Differential Geometry, Topology,
and special structures

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 Spring 2023 DGT Seminar Spring 2022, DGT Seminar Fall 2022
Here is a link to the formerly named seminar of similar scope: ACG seminar

Fall 2023: (all meetings in person)

9/1: Prof. Mehdi Lejmi (BCC, CUNY)
Title: Generalized almost-Kahler Ricci solitons

Abstract: In this talk, we introduce generalized almost-Kahler solitons metrics. In the symplectic Fano case, they are obstructions to the existence of constant Hermitian scalar curvature almost-Kahler metrics. In this talk, we will discuss their deformations, existence and construct some examples. This is a joint work with Michael Albanese and Giuseppe Barbaro.

9/8: no meeting this day

9/15: CUNY is closed this day

9/22: Prof. Teng Fei (Rutgers)
Title: Recent Progress on Type IIA Flow

Abstract: Motivated by Type IIA superstring theory, the Type IIA flow is a weakly parabolic PDE of 3-forms defined on 6-dimensional symplectic Calabi-Yau manifolds. In this talk, we explore various geometric and analytic aspects of the Type IIA flow. In particular, we show that the Type IIA flow is dynamically stable, and we apply this result to prove the stability of Kahler Calabi-Yau manifolds under symplectic deformations. This is based on joint work with Phong, Picard, and Zhang.

9/29: Prof. Scott Wilson (CUNY)
Title: A higher-homotopical BV-structure on the differential forms of (almost) complex manifolds with compatible metric.

Abstract: In 1985 Koszul showed that the differential forms of a symplectic manifold has an additional second order operator; part of what is now called a differential BV-algebra. Subsequent work by Getzler, Barannivok-Kontseivich, and Manin describe this structure as a (genus zero) cohomological field theory on the de Rham cohomology, i.e. an action of the compactified moduli space of (genus zero) Riemann surfaces with marked points. Such structures, also known as (formal) Frobenius manifolds, or hypercommutative algebras, have numerous connections with the A-model and mirror symmetry.
In this talk I'll explain a natural generalization of this to (almost) complex manifolds with compatible metrics, using a higher-homotopical notion of BV-algebras. This relies on generalizations of the Kahler identities to the complex-Hermitian case (due to Demailly) and to the almost-complex Hermitian case (recently obtained by Fernandez and Hosmer). My goal is to explain the setup, establish the existence of the higher-homotopy BV-structure, and give some explicit examples of complex manifolds where these higher operations are non-zero.
This constitutes work in progress with Joana Cirici.

10/6: Shuang Liang (Columbia U.)
Title: Continuity equation on Hermitian manifolds and its application

Abstract: I will discuss the continuity equation on Hermitian manifolds, introduced by La Nave-Tian and extended to Hermitian metrics by Sherman-Weinkove. We use the Calabi estimates to show that on a compact complex manifold, the Chern scalar curvature of a solution must blow up at a finite-time singularity. Additionally, starting from certain classes of initial data on Oeljeklaus-Toma manifolds, we prove Gromov-Hausdorff and smooth convergence of the metric to a particular non-negative (1,1)-form. This is joint work with Xi Sisi Shen and Kevin Smith.

10/13: Prof. Richard Hind (U. Notre Dame)
Title: TBA


10/20: Prof. Mahmoud Zeinalian (Lehman, CUNY)
Title: TBA


10/27: Nikita Klemyatin (Columbia U.)
Title: TBA


11/3: Prof. Kevin Sackel (UMass Amherst)
Title: TBA


11/10: Prof. Anna Fino (Florida Int. U.)
Title: TBA


11/17: Prof. David Pham (QCC, CUNY)
Title: TBA


11/24: no meeting (Thanksgiving)


12/1: TBA


12/8: TBA

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