Differential Geometry, Topology,
and special structures
Seminar

Department of Mathematics
The Graduate Center of CUNY

Fridays 11:45noon - 1:45pm
Location: Room 5417
Organizers: Luis Fernandez, Mehdi Lejmi, Scott Wilson

Scope: this seminar is devoted to geometric structures on manifolds, and topological properties of them. We are particularly interested in (almost) complex manifolds, symplectic topology and geometry, (almost) Kahler geometry, and spin^c geometry, as well the tools of algebraic topology and geometric analysis that have proven useful in studying such structures. The goal is expose faculty and students to present day research and foundational material in a relaxed format that includes traditional lectures with an actively engaged audience.
Here is a link to previous semesters: DGT Seminar Fall 2023, DGT Seminar Spring 2023 DGT Seminar Fall 2022, DGT Seminar Spring 2022
Here is a link to the formerly named seminar of similar scope: ACG seminar

1/26: no meeting this day


2/2: Prof. Scott Wilson (CUNY)
Title: On the algebraic structure of symplectic manifolds with Lagrangian subbundles.

Abstract: A symplectic manifold equipped with an integrable Lagrangian splitting of its tangent bundle inherits a tremendous amount of interesting structure on the de Rham complex. Among these include a bigrading, a Frolicher-type spectral sequence, two bigraded-component differential BV-alegbras, and induced actions of the homology of moduli space of genus zero curves on the de Rham cohomology. Various generalizations of all this hold if we drop the integrability assumptions (of the Lagrangians, or the symplectic structure) and in particular yield higher homotopical BV-algebras. I'll describe the general setup, and give computable examples where the cohomology operations are non-trivial. This story is similar (and perhaps mirror) to the complex-Hermitian discussion reported on last semester, and is ongoing work with Joana Cirici.


2/9: Prof. Marco Castronovo (Columbia U.)
Title: Decoupling Fukaya categories.

Abstract: A natural problem in symplectic topology is to classify Lagrangian submanifolds up to Hamiltonian isotopy. There is growing evidence that this is impossible, but one can hope to have a coarser classification by proving that finitely many Lagrangians generate the Fukaya category. I will illustrate some concrete examples where we know how to do this, some in which we don't, and a new technique called decoupling that could partially bridge the gap.


2/16: Prof. Spiro Karigiannis (U. Waterloo)
Title: A curious system of second order nonlinear PDEs for U(m)-structures on manifolds.

Abstract: Compact Kahler manifolds possess a number of remarkable properties, such as the Kahler identities, the del-delbar-lemma, and the relation between Betti numbers and Hodge numbers. I will discuss an attempt in progress to generalize some of these ideas to more general compact U(m)-manifolds, where we do not assume integrability of the almost complex structure nor closedness of the associated real (1,1)-form. I will present a system of second order nonlinear PDEs for such a structure, of which the Kahler structures form a trivial class of solutions. Any compact non-Kahler solutions to this second order system would have properties that are formally similar to the above-mentioned properties of compact Kahler manifolds, including relations between cohomological (albeit non-topological) data. This is work in progress with Xenia de la Ossa (Oxford) and Eirik Eik Svanes (Stavanger).


2/23: Prof. Jiahao Hu (CUNY)
Title: Introduction to spin, spin^c and spin^h

Abstract: I will introduce the notion of spin, spin^c and spin^h manifolds and survey some of their properties and applications in geometry and topology. We will see that they share a unifying theme in representation theory and index theory, but play very different roles in geometry.


3/1: Prof. Gueo Grantcharov (Florida Int. U.)
Title: Hermitian metrics on non-Kaehler manifolds admitting special holomorphic fibrations.

Abstract: I'll briefly review the existence of special Hermitian metrics - balanced, pluriclosed, and CYT, on two types of fibered complex spaces. The first type is the principal bundle with complex tori as fibers over Kaehler base, and the second is a suspension of Kaehler space over a complex torus. These types generalize the simplest nil- and solvmanifold examples. Then I'll focus on the second type and its relations to some open questions in Hermitian geometry.


3/8: Prof. Aleksandar Milivojevic (U. Waterloo)
Title: Formality and dominant maps

Abstract: In the mid 70's, Deligne-Griffiths-Morgan-Sullivan demonstrated a strong topological condition a closed manifold would have to satisfy if it were to carry a Kahler complex structure. Namely, the manifold would have to be formal, in the sense of its de Rham algebra of forms being weakly equivalent to its cohomology. In particular, there can be no non-trivial Massey products on a compact Kahler manifold. The salient underlying property of compact Kahler manifolds which implies formality is preserved under surjective holomorphic maps. It turns out that formality itself is preserved under non-zero degree continuous maps of spaces satisfying Poincare duality on their rational cohomology. I will explain the key components of this argument and show how one can then apply it to various situations; we recover several results that ensure formality in the presence of certain Riemannian metrics. This is joint work with Jonas Stelzig and Leopold Zoller.


3/15: Prof. Carlo Scarpa (Universite de Quebec a Montreal)
Title: TBA

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3/22: Spencer Cattalani (Stony Brook U.)
Title: TBA

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3/29: no meeting (no classes this day)


4/5: Prof. Tamas Darvas (U. Maryland)
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4/12: TBA
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4/19: Mathew George (The Ohio State University)
Title: TBA

Abstract: TBA


4/26: no meeting (Spring Break)


5/3: Prof. Yann Rollin (Nantes U.)
Title: TBA

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5/10: Prof. Yang Li (MIT)
Title: TBA

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