Spring 2025: (all meetings in person)
|
1/31: Prof. Teng Fei (Rutgers)
Title: Degeneration of Calabi-Yau 3-folds and 3-forms.
Abstract: We study the geometries associated to various 3-forms on a symplectic 6-manifold of different orbital types. As an application, we demonstrate how this can be used to find Lagrangian foliations and other geometric structures of interest arising from certain degeneration of Calabi-Yau 3-folds.
|
2/7: Prof. Mehdi Lejmi (CUNY)
Title: HKT balanced manifolds.
Abstract: In this talk, we discuss some properties of HKT (hyperkahler with torsion) balanced manifolds. First, we will study the Lie algebra of hyperholomorphic vector fields on such manifolds. Then, we will prove a deformation result, namely we prove that the HKT balanced condition is an open condition in the HKT cone. This is a joint work with Giovanni Gentili.
|
2/14: Prof. Barry Minemyer (Commonwealth University)
Title: Complex Hyperbolic Gromov-Thurston Metrics.
Abstract: : In 1987 Gromov and Thurston developed the first Riemannian manifolds that are not homotopy equivalent to a hyperbolic manifold but admit a Riemannian metric that is ϵ-pinched for any given ϵ>0. The manifolds that they construct are branched covers of hyperbolic manifolds, and to construct the metric they perform a sort of "geometric surgery" about the ramification locus. In 2022 Stover and Toledo proved the existence of similar branched cover manifolds built out of complex hyperbolic manifolds, and via a result of Zheng these manifolds admit a negatively curved Kahler metric. In this talk we will discuss how to construct a (not Kahler) Riemanain metric on these Stover-Toledo manifolds which is ϵ-close to being 1/4-pinched for any prescribed ϵ>0.
|
2/21: Prof. Christopher Seaton (Skidmore College)
Title: Distinguishing between singular symplectic quotients using invariant theory.
Abstract: Given a Hamiltonian action of a compact Lie group on a symplectic manifold with equivariant moment map, the Marsden-Weinstein symplectic quotient is defined to be the quotient of the zero-fiber of the moment map by the group action. In this case, zero is assumed to be a regular value of the moment map so that the zero-fiber is a manifold on which the group acts freely and properly. When zero is not a regular value, the zero-fiber of the moment map is no longer a manifold, and its quotient by the group action is a singular space. By the results of Sjamaar-Lerman, such a singular symplectic quotient has the structure of a symplectic stratified space, a disjoint union of smooth symplectic manifolds, as well as a Poisson algebra of smooth functions.
The local structure of a singular symplectic quotient can be understood by considering the case of a unitary representation of a compact Lie group, thought of as a real symplectic vector space. In this case, there is a unique homogeneous quadratic moment map for which zero is generally a singular value. There are many examples of non-isomorphic groups and representations that yield singular symplectic quotients that are symplectomorphic via a symplectomorphism that preserves their algebraic structures. We will discuss the use of the polynomial invariants of the representation and topological methods to study the smooth structures of symplectic quotients, identify symplectomorphic examples, and distinguish between non-symplectomorphic cases.
|
2/28: Prof. Gueo Grantcharov (Florida International U.)
Title: Isotropic Killing vector fields and structures on complex surfaces
Abstract: In a 4-dimensional vector space with scalar product of signature (2,2) and fixed orientation, two independent vectors spanning a maximal isotropic (null) plane determine a canonical action of the para-quaternioins. We noticed that on an oriented 4-manifold with such pseudo-Riemannian metric, existence of two isotropic (null) Killing vector fields leads to integrability of the induced structure - called para-hypercomplex, and the metric is anti-selfdual. Using the Kodaira classification one can describe the topology of the underlying 4-manifold in the compact case. In the talk examples of such structures on several of the 4-manifolds will be provided and some restrictions for a compact complex surface to admit split signature Hermitian metric with one non-vanishing null Killing vector field will be established. The talk is based on a joint work with J. Davidov and O. Mushkarov
|
2/28: Prof. Charles Cifarelli (Stony Brook University)
Title: Explicit complete Calabi-Yau metrics and Kähler-Ricci solitons.
Abstract: Since Calabi's original paper, the Calabi Ansatz has been central for constructions in Kähler geometry. Calabi himself used it to construct complete Ricci-flat metrics on the total space of the canonical bundle of a Kähler-Einstein Fano manifold B, generalizing some well-known examples coming from physics. Over the years, work of Koiso, Cao, Feldman-Ilmanen-Knopf, Futaki-Wang, Chi Li, and others have shown that the Calabi Ansatz can be used to produce complete Kähler-Ricci solitons, important singularity models for the Kähler-Ricci flow, on certain line bundles over B. In this talk, I'll explain a generalization of these results to the total space of some higher-rank direct-sum vector bundles over B. In our case the Calabi Ansatz is not suitable, and we instead use the theory of hamiltonian 2-forms, introduced by Apostolov-Calderbank-Gauduchon-Tønneson-Friedman. The construction produces new examples of complete shrinkers, steadies, and Calabi-Yau metrics, as well as recovering some known ones.
|
3/7: Prof. Kazuo Akutagawa (Chuo University, Tokyo)
Title: Harmonic maps from the product of hyperbolic spaces
to hyperbolic spaces.
Abstract: In this talk, we will consider the asymptotic Dirichlet problem
for harmonic maps from the product $\mathbb{H}^{m_1} \times \mathbb{H}^{m_2}$ of two hyperbolic spaces
to hyperbolic spaces.
It remarks that $\mathbb{H}^{m_1} \times \mathbb{H}^{m_2}$ is a higher rank symmetric space of noncompact type.
We first show uniqueness and non-existence results, particularly the existence of such harmonic maps (with some natural conditions)
implies that it must be $m _1 = m_2 = 2$.
We also show an existence result for harmonic maps from $\mathbb{H}^2 \times \mathbb{H}^2$
to hyperbolic spaces.
This is a joint work with Yoshihiko Matsumoto.
|
3/14: Prof. Joshua Jordan (Iowa)
Title: Generalized Geometry and Pluriclosed Flow
Abstract: Pluriclosed flow is a Hermitian curvature flow introduced by Streets and Tian in 2010 which preserves the pluriclosed condition along the flow. I will discuss some joint work with Mario Garcia-Fernandez and Jeff Streets using the framework of generalized complex geometry, as introduced by Hitchin and Gualtieri in the early '00s, to obtain some results on long-time existence and convergence for the flow and obstructions to the existence of Bismut-Ricci flat metrics. If there is time, I will discuss some more recent work with Hao Fang solving a related problem on a class of generalized Kahler surfaces.
|
3/21: Prof. Dror Varolin (Stony Brook University)
Title: Deformation of Bergman Spaces
Abstract: To a holomorphic vector bundle $E \to X$ with Hermitian metric $\mathfrak{h}$ over a complex $n$-manifold $X$ one can associated its so-called
$\textit{Bergman space}$:
\[
\mathscr{H} (\mathfrak{h}) := \left \{ f \in H^0(X, \mathcal{O} _X(K_X \otimes E)) \ ;\ \int _X |f|^2_{\mathfrak{h}} < +\infty \right \}.
\]
A smooth holomorphic family of Hermitian vector bundles thus yields a family of Hilbert spaces. This family of Hilbert spaces is similar to a holomorphic vector bundle in certain ways, but need not be locally trivial. Nevertheless, one can associate to such an object a notion of curvature which recovers the curvature of the Chern connection of the $L^2$ metric when the family of Hilbert spaces is a holomorphic Hilbert bundle. In this talk we shall focus on the case in which the holomorphic family of complex manifolds is smooth, proper and K\"ahler. We shall explain this construction, prove a positivity-of-curvature theorem, and discuss some of its consequences.
|
3/28: Prof. Scott Wilson (CUNY)
Title: Solvmanifolds, Massey products, and A-infinity structures.
Abstract: I will survey the construction of some explicit manifolds with nontrivial triple and quadruple Massey products. Particularly notable is a recent example of a 6-dimensional symplectic manifold that does not admit a Kahler structure, due the presence of a nontrivial quadruple Massey product. With a view towards understanding the naturality of higher Massey products in the category of A-infinity algebras, I'll also give a gentle introduction to A-infinity structures and A-infinity morphisms, and indicate how these low order Massey products are understood in this formalism. This is preparatory material for ongoing work with Kate Poirier and Thomas Tradler.
|
4/4: Prof. David Pham (Queensborough, CUNY)
Title: Some Applications of Samelson's construction in Complex Geometry
Abstract:
In 1954, Samelson showed that every even dimensional compact Lie group admits a left-
invariant integrable almost complex structure. Almost simultaneously, Wang found
sufficient and necessary conditions for when an even dimensional homogenous manifold
admits an invariant integrable almost complex structure. In this talk, we
present some recent applications of Samelson’s construction in the setting of SKT geometry.
Specifically, we construct a family of left-invariant SKT metrics on the exceptional Lie group G2
where G2 is equipped with a Samelson complex structure. This work is motivated by a 2023 paper of Fino
and Grantcharov where (among other things) they carried out a similar calculation for the Lie
group SO(9). In addition, we also show that Samelson’s construction can be suitably modified to
yield a left-invariant integrable almost complex structure on the tangent Lie group of any compact Lie group
(odd or even dimensional). We then move from Lie groups to the more general setting of homogeneous
reductive manifolds which are equipped with invariant complex structures. We review the correspondence between
invariant structures on homogeneous manifolds and their counterparts at the Lie algebra level. We give a
number of actual examples of homogeneous Hermitian manifolds (both Kahler and non-Kahler). We then consider the
problem of calculating and extracting information from the curvature of Gauduchon connections on a homogeneous
Hermitian manifold. This part of the talk is a work in progress.
|
4/11: no meeting
|
4/18: no meeting- Spring break
4/25: tba
Title:
Abstract:
|
5/2: Prof. Ljudmila Kamenova (Stony Brook University)
Title:
Abstract:
| |
