Informal Geometric Analysis Seminar

University of Maryland

ORGANIZED BY: D. Cristofaro-Gardiner, T. Darvas, Y. A. Rubinstein.

PREVIOUS YEARS: 2012-2013, 2013-2014, 2014-2015, 2015-2016, 2016-2017, 2017-2018, 2018-2019, 2019-2020, 2020-2021, 2021-2022, 2022-2023.

- September 26

Michael Hutchings (UC Berkeley)

*Title: Obstruction bundle gluing*

Abstract: Obstruction bundle gluing is a technique which can be used for the foundations of topological invariants that count holomorphic curves (or solutions to other PDEs) in situations where transversality fails, but not too badly. This talk will give an introduction to obstruction bundle gluing and work out a simple example that arises in defining embedded contact homology (the simplest nontrivial case of proving that the differential squares to zero). Based on joint work with Cliff Taubes. - October 10

Yaxiong Liu (UMD)

*Title: The eigenvalue problem of complex Hessian operators*

Abstract: In a very recent pair of nice papers of Badiane and Zeriahi, they consider the eigenvalue problem of complex Monge-Ampere and complex Hessian, and show that the C^{1,\bar{1}}-regularity of eigenfunction for MA and C^alpha-regularity for complex Hessian. They posed a question about the C^{1,1}-regularity. We give a positive answer and show the C^{1,1}-regularity of eigenfunction. This is a joint work with Jianchun Chu and Nicholas McCleerey. - October 24, Tamas is away
- November 7

Siarhei Finski (CNRS)

*Title: On the geometry at infinity on the space of Kahler potentials and submultiplicative filtrations*

Abstract: For a complex projective manifold polarised by an ample line bundle, we study the geometry at infinity on the space of all positive metrics on the line bundle. We show that this geometry is related to some asymptotic properties of submultiplicative filtrations on the section ring of the polarisation. This establishes a certain metric relation between test configurations, filtrations and geodesic rays in the space of Kahler metrics. - November 14

Aaron Kennon (UCSB)

*Title: Progress towards long-time existence and convergence of geometric flows of G2-structures*

Abstract: A primary goal motivating the study of geometric flows of G2-structures is to better understand which 7-manifolds admit certain types of these metrics. Of particular interest are the cases of G2-holonomy metrics and nearly-parallel G2-structures, both of which are intricately related to broader themes in differential geometry. I will survey what is known for specific promising flows of G2-structures, what would be desirable to prove, and the relevance of some of my work on the Laplacian flow and Laplacian coflow specifically to the existence of G2-holonomy metrics and nearly-parallel G2-structures, respectively. - November 28, ONLINE

Slawomir Dinew (Krakow)

*Title: Calabi-Yau equations on hypercomplex manifolds*

Abstract: Given the spectacular success of complex geometry it is tempting to try to generalize what is possible over quaternionic variables. As it turns out the notion of a quaternionic manifold has to be different in order to have rich theory. In the talk we shall briefly describe the "right" notion -that is the hypercomplex manifolds and various special cases. Then we shall describe the quaternionic analogue of the Calabi-Yau equation and discuss its solvability in special cases. - December 5

Henri Guenancia (CNRS, Toulouse)

*Title: Diameters of compact Kahler manifolds*

Abstract: Given a compact Kahler manifold (X, omega), I'll explain how one can quantitatively bound its diameter solely in terms of the volume form attached to \omega. The results partially generalize earlier results by Fu-Guo-Song, Y. Li and Guo-Phong-Song-Sturm and rely only on complex analytic methods (and don't involve riemannian geometry arguments). If time permits, I'll discuss how one could generalize those estimates in the case of singular varieties. This is based on joint work with V. Guedj and A. Zeriahi. - January 30

Zhenhua Liu (Princeton)

*Title: General behavior of area-minimizing subvarieties*

Abstract: We will review some recent progress on the general geometric behavior of homologically area-minimizing subvarieties, namely, objects that minimize area with respect to homologous competitors. They are prevalent in geometry, for instance, as holomorphic subvarieties of a Kahler manifold, or as special Lagrangians on a Calabi-Yau, etc. A fine understanding of the geometric structure of homological area-minimizers can give far-reaching consequences for related problems. Camillo De Lellis and his collaborators have proven that area-minimizing integral currents have codimension two rectifiable singular sets. A pressing next question is what one can say about the geometric behavior of area-minimizing currents beyond this. Almost all known examples and results point towards that area-minimizing subvarieties are subanalytic, generically smooth, and calibrated. It is natural to ask if these hold in general. In this direction, we prove that all of these properties thought to be true generally and proven to be true in special cases are totally false in general. We prove that area-minimizing subvarieties can have fractal singular sets. Smoothable singularities are non-generic. Calibrated area minimizers are non-generic. Consequently, we answer several conjectures of Frederick J. Almgren Jr., Frank Morgan, and Brian White from the 1980s. - March 5

Bin Guo (Rutgers)

*Title: TBA*

Abstract: TBA - March 12

Nick McCleerey (Purdue)

*Title: Local Lelong Numbers for m-Subharmonic Functions*

Abstract: We discuss work in progress on defining a local notion of the Lelong number of an m-subharmonic function along a complex submanifold. We then outline an application of our definition to some singularity-type envelopes. - March 13, special time and location at JHU, joint with Hopkins-Maryland geometry seminar

Hamid Hezari (UC Irvine)

*Title: The inverse spectral problem for ellipses*

Abstract: This talk is about Kac's inverse problem from 1966: "Can one hear the shape of a drum?" The question asks whether the frequencies of vibration of a bounded domain determine the shape of the domain. First we present a quick survey on the known results. Then we discuss the key connection between eigenvalues of the Laplacian and the dynamics of the billiard, which is governed by the so called "Poisson Summation Formula". Finally we discuss our main theorem that "one can hear the shape of nearly circular ellipses". This was a joint work with Steve Zelditch (9/13/1953-9/11/2022). - March 14, special \location: MTH 2300

Oliver Edtmair (UC Berkeley)

*Title: Symplectic capacities of convex domains*

Abstract: TBA - March 19, Spring break, no talk
- March 26, Tamas is away
- April 4, 3:30 PM, MATH1311

Gautam Iyer

*Title: One example of Residual Diffusivity*

Abstract: TBA - April 9, Tamas is away
- April 15, 2 PM, MATH3206, joint between Hopkins-Maryland geometry seminar and Alg Geom Seminar

Ravi Vakil (Stanford)

*Title: TBA*

Abstract: TBA - April 30

Yanir Rubinstein (UMD)

*Title: Tian's stabilization problem - algebraic meets complex & convex geometry*

Abstract: Coercivity thresholds are a central theme in geometry. They appear classically in the Yamabe problem (constant scalar curvature in a conformal class), in the Nirenberg problem (prescribed curvature on the 2-sphere), and in numerous problems on determining best constants in Sobolev embeddings and related functionals inequalities. In 1980's Aubin and Tian introduced the first such thresholds in the Kahler-Einstein problem and their study has been a central and still very active field. In 1988 Tian observed that these thresholds have quantum versions and he posed the so-called Stabilization Problem: do the equivariant quantum thresholds become constant (and hence equal to the classical thresholds)? - May 7

Huai-Dong Cao (Lehigh)

*Title: Geometry of four-dimensional Ricci solitons with (half) nonnegative isotropic curvature*

Abstract: Ricci solitons, introduced by R. Hamilton in the mid-80s, are self-similar solutions to the Ricci flow and natural extensions of Einstein manifolds. They often arise as singularity models and hence play a significant role in the Ricci flow. In this talk, I will present some recent progress on classifications of 4-dimensional gradient Ricci solitons with nonnegative (or half nonnegative) isotropic curvature. This talk is based on my joint work with Junming Xie - May 14 (Brinn Center 10:30 AM, joint with conference "SCV, Complex Geometry and related PDEs")

Antonio Trusiani (Chalmers)

*Title: Singular cscK metrics on smoothable varieties*

Abstract: We extend the notion of cscK metrics to singular varieties. We establish the existence of these canonical metrics on Q-Gorenstein smoothable klt varieties when the Mabuchi functional is coercive, these arise as a limit of cscK metrics on close-by fibres. The proof relies on developing a novel strong topology of pluripotential theory in families and establishing uniform estimates for cscK metrics. A key point is the lower semicontinuity of the coercivity threshold of Mabuchi functional along degenerate families of normal compact Kahler varieties with klt singularities. The latter suggests the openness of (uniform) K-stability for general polarized families of normal projective varieties. This is a joint work with Chung-Ming Pan and Tat Dat To

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