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.
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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 is work in progress with Joana Cirici.
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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.
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10/13:
Prof. Richard Hind (U. Notre Dame)
Title: Symplectic geometry of concave domains
Abstract: A concave domain will be a subset of 4 dimensional space which is invariant under the standard torus action and whose moment image is the region under the graph of a convex function. We investigate these domains and symplectic embeddings between them.
If the moment image has finite area then our domain is symplectomorphic to a bounded domain, although not necessarily one with smooth boundary, and we obtain lower bounds on the Minkowski dimension in terms of the decay rate of the convex function. More generally, this decay rate obstructs volume preserving symplectic embeddings, and our concave domains provide the first examples of symplectic manifolds without packing stability.
Our main tool will be the Embedded Contact Homology capacities, and in particular their subleading asymptotics. This is joint work with Dan Cristofaro-Gardiner.
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10/20: Prof. Mahmoud Zeinalian (Lehman, CUNY)
Title: The Chern character after Toledo-Tong and Green
Abstract: I will review some of the basic machinery used in a formulating characteristic classes for coherent sheaves on complex manifolds. The main ideas go back to the fundamental work of Toledo and Tong in the 70s. A natural extension of their ideas leads to defining these invariants for higher stacks. I will showcase some of the main tools and concepts without methodically entering the subject of higher stacks, making the talk appealing to classical differential geometers. This is based on joint works with T. Tradler, M. Miller, C. Glass, and T. Hosgood.
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10/27: Nikita Klemyatin (Columbia U.)
Title: The compactness theorems for Type IIA and Type IIB flows
Abstract: In this talk, we discuss two geometric flows that arise naturally in physics. One of them is the flow of almost complex structures on a symplectic 6-manifold with a nondegenerate 3-form, and the other is the flow of conformally balanced metrics on a non-Kahler Calabi-Yau manifold. We explain the background geometry, the properties of these flows, and what they have in common. Also, we state, and, if time permits, prove the compactness theorems for these flows, which are similar to Hamilton's compactness theorem for the Ricci flow.
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11/3: Prof. Kevin Sackel (UMass Amherst)
Title: Exact submanifolds and deformations of the de Rham complex
Abstract: Suppose M is a manifold with a chosen real cohomology
class c. By an exact submanifold for the pair (M,c), we mean a
submanifold X of M such that the restriction of c to X is zero. We may
ask whether there exists an exact submanifold representing a given
integral homology class Z (of M). For each such Z, we describe an obstruction theory that can be built by deforming the de Rham complex. The obstructions are completely computable and relate to higher Massey products. For applications, we provide some explicit computations which demonstrate the rarity of exact hypersurfaces in symplectic manifolds.
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11/10: Prof. Anna Fino (Florida Int. U.)
Title: Balanced Hermitian metrics
Abstract: A Hermitian metric on a complex manifold is called balanced if its fundamental form is co-closed.
In the talk I will give a general overview about balanced metrics and I will present some results in relation to geometric flows and the Hull-Strominger system.
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11/17: Prof. David Pham (QCC, CUNY)
Title: Left-invariant Hermitian connections on Lie groups
Abstract: In this talk, we will discuss some recent work related to left-invariant Hermitian connections on Lie groups with almost Hermitian structures. In the first part of the talk, we will give a general review of Hermitian connections on arbitrary almost Hermitian manifolds where we will derive the general torsion formula for a Hermitian connection. From this, we also derive the torsion formula for Gauduchon connections. (Our formulas differ from those of Gauduchon but are shown to be equivalent.) After the review, we focus on the special case of left-invariant Hermitian connections on Lie groups equipped with left-invariant almost Hermitian structures. In this setting, we derive explicit formula for the components of the torsion for every left-invariant Hermitian connection with respect to a convenient frame of left-invariant vector fields on the Lie group. As the resulting formulas are rather complicated, we apply them to almost Hermitian structures where the almost complex structure is totally real in the sense of Cleyton et al. In this particular case, the torsion formula simplifies considerably and some curvature results are obtained. Surprisingly, strong Kahler geometry with torsion makes an appearance in this non-integrable setting. This is joint work with Fei Ye.
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11/24:
no meeting (Thanksgiving)
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12/1: no meeting scheduled
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12/8: Abdellah Lahdili (Universite de Quebec a Montreal)
Title: The Einstein-Hilbert functional and K-stability
Abstract: In this talk, I will discuss a correspondence between the problem of finding a constant scalar curvature Kahler metric (cscK for short)
on a polarized complex manifold and the CR-Yamabe problem on the associated circle bundle. I will then use a CR version of the Einstein-Hilbert functional
to show K-semistability of polarised complex manifolds admitting cscK metrics. This is a joint work with Eveline Legendre and Carlo Scarpa.
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