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: [Cancelled or postponed] Prof. Andrei Caldararu (U. Wisconsin)
Title: A gentle introduction to Categorical Enumerative Invariants (CEI)

Abstract: Mirror symmetry comes in two distinct flavors: enumerative (matching counts of curves on a Calabi-Yau manifold with invariants obtained from a variation of Hodge structures) and homological (stated as an equivalence between a Fukaya category and a family of derived categories). In 1994 Kontsevich predicted that the former incarnation of mirror symmetry should tautologically follow from the latter: there should exist enumerative invariants associated to a Calabi-Yau category, which specialize to curve counts for the Fukaya category, and to the invariants obtained from a variations of Hodge structures for the family of derived categories. Kontsevich's prediction has come closer to reality in recent years, following the introduction of CEI in works of Costello (2005) and Caldararu-Costello-Tu (2020). I will give a general introduction to the subject and discuss open problems and future directions of study. No previous knowledge of mirror symmetry, symplectic geometry, or Calabi-Yau geometry required.


10/25: Prof. Ethan Addison (Stony Brook U.)
Title: A Flowing Construction to Generalize Poincare-Type Metrics

Abstract: Poincare-type metrics are a flavor of cusp metrics on the complement of a divisor in a compact Kahler manifold exhibiting many friendly geometric properties. Yet, as shown by H. Auvray in the context of Calabi's extremal metrics, they have certain limitations owing to their sensitivity to the geometry of the ends. We introduce a construction called gnarling which augments Poincare-type metrics by incorporating certain holomorphic flows along the divisor. After outlining several aspects of these gnarled metrics, including a key growth estimate, we show their utility in perturbing classes of cscK Poincare-type metrics.


11/1: Prof. Lucia Martin-Merchan (U. Waterloo)
Title: Compact holonomy G_2 manifolds need not be formal

Abstract: The connection between holonomy and rational homotopy theory was discovered by Deligne, Griffiths, Morgan, and Sullivan, who proved that compact Kahler manifolds are formal. This led to the conjecture that compact manifolds with special and exceptional holonomy should also be formal. In this talk, we discuss an example from arXiv:2409.04362 that disproves the conjecture for holonomy G_2 manifolds. The construction uses the orbifold resolution techniques developed by Joyce and Karigiannis. Non-formality is a consequence of the configuration of the singular locus.


11/8: 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.


11/15: Prof. Guangbo Xu (Rutgers U.)
Title: Quantum Kirwan map for symplectic quotients

Abstract: Various cohomological operations on symplectic manifolds can be "quantum deformed" using pseudoholomorphic curves or similar objects. By deforming the cup product, one obtains the quantum cohomology. Suppose a Lie group acts on a symplectic manifold V resulting in a smooth symplectic quotient X, then under certain monotonicity condition, we prove that the Kirwan map, which sends equivariant cohomology classes of V to the cohomology of X, can also be deformed by gauge theoretic objects called "vortices". This generalizes the result of Woodward for the algebraic case and solved a conjecture of Salamon under the monotonicity assumption. If time permits, I will talk about the analogous situation for Steenrod operations, which can be regarded as an "equivariant extension" of cup product.


11/22: Prof. Qi Yao (Stony Brook U.)
Title: The Asymptotic behavior of the Solution to Homogeneous Complex Monge-Ampere Equation on ALE Kahler manifolds

Abstract: Initiated by Mabuchi, Semmes and Donaldson, homogeneous complex Monge-Ampere (HCMA) equations become a central topic in understanding uniqueness and existence of canonical metrics in Kahler classes. Under the setting of asymptotically locally Euclidean (ALE) Kahler manifolds, one of the main difficulties is the asymptotic behaviors of solutions to HCMA equations. In this talk, I will give an introduction to canonical metric problems under the setting of ALE Kahler manifolds and discuss the recent progress on this problem. I will present a new result on the asymptotic behavior of HCMA solutions and outline the technical methods used in proving it. (This result will be available on arXiv shortly.)


12/6: Prof. Janet Talvacchia (Swarthmore College)
Title: Analogs of coKahler and Sasakian Structures in generalized geometry

Abstract: I'll discuss under what conditions one can form generalized Kahler structures from products of odd dimensional generalized contact structures.


12/13: informal meeting
Title: students and faculty welcome






Powered by MathJax