1/26: no meeting this day

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 include a bigrading,
a Frolichertype spectral sequence, two bigradedcomponent
differential BValegbras, and induced actions of the homology of
moduli space of genus zero curves on the de Rham cohomology.
Various generalizations of all this hold if we drop the integrability
assumptions (of the Lagrangians, or the symplectic structure)
and in particular yield higher homotopical BValgebras. I'll describe the general
setup, and give computable examples where the cohomology
operations are nontrivial.
This story is similar
(and perhaps mirror) to the complexHermitian discussion
reported on last semester, and is ongoing work with Joana Cirici.

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.

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 deldelbarlemma, 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 nonKahler solutions to this second order system would have properties that are formally similar to the abovementioned properties of compact Kahler manifolds, including relations between cohomological (albeit nontopological) data. This is work in progress with Xenia de la Ossa (Oxford) and Eirik Eik Svanes (Stavanger).

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.

3/1: Prof. Gueo Grantcharov (Florida Int. U.)
Title: Hermitian metrics on nonKaehler 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.

3/8: Prof. Aleksandar Milivojevic (U. Waterloo)
Title: Formality and dominant maps
Abstract: In the mid 70's, DeligneGriffithsMorganSullivan 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 nontrivial 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 nonzero 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.

3/15: Prof. Carlo Scarpa (Universite de Quebec a Montreal)
Title: TBA
Abstract:

3/22: Spencer Cattalani (Stony Brook U.)
Title: TBA
Abstract:

3/29: no meeting (no classes this day)

4/5: Prof. Tamas Darvas (U. Maryland)
Title:
Abstract:

4/12: TBA
Title:
Abstract:

4/19: Mathew George (The Ohio State University)
Title: TBA
Abstract: TBA

4/26: no meeting (Spring Break)

5/3: Prof. Yann Rollin (Nantes U.)
Title: TBA
Abstract:

5/10: Prof. Yang Li (MIT)
Title: TBA
Abstract:
