Differential Geometry, Topology,
and special structures

Department of Mathematics
The Graduate Center of CUNY

Fridays 11:45noon - 1:45pm
Location: Room 6495
Organizers: Luis Fernandez, Mehdi Lejmi, Scott Wilson

Scope: this seminar is devoted to geometric structures on manifolds, and topological properties of them. We are particularly interested in almost complex manifolds, including complex manifolds, symplectic topology and geometry, (almost) Kahler geometry, and spin_c geometry, as well the tools of algebraic topology and geometric analysis that have proven useful in studying such structures. The goal is expose students and faculty to present day research and foundational material in a relaxed format that includes traditional lectures with an actively engaged audience.
Here is a link to the formerly named seminar of similar scope: ACG seminar

Spring 2022: (all meetings in person)

2/4: To all students (and faculty)
Title: Please see invitation on seminar website

Abstract: (Invitation) PhD students interested in geometry and topology, who have completed the first-year coursework in these topics, are all invited to attend the Differential Geometry and Topology Seminar on Fridays, 11:45-1:45. The seminar is devoted to the study of smooth manifolds, and particularly special structures on them, such as metrics, (almost) complex, and (almost) symplectic structures, as well as the interplay between these and fundamental structures in topology, such as cohomology and homotopy. This coming semester, Spring 2022, the faculty organizers will give introductory lectures on material that builds off the Differential Geometry course sequence, bringing students up to cutting-edge research, as well as open problems in the field. The format of the seminar is informal, with an opportunity to ask many questions and learn the fundamentals in depth. Additional speakers will be scheduled later in the semester, whose talks will become accessible via the introductory lectures. The first meeting 2/4 will be a welcome back and introduction with time to meet, talk math, and plan topics. The three part lecture series will start on 2/18.

2/8: CUNY uses this Tuesday as a Friday schedule. We will not meet this day.

2/11: CUNY is closed this day.

2/18: Prof. Scott Wilson
Title: Elementary introduction to structures on manifolds.

Abstract: I will survey the funadmental triality of rank-2 tensors on manifolds, with a focus on pointwise/local/global existence (or obstructions) and indicate some interesting open problems and recent results in each direction.

2/25: Prof. Mehdi Lejmi
Title: Holomorphic vector fields on Kahler manifolds

Abstract: We study real holomorphic and Killing vector fields on compact Kahler manifolds. If time permits, we will discuss the structure of the Lie algebra of homolomorphic vector fields on constant scalar curvature Kahler manifolds (Lichnerowicz Theorem) and on extremal Kahler manifolds (Calabi Theorem).

3/4: Prof. Luis Fernandez
Title: Generalized Kahler identities for almost complex manifolds (joint work with Sam Hosmer)

Abstract: I will start by quickly reviewing the bigraded complex on the complexified exterior bundle of an almost complex manifold, as well as the different operators on this complex and their relationships. In particular, when the manifold is Kahler, the commutator relations between the d operator and the Lefschetz operator L and its adjoint give the classical Kahler identities, which have important consequences. I will then introduce the Clifford bundle of a manifold, the bigraded complex in the complexified Clifford bundle, the Dirac operator, and other operators in this complex. By computing the commutator between the Dirac operator and one of these operators and translating the results to the exterior bundle, the commutation relations obtained give a clean generalization of the Kahler identities to an arbitrary almost complex manifold. This extends the generalizations of the Kahler identities obtained by Demailly for complex manifolds and Cirici-Wilson for almost-Kahler manifolds. (Joint work with Sam Hosmer)

3/11: Prof. Aleksandar Milivojevic (MPI Bonn)
Title: Approaching Hirzebruch's prize question via rational surgery

Abstract: In his 1990's book on manifolds and modular forms, Hirzebruch asked whether there exists a 24-dimensional spin manifold satisfying certain conditions on its Pontryagin classes, motivated by the observation that one could compute the dimensions of the irreducible representations of the Monster group via certain characteristic numbers of such a manifold. Hopkins and Mahowald showed in the early 2000's that such manifolds exist by understanding the bordism theory of manifolds admitting string structures (a further "lift" of the special orthogonal group beyond spin). In this talk, I will present an alternative, relatively elementary construction of a manifold as asked for by Hirzebruch, using an adaptation of results of Sullivan from the 1970's to spin manifolds. The construction will touch upon rational homotopy theory, surgery, index theory and characteristic classes, and involves some basic computer-assisted number theory when all the machinery is set up.

3/18: Prof. Christoforos Neofytidis (Ohio State U.)
Title: Topological Kodaira dimension, mapping degree and the simplicial volume

Abstract: In a 1978 lecture at CUNY Graduate Center, Gromov suggested to study the domination relation, i.e. the transitive relation defined by existence of maps of non-zero degree, as a partial ordering of the homotopy types of closed manifolds of the same dimension. Gromov and, independently, Milnor-Thurston asked which numerical homotopy invariants are monotone with respect to the domination relation. A prominent example of such invariant is the simplicial volume. In this talk, we introduce a type of Kodaira dimension and study its monotonicity with respect to the domination relation, as well as its relation to the simplicial volume. Joint work with Weiyi Zhang.

3/25: Prof. Jonas Stelzig (LMU, Munich)
Title: Linear combinations of cohomological invariants of compact complex manifolds.

Abstract: Some basic invariants of compact complex manifolds are Hodge and Chern numbers. A priori, the former depend on the complex structure and the latter on the almost complex structure. In the 50's, Hirzebruch asked which linear combinations of these are actually topological invariants. In the first part of the talk, I'll the answer to this question and some related ones, building on work of Kotschick and Schreieder in the Kahler case. In the second part, I'll explain how the numbers are just the tip of an iceberg on non-Kahler manifolds and introduce methods that allow to systematically treat (and partially answer) the more general questions involving the whole iceberg.

4/1: Guiseppe Barbaro (Sapienza U., Rome)
Title: Cohomological aspects of the stability of the pluriclosed flow on compact simply-connected simple Lie groups of rank two

Abstract: We compute the (1,1)-Aeppli cohomology of compact simply-connected simple Lie groups of rank two. In particular, we verify that they are of dimension one and generated by the classes of the Bismut flat metrics coming from the Killing forms. This yields a result on the stability of the pluriclosed flow on these manifolds.

4/8: Chi Li (Rutgers U.)
Title: Polarized Hodge structures for Clemens manifolds

Abstract: A conifold transition is a geometric transformation that is used to connect different moduli spaces of Calabi-Yau threefolds. Let X be a projective Calabi-Yau threefold. A conifold transition first contracts X along disjoint rational curves with normal bundles of type (-1,-1), and then smooths the resulting singular complex space Z to a new compact complex manifold Y. Such Y is called a Clemens manifold and can be non-Kahler. We prove that any small smoothing Y of Z satisfies ddbar-lemma. We also show that the resulting pure Hodge structure of weight three is polarized by the cup product. This answers some questions of R. Friedman. The proof uses the theory of limiting mixed Hodge structures and basic linear algebra.

4/15: Prof. Jonas Stelzig (LMU, Munich)
Title: Bigradings and complex manifolds

Abstract: The vector space of complex-valued differential forms on a complex manifold comes equipped with a bigrading, two anticommuting differentials, and a multiplication for which these are derivations. Abstracting this structure, one arrives at the notions of 'double complex' (without the multiplication) and 'commutative bigraded bidifferential algebra'.
I will survey some recent work on these structures, always with applications to complex geometry in mind. In particular, I will present a strong notion of quasi-isomorphism and how it allows one to define a 'holomorphic' homotopy theory for compact complex manifolds. Throughout, I will try to raise questions I consider interesting, some of which have an answer, and some of which do not (yet).

4/29: Prof. Scott Wilson
Title: Revisiting a proof of formality of Kahler manifolds, with a view towards complex manifolds and special metrics (d-dc + 3)

Abstract: From one such proof of formality we obtain a new long exact sequence, defined for all complex manifolds, that relates de Rham cohomology to two other complex analytic groups. These groups relate in a transparent way to other well-studied notions, such as the Bott-Chern and Aeppeli cohomologies, and pure Hodge structures, while yielding new numerical (in)equalities involving Betti numbers as well.
A mild weakening of the so-called d-dc condition, that we call (d-dc + 3), is equivalent to the vanishing of the connecting homomorphism in this sequence. Moreover, it implies E1-degeneration and is stable under many blowups. I'll discuss numerous examples: all compact complex surfaces, higher Hopf surfaces (but not all products of odd spheres), certain twistor spaces, and complex manifolds which admit locally conformal Kahler metrics satisfying a certain parallel-ness condition (i.e. Vaisman metrics).
This gives a new complex analytic obstruction to the existence of certain compatible metrics, and highlights a metric condition that implies E1-degeneration and much more (namely, d-dc + 3).
This is joint work with Jonas Stelzig.

5/6: Informal discussion
Faculty and students

Abstract: we will discuss ideas related to previous talks. Next week is the last talk of the semester.

5/13: In preference to Categorical Semantics of Entropy workshop, seminar cancelled.
see: https://itsatcuny.org/calendar/220513/scse for information on the workshop's friday schedule.

Note the workshop begins on Wednesday May 11th: https://itsatcuny.org/calendar/220511/tutorial-scse.