Differential Geometry, Topology,
and special structures
Seminar

Department of Mathematics
The Graduate Center of CUNY

Fridays 11:45am - 1:45pm
Location: Room 4419
Organizers: Luis Fernandez, Mehdi Lejmi, Scott Wilson, Marco Castronovo

Scope: This seminar is devoted to geometric structures on manifolds, and their topological properties. We are particularly interested in (almost) complex manifolds, symplectic topology and geometry, (almost) Kahler geometry, and spin^c geometry, as well as 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 2024, 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

8/30: no meeting this day


9/6: no meeting this day


9/13: Prof. Marco Castronovo (Columbia U.)
Title: Introduction to symplectic topology.

Abstract: This talk will be an overview of contemporary symplectic topology, geared towards graduate students. I will start with some basic definitions and examples, then list a few representative questions. Pseudo-holomorphic curves are the main tool used to study such questions. I will explain what they are, and how they can be used to deform and categorify the invariants of classical algebraic topology.


9/20: Prof. Semon Rezchikov (Princeton U.)
Title: Cyclotomic Structure in Symplectic Topology

Abstract: Symplectic cohomology is a fundamental invariant of a symplectic manifold M with contact type boundary that is defined in terms of dynamical information and counts of pseudoholomorphic genus zero curves, and carries algebraic structures that parallel the algebraic structures on the Hochschild (co)homology of the Fukaya category of M. We show, under natural topological assumptions, that the symplectic cohomology is the homology of a genuine p-cyclotomic spectrum in the sense of Nikolaus-Scholze. The cyclotomic structure arises geometrically from the map which sends loops in M to their p-fold iterates. The data of this refinement is expected to produce many new algebraic structures of an arithmetic nature on symplectic cohomology, analogously to the way that prismatic cohomology refines the de Rham cohomology of a variety. The methods involved use ideas from genuine equivariant stable homotopy theory to avoid the issues associated to the failure of equivariant transversality. If time permits, we will discuss some expected connections to string topology.


9/27: Sebastian Haney (U. Columbia)
Title: Infinity inner products and open Gromov-Witten invariants.

Abstract: The open Gromov-Witten (OGW) potential is a function defined on the Maurer-Cartan space of a closed Lagrangian submanifold in a symplectic manifold with values in the Novikov ring. From the values of the OGW potential, one can extract open Gromov-Witten invariants, which count pseudoholomorphic disks with boundary on the Lagrangian. Existing definitions of the OGW potential only allow for the construction of OGW invariants with values in the real or complex numbers. In this talk, we will present a construction of the OGW potential which gives invariants valued in any field of characteristic zero. The main algebraic input for our construction is an infinity inner product, which comes from a proper Calabi-Yau structure on the Fukaya category. If time permits, we will also discuss a partial extension of our results that give OGW invariants valued in fields of positive characteristic, and connections to homological mirror symmetry.


10/4: no meeting (no classes at CUNY)


10/11: no meeting (no classes at CUNY)


10/18: Prof. Andrei Caldararu (U. Wisconsin)
Title: TBA

Abstract: TBA


10/25: Prof. Ethan Addison (Stony Brook U.)
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11/1: Prof. Lucia Martin-Merchan (U. Waterloo)
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11/8:
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11/15: Prof. Guangbo Xu (Rutgers U.)
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11/22: Prof. Qi Yao (Stony Brook U.)
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12/6: Prof. Janet Talvacchia (Swarthmore College)
Title: TBA

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12/13:
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