The Complex Analysis and Geometry Seminar will run on Fridays, 10:30-11:30 am, in Hill 705. All are welcome!

The seminar is organized by Purvi Gupta, Xiaojun Huang and Jian Song. .

### Upcoming Talks (Fall 2019)

• November 21, 2019.
Max Hallgren, Cornell University

Title. Entropy convergence of Ricci flows with a type-I scalar curvature bound.

• December 06, 2019.
Dan Burns, University of Michigan, Ann Arbor
Talk Cancelled.

### Past Talks

• November 15, 2019.

Note: Two talks on the same day.
Zhiqin Lu, University of California, Irvine
Time: 10:30 am
Venue: HLL 705

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

Abstract. TBA

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Min Ru, University of Houston
Time: 2:30 pm
Venue: TBA

Title.On the beta-constant which appeared in the Ru-Vojta theorem.

Abstract. In this talk, I'll first discuss the recent theorem which I proved, joint with Paul Vojta, in the theory of Diophantine approximation. I will then compare the constant $\delta(L, D)$ appeared in that theorem with the constant $\delta(L)$ introduced by Blum-Jonsson. The constant $\delta(L)$ was used to describe the K-stability for the ${\Bbb Q}$-Fano varieties, and it seems that there are some mysterious connections to our theorem.

• October 29, 2019.

Note: Unusual day and time.
Nikhil Savale, University of Cologne
Time: 11:00 am
Venue: HLL 705

Title. Bochner Laplacian and Bergman kernel expansion of semi-positive line bundles on a Riemann surface.

Abstract. We generalize the results of Montgomery for the Bochner Laplacian on high tensor powers of a line bundle. When specialized to Riemann surfaces, this leads to the Bergman kernel expansion and geometric quantization results for semi-positive line bundles whose curvature vanishes at finite order. The proof exploits the relation of the Bochner Laplacian on tensor powers with the sub-Riemannian (sR) Laplacian.

• October 25, 2019.
Slawomir Kolodziej, Jagiellonian University, Kraków

Title. On volumes of nef classes on compact Hermitian manifolds.

Abstract. This is joint work with V. Tosatti. We give a partial answer to the conjecture on the volume of a nef class on a compact Hermitian manifold. In particular, we show that the conjecture is true for any semi-positive class and for any nef class in dimension 2.

• October 18, 2019.

Note: Special location and time. Held jointly with Temple University.
Laurent Stolovitch, Université de Nice Sophia-Antipolis
Time: 11:00 am
Venue: Room 617 Wachman Hall, Temple University

Title. Equivalence of Cauchy-Riemann manifolds and multisummability theory.

Abstract. We prove that if two real-analytic hypersurfaces in $\mathbb{C}^2$ are equivalent formally, then they are also $\mathcal{C}^\infty$ CR-equivalent at the respective point. As a corollary, we prove that all formal equivalences between real-algebraic Levi-nonflat hypersurfaces in $\mathbb{C}^2$ are algebraic (in particular are convergent). The result is obtained by using the recent CR - DS technique, connecting degenerate CR-manifolds and Dynamical Systems, and employing subsequently the multisummability theory of divergent power series used in the Dynamical Systems theory. This is a joint work with I. Kossovskiy and B. Lamel.

• September 27, 2019.
Debraj Chakrabarti, Central Michigan University

Title. An Alexander-Pontrjagin type duality in the $L^2$ theory of the $\overline{\partial}$-operator.

Abstract. Hörmander's $L^2$ estimates on the $\overline{\partial}$-equation on bounded pseudoconvex domains are a cornerstone of modern complex analysis. Subsequently, attempts were made by Folland-Kohn, Shaw, Hörmander and others to extend these $L^2$ methods to appropriate classes of nonpseudoconvex domains. In this talk, we discuss an approach to the $\overline{\partial}$-problem on an annulus between two weakly pseudoconvex domains which is inspired by Alexander-Pontrjagin duality in topology. This is ongoing joint work with Phil Harrington of Arkansas.

• September 27, 2019.
Note: This is a departmental colloquium talk.

Steven Krantz, Washington University, St. Louis
Time: 4:00 pm
Venue: HLL 705

Title.Analysis on the Worm Domain.

Abstract. The Diederich-Fornaess worm domain is a domain in complex space with very special geometric and analytic properties. Anyone with a basic course in complex analysis can understand the worm. But working with the worm is a challenge. It is subtle and usually quite difficult. In joint work with Marco Peloso and Caterina Stoppato we have been able to study the Bergman kernel for the worm and to learn some of its fundamental properties. This will be an expository lecture, and will be accessible to a broad audience.

• April 26, 2019.
Note: Two talks on the same day.
Ben Weinkove, Northwestern University

Time: 10:30 am
Venue: HLL 705

Title. Complex Monge-Ampere equations on manifolds

Abstract. More than 40 years ago, Yau solved the complex Monge-Ampere equation on compact Kahler manifolds. I will give an overview of some extensions of this result including the case of non-Kahler manifolds and behavior of solutions when the equation degenerates. I will also indicate some open problems and new directions.

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John Erik Fornaess, Norwegian University of Science and Technology
Time: 2:30 pm
Venue: HLL 705

Title.The squeezing function

Abstract. In complex analysis the most important domain is the unit disc. In fact all domains (at least simply connected and bounded) are biholomorphic, i.e. analytically equivalent, to the disc. In higher dimension, the natural analogue is the unit ball. But in higher dimension, the general domain is not biholomorphic to the ball. A basic question is then how well a general domain can be approximated by the ball. If we have a ball B_r of radius r<1 contained in the unit ball B_1, then a domain U in B_1 containing B_r is said to be squeezed between the two balls. The larger we can choose r, the closer the domain U is to the ball.

• April 19, 2019.
Berit Stensones, Norwegian University of Science and Technology

Title. Examples of Automorphisms with Fatou components with complicated topology

Abstract. It is known that in 1 variable the Fatou Component of an entire map will have to be simply connected. Further one can not have points that under iterations of a map spirals in toward a fixed point, the so called snail lemma. We shall show that in higher dimensions the situation is very different.

• April 12, 2019.
Ved Datar, Indian Institute of Science, Bangalore

Title. Adiabatic limits of ASD connections on collapsing K3 surfaces

Abstract.I will talk about recent joint work with Adam Jacob and Yuguang Zhang on a convergence result for a family of Yang-Mills connections over an elliptic K3 surface as the fibres collapse. We show that outside a finite number of fibres, the curvatures of the connections are uniformly bounded (with respect to fixed background metrics), and that the connections converge in L^{1,p} to a limiting connection whose restriction to a generic fibre is a flat connection. This solves a conjecture of Fukaya in dimension two, and is a vector bundle analog of the work of Gross-Wilson and Gross-Tosatti-Zhang on the collapsing behaviour of Ricci flat metrics in degenerating Kahler classes.

• February 15, 2019.
Shiferaw Berhanu, Temple University

Title. Unique continuation at the boundary for a class of elliptic equations

Abstract. We will present some recent results on unique continuation at the boundary for a class of second order elliptic partial differential equations and the biharmonic operator. The works are inspired by a theorem of X. Huang et al for holomorphic functions which also motivated M. S. Baouendi and L. Rothschild to prove generalizations for harmonic functions.

• March 15, 2019.
Andrew Zimmer, Louisiana State University, Baton Rouge

Title. The geometry of domains with negatively pinched Kaehler metrics

Abstract. Every bounded pseudoconvex domain in $\mathbb{C}^n$ has a natural complete metric: the Kaehler-Einstein metric constructed by Cheng-Yau. When the boundary of the domain is strongly pseudoconvex, Cheng-Yau showed that the holomorphic sectional curvature of this metric is asymptotically a negative constant. In this talk I will describe some converses to this result, including the following: if a smoothly bounded convex domain has a complete Kaehler metric with close to constant negative holomorphic sectional curvature near the boundary, then the domain is strongly pseudoconvex. This is joint work with F. Bracci and H. Gaussier.

• March 08, 2019.
Xin Dong, University of California, Irvine

Title. Bergman and Suita metrics

Abstract. For any open Riemann surface $X$ admitting Green functions, the Suita conjecture states that the Gaussian curvature of the Suita metric induced by the logarithmic capacity is bounded from above by $-4$, and the curvature is equal to $-4$ at some point if and only if $X$ is biholomorphic to the unit disc less a (possibly) closed polar subset. We talk about our new proof of the above equality part by using the plurisubharmonic variation properties of the Bergman kernels. We also relate this with a joint work with Bun Wong on the holomorphic sectional curvature of the Bergman metric on manifolds.

• October 12, 2018.
Rasul Shafikov, University of Western Ontario, Canada

Title. On rationally convex embeddings and immersions of real manifolds in complex spaces.

Abstract. A classical result of Duval-Sibony characterizes rationally convex totally real embeddings of real manifolds into $\mathbb{C}^n$ as those that are isotropic with respect to some Kahler form. In this talk I will describe some generalizations of this result for topological embeddings and immersions, and will discuss some applications.

• October 05, 2018.
Sean Curry, University of California, San Diego

Title. Strictly pseudoconvex domains in $\mathbb{C}^2$ with obstruction flat boundary.

Abstract. A bounded strictly pseudoconvex domain in $\mathbb{C}^n$, $n>1$, supports a unique complete Kahler-Einstein metric determined by the Cheng-Yau solution of Fefferman's Monge-Ampere equation. The smoothness of the solution of Fefferman's equation up to the boundary is obstructed by a local curvature invariant of the boundary called the obstruction density. In the case $n=2$ the obstruction density is especially important, in particular in describing the logarithmic singularity of the Bergman kernel. For domains in $\mathbb{C}^2$ diffeomorphic to the ball, we motivate and consider the problem of determining whether the global vanishing of this obstruction implies biholomorphic equivalence to the unit ball. (This is a strong form of the Ramadanov Conjecture.)