produce special-enough sounds to determine their source. In this talk, I will outline my proof that some very special drums (principal congruence covers of certain shimura curves) are determined by the way they sound.

Apr. 25, 7 pm (Unusual time): Kaufman, Lucas, Center for Complex Geometry, Instutute for Basic Science, Korea.
Title:Random walks on SL_2(C) via holomorphic dynamics .
AbstractGiven a sequence of random 2 by 2 complex matrices, it is a classical problem to study the statistical properties of their product. This theory dates back to fundamental works of Furstenberg, Kesten, etc. and is still an active research topic. In this talk, I intend to show how methods from complex analysis and analogies with holomorphic dynamics offer a new point of view to this problem. This is used to obtain several new limit theorems for these random processes, often in their optimal version. This is based on joint works with T.-C. Dinh and H. Wu.
Zoom meeting: https://purdue-edu.zoom.us/j/98781403290?pwd=RTAwRExWcTd0K2JpUDNzNlJwSlFiZz09 > Meeting ID: 987 8140 3290> Passcode:523599