Informal Geometric Analysis Archives for Fall 2024 to Spring 2025
Obstruction bundle gluing
When: Tue, September 26, 2023 - 3:30pmWhere: Kirwan Hall 1313
Speaker: Michael Hutchings (UC Berkeley) -
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.
The eigenvalue problem of complex Hessian operators
When: Tue, October 10, 2023 - 3:30pmWhere: Kirwan Hall 1313
Speaker: Yaxiong Liu () -
Abstract: 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.
On the geometry at infinity on the space of Kahler potentials and submultiplicative filtrations
When: Tue, November 7, 2023 - 3:30pmWhere: Kirwan Hall 1313
Speaker: Siarhei Finski (CNRS) -
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.
Progress towards long-time existence and convergence of geometric flows of G2-structures
When: Tue, November 14, 2023 - 3:30pmWhere: Kirwan Hall 1313
Speaker: Aaron Kennon (UCSB) -
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.
Calabi-Yau equations on hypercomplex manifolds
When: Tue, November 28, 2023 - 3:30pmWhere: https://umd.zoom.us/j/4131293393
Speaker: Slawomir Dinew (Jagellonian University, Krakow) -
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.
Diameters of compact Kahler manifolds
When: Tue, December 5, 2023 - 3:30pmWhere: Kirwan Hall 1313
Speaker: Henri Guenancia (CNRS, Toulouse) -
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.
General behavior of area-minimizing subvarieties
When: Tue, January 30, 2024 - 3:30pmWhere: Zoom
Zhenhua Liu (Princeton)
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.
zoom link: https://umd.zoom.us/j/4131293393
Geometric estimates in Kahler geometry
When: Tue, March 5, 2024 - 3:30pmWhere: Kirwan Hall 1313
Speaker: Bin Guo (Rutgers) -
Abstract: We will discuss the role of complex Monge-Ampere equations as auxiliary equations in deriving sharp analytic and geometric estimates in Kahler geometry. By studying Green's functions, we will explore how to derive estimates for diameters and establish uniform Sobolev inequalities on Kähler manifolds, which depend only on entropy of the volume form and are independent of the lower bound of the Ricci curvature. This talk is based on joint works with D. H. Phong, J. Song, and J. Sturm.
Local Lelong Numbers for m-Subharmonic Functions
When: Tue, March 12, 2024 - 3:30pmWhere: Kirwan Hall 1313
Speaker: Nick McCleerey (Purdue University) -
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.
Symplectic capacities of convex domains
When: Thu, March 14, 2024 - 3:30pmWhere: Kirwal Hall 2300
Speaker: Oliver Edtmair (UC Berkeley) -
Ribbon knots, slice knots and quotients of knot groups
When: Thu, March 28, 2024 - 3:30pmWhere: Kirwan Hall 2300
Speaker: Alexandra Kjuchukova (Notre Dame) -
Abstract: A knot K in $S^3$ is slice if it bounds a smooth disk in the four-ball; and is ribbon if this disk can be chosen to have no local maxima with respect to the radial height function on B^4. An old question of Fox asks if every slice knot is ribbon.
Let K be a knot equipped with a dihedral quotient of \pi_1(S^3\K). I'll explain how to extract an invariant of K from this data, using the signature of a certain 4-manifold which is a branched cover of B^4 over a properly embedded surface F with boundary K. I'll describe how this invariant can obstruct K from bounding, in alternate cases, either ribbon or slice disks in the four-ball. And, I'll give a necessary and sufficient condition for the existence of such a surface.
Tian's stabilization problem - algebraic meets complex & convex geometry
When: Tue, April 30, 2024 - 3:30pmWhere: Kirwan Hall 1313
Speaker: Yanir Rubinstein (UMD) -
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)? Cheltsov conjectured that these invariants coincide with the algbero-geometric log canonical thresholds (lct), and this was verified by Demailly (2008). The best result so far has been
Birkar's theorem (2019) that shows that the quantum lcts
are constant along a subsequence in the absence of group actions.
Over the past 20 years, previous works have claimed a solution to
Tian's problem in the toric case, but it turns out that they assume
without justification monotonicity of these invariants in the
quantization parameter.
In joint work with C. Jin we offer a new approach
and solve Tian's problem in the toric case. Surprisingly, the
equivariant lcts are constant already from the first quantum level.
For more general Grassmannian lcts we offer counterexamples to stabilization and determine when it holds. The key new ideas are understanding the effect of finite group actions on these invariants,
and relating these thresholds to support and gauge functions from
convex geometry. Time permitting I will discuss extensions and generalizations to other invariants.
Geometry of four-dimensional Ricci solitons with (half) nonnegative isotropic curvature
When: Tue, May 7, 2024 - 3:30pmWhere: Kirwan Hall 1313
Speaker: Hui Dong Cao (Lehigh) -
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.
Singular cscK metrics on smoothable varieties
When: Tue, May 14, 2024 - 10:30amWhere: Brinn Center
Speaker: Antonio Trusiani (Chalmers University) -
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 Kähler 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 Tô.
The Aleksandrov--Fenchel inequality and applications
When: Thu, June 13, 2024 - 3:30pmWhere: Kirwan Hall 1313
Speaker: Vasanth Pidaparthy (University of Maryland) -
Abstract: We will present the Alexandrov--Fenchel inequality along with applications to Brunn--Minkovski theory.