Spring 2020:
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1/31: Organizational meeting
Abstract: This semester, in addition to regular talks on
research, there will be several survey talks and expositions on research
papers. In this meeting we'll decide on these topipcs and the schedule, and of
course talk math as time permits.
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2/7: Prof. Scott Wilson
Title: Survey of formality of Kahler manifolds, with a view towards complex manifolds.
Abstract: I will survey the celebrated theorem of Deligne, Griffiths,
Morgan, and Sullivan, that compact Kahler manifolds are formal. This in
particular implies that the rational homotopy groups can be computed formally from
the cohomology ring, and that all Massey products vanish. The proof,
which relies upon the Kahler identities, shows that the there is an equivalance between the
de Rham algebra of differential forms and the cohomology. The one new
observation I will make is that every integrable complex structure and choice
of metric determines a natural morphism between the
differential forms and the cohomology, even in the non-Kahler case. It is an open problem to study the
extent to which such morphisms determine topological information about
the underlying complex manifold.
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2/14: Aleksandar Milivojevic
Title: The rational topology of closed almost complex manifolds
Abstract: Given an even-dimensional oriented closed smooth manifold,
there are topological obstructions to it admitting an almost complex (AC)
structure. The most prominent of these come from (a) the requirement that the Pontryagin class must have a "square root" in the form of Chern classes, and (b) various congruences among the tentative Chern numbers coming from Hirzebruch-Riemann-Roch/Atiyah-Singer index theory.
If we start with only a (simply connected) topological space with no additional structure, we can ask whether there exists a closed AC manifold rationally equivalent to the space (i.e. admitting a map to a "rationalized" version of the space which induces an isomorphism on rational homology). A first necessary condition is that the space satisfies Poincare duality on its rational cohomology, giving it a rational cohomological dimension which we will assume is at least six. If this topological space can support "Chern classes" in its rational cohomology whose associated Chern numbers satisfy the congruences from point (b) above, along with some further conditions on the middle degree nondegenerate pairing if the dimension is divisible by 4, then there is indeed such a closed AC manifold with these Chern classes.
I will elaborate on this result, and will work through an example illustrating how number theory quickly enters the discussion. Further, I will comment on a manifestation of the result: the existence of an AC manifold realizing a given rational homotopy type depends only on the rational cohomology algebra of the homotopy type. Recalling last week's talk discussing formality, this in particular implies that every rational algebra realized by some AC manifold is also realized by a formal one. But perhaps the interest lies in the opposite direction: once an algebra is known to be realized by an AC manifold, one can at will perturb the algebra within the category of differential graded algebras (while preserving the cohomology algebra) to obtain closed AC manifolds with highly complicated topology.
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2/21: Group discussion
Title: open problems and questions: all faculty and students welcome.
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2/28: Qian Chen, CUNY
Title: Spectral Sequences for Cohomologies on Almost Complex Manifolds
Abstract: In 2017, Chan, Karigiannis, and Tsang introduced new cohomology groups for almost complex manifolds. In the integrable case, one of these groups was previously introduced by Deligne, Griffiths, Morgan, and Sullivan, and played an important role in the proof of formality of Kahler manifolds. In this talk I will show there are spectral sequences for computing these cohomologies, and explain how they fit in a natural diagram involving De Rham and Bott-Chern cohomologies, and the spectral sequence for Dolbeault cohomology introduced by Cirici-Wilson. I'll also compute several examples for nil-manifolds to show these spectral sequences are in general very interesting.
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3/6: Prof. Bianca Santoro, City College, CUNY
Title: The space of Kahler metrics
Abstract: TBA
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3/13: Raymond Puzio
Title: TBA
Abstract: TBA
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3/20: Prof. David Pham, QCC CUNY
Title: TBA
Abstract: TBA
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3/27: Prof. Yury Ustinovsky, NYU
Title: TBA
Abstract: TBA
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4/3: Sam Hosmer
Title: TBA
Abstract: TBA
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4/10:
Title: no meeting
4/17:
Title: no meeting
4/24: Prof. Luis Fernandez, BCC CUNY
Title: TBA
Abstract: TBA
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5/1:
Title: TBA
Abstract: TBA
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5/8:
Title: TBA
Abstract: TBA
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