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