1/26: no meeting this day
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2/2: Prof. Scott Wilson (CUNY)
Title: On the algebraic structure of
symplectic manifolds with Lagrangian subbundles.
Abstract: A symplectic manifold equipped with an integrable
Lagrangian splitting of its tangent bundle inherits a tremendous amount
of interesting structure on the de Rham complex. Among these includes a bigrading,
a Frolicher-type spectral sequence, two bigraded-component
differential BV-alegbras, and induced actions of the homology of
moduli space of genus zero curves on the de Rham cohomology.
Various generalizations hold if we drop the integrability
assumptions (of the Lagrangians, or the symplectic structure)
and in particular yield higher homotopical BV-algebras. I'll describe the general
setup, and give computable examples where the cohomology
operations are non-trivial.
This story is similar
(and perhaps mirror) to the complex-Hermitian discussion
reported on last semester, and is ongoing work with Joana Cirici.
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2/9: Prof. Marco Castronovo (Columbia U.)
Title: Decoupling Fukaya categories.
Abstract: A natural problem in symplectic topology is to classify Lagrangian submanifolds up to Hamiltonian isotopy. There is growing evidence that this is impossible, but one can hope to have a coarser classification by proving that finitely many Lagrangians generate the Fukaya category. I will illustrate some concrete examples where we know how to do this, some in which we don't, and a new technique called decoupling that could partially bridge the gap.
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2/16: Prof. Spiro Karigiannis (U. Waterloo)
Title: A curious system of second order nonlinear PDEs for U(m)-structures on manifolds.
Abstract: Compact Kahler manifolds possess a number of remarkable properties, such as the Kahler identities, the del-delbar-lemma, and the relation between Betti numbers and Hodge numbers. I will discuss an attempt in progress to generalize some of these ideas to more general compact U(m)-manifolds, where we do not assume integrability of the almost complex structure nor closedness of the associated real (1,1)-form. I will present a system of second order nonlinear PDEs for such a structure, of which the Kahler structures form a trivial class of solutions. Any compact non-Kahler solutions to this second order system would have properties that are formally similar to the above-mentioned properties of compact Kahler manifolds, including relations between cohomological (albeit non-topological) data. This is work in progress with Xenia de la Ossa (Oxford) and Eirik Eik Svanes (Stavanger).
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2/23: Prof. Jiahao Hu (CUNY)
Title: Introduction to spin, spin^c and spin^h
Abstract: I will introduce the notion of spin, spin^c and spin^h manifolds and survey some of their properties and applications in geometry and topology. We will see that they share a unifying theme in representation theory and index theory, but play very different roles in geometry.
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3/1: Prof. Gueo Grantcharov (Florida Int. U.)
Title: Hermitian metrics on non-Kaehler manifolds admitting special holomorphic fibrations.
Abstract: I'll briefly review the existence of special Hermitian metrics - balanced, pluriclosed, and CYT, on two types of fibered complex spaces. The first type is the principal bundle with complex tori as fibers over Kaehler base, and the second is a suspension of Kaehler space over a complex torus. These types generalize the simplest nil- and solvmanifold examples. Then I'll focus on the second type and its relations to some open questions in Hermitian geometry.
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3/8: Prof. Aleksandar Milivojevic (U. Waterloo)
Title: Formality and dominant maps
Abstract: In the mid 70's, Deligne-Griffiths-Morgan-Sullivan demonstrated a strong topological condition a closed manifold would have to satisfy if it were to carry a Kahler complex structure. Namely, the manifold would have to be formal, in the sense of its de Rham algebra of forms being weakly equivalent to its cohomology. In particular, there can be no non-trivial Massey products on a compact Kahler manifold. The salient underlying property of compact Kahler manifolds which implies formality is preserved under surjective holomorphic maps. It turns out that formality itself is preserved under non-zero degree continuous maps of spaces satisfying Poincare duality on their rational cohomology. I will explain the key components of this argument and show how one can then apply it to various situations; we recover several results that ensure formality in the presence of certain Riemannian metrics. This is joint work with Jonas Stelzig and Leopold Zoller.
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3/15: Prof. Carlo Scarpa (Universite de Quebec a Montreal)
Title: Canonical representatives of complexified Kahler classes
Abstract: One of the classical problems in Differential Geometry is to fix a canonical choice of a Riemannian metric on any given manifold. This is usually done by imposing conditions on the curvature of the metric and other data (e.g. the conformal class of the metric), which are translated into PDEs for a local potential of the metric. In this talk, we will begin by giving an overview of the particular characteristics of this problem in the context of compact complex manifolds and Kahler metrics. We will then see how similar ideas can be fruitfully applied to a problem motivated by constructions appearing in mirror symmetry, namely the existence of canonical representatives for a complexified Kahler class on a compact complex manifold. I will explain why a natural choice of PDEs in this case is a particular coupling of the deformed Hermitian Yang-Mills equation and the constant scalar curvature equation. We will then see how to prove the existence of canonical representatives in some special cases and talk about the obstructions to the existence of these representatives. Based in part on joint work with Jacopo Stoppa.
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3/22: Spencer Cattalani (Stony Brook U.)
Title: Complex Cycles and Symplectic Topology
Abstract: Among all almost complex manifolds, those which are tamed by symplectic forms are particularly well studied. What geometric properties characterize this class of manifolds? That is, given an almost complex manifold, how can one tell whether it is tamed by a symplectic form? By a 1976 result of D. Sullivan, this question can be answered by studying complex cycles. I will explain what complex cycles are and their role in two recent results, which confirm speculations posed by M. Gromov in 2000 and 1985, respectively. The first is that an almost complex manifold admits a taming symplectic structure if and only if it satisfies a certain bound on the areas of coarsely holomorphic curves. The second is that an almost complex 4-manifold which has many pseudoholomorphic curves admits a taming symplectic structure. This leads to an almost complex analogue of D. McDuff's classification of rational symplectic 4-manifolds.
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3/29: no meeting (no classes this day)
4/5: Prof. Tamas Darvas (U. Maryland)
Title: The trace operator of quasi-plurisubharmonic functions on compact Kahler manifolds
Abstract: We introduce the trace operator for quasi-plurisubharmonic functions on compact Kahler manifolds, allowing to study the singularities of such functions along submanifolds where their generic Lelong numbers vanish. Using this construction we obtain novel Ohsawa-Takegoshi extension theorems and give applications to restricted volumes of big line bundles.
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4/12: Prof. Michael Albanese (U. Waterloo)
Title: Aspherical Complex Surfaces
Abstract: The collection of manifolds is vast and diverse. One class that we can hope to understand are those which have contractible universal cover, namely aspherical manifolds. These manifolds are determined up to homotopy equivalence by their fundamental group. There are several conjectures related to the Euler characteristic and signature of aspherical manifolds, namely the Hopf conjecture, the Singer conjecture, and in dimension 4, the Gromov-Luck inequality. We discuss these conjectures in the setting of aspherical complex surfaces. This is joint work with Luca Di Cerbo and Luigi Lombardi.
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4/19: Mathew George (The Ohio State University)
Title: Fully nonlinear PDEs of real forms on Hermitian manifolds
Abstract: Over many decades fully nonlinear PDEs, especially
the Monge-Ampere equation played a central role in the study of
complex manifolds. Most previous works focused on problems that can be
expressed as equations involving real (1, 1)-forms. Motivated by questions in complex geometry involving real (p,p) forms for p>1,
we introduce a nonlinear PDE theory involving real (p,p) forms on
Hermitian manifolds. In this talk, I will discuss the basic setup and explain some of the main challenges in working with such equations. The existence of solutions is shown for a large class of these equations. This is a joint work with Bo Guan.
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4/26: no meeting (Spring Break)
5/3: Prof. Yann Rollin (Nantes U.)
Title: Polyhedral symplectic geometry and moment maps
Abstract: I will discuss several moment map constructions with
applications to smooth symplectic geometry and polyhedral symplectic
geometry. In particular:
--symplectic diffeomorphisms of the 4-torus can
be interpreted as certain zeroes of a hyperKahler moment map geometry
--immersed Lagrangian surfaces in the symplectic affine 4-space can be understood as certain zeroes of a Kahler moment map geometry.
These constructions give interesting moment map flows that can be readily adapted to the polyhedral setting. In this case, the evolution equations are ODEs and define retractions of the moduli spaces. In a sense, they have the best possible expected behavior and lead to numerical examples of polyhedral symplectic maps of the 4-torus, or polyhedral Lagrangian surfaces.
References: https://arxiv.org/abs/2404.11347 https://arxiv.org/abs/2110.02679
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5/10: Prof. Yang Li (MIT)
Title: Metric SYZ conjecture
Abstract: One possible interpretation of the SYZ conjecture is that for a polarized family of CY manifolds near the large complex structure limit, there is a special Lagrangian fibration on the generic region of the CY manifold. Generic here means a set with a large percentage of the CY measure, and the percentage tends to 100% in the limit. I will discuss my recent progress on this version of the SYZ conjecture, with some emphasis on the Fermat hypersurface case.
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