Differential Geometry, Topology,
and special structures

Department of Mathematics
The Graduate Center of CUNY

Fridays 11:45noon - 1:45pm
Location: Room 6417
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 previous semesters: DGT Seminar Spring 2022
Here is a link to the formerly named seminar of similar scope: ACG seminar

Fall 2022: (all meetings in person)

8/26: no meeting

9/2: College closed (Labor day)

9/9: Jiahao Hu (Stony Brook University)
Title: The four components of d on almost complex manifolds

Abstract: The exterior differential d on complex-valued differential forms of complex manifolds decomposes into the Cauchy-Riemann operator and its complex conjugate. Meanwhile on almost complex manifolds, the exterior d in general has two extra components, thus decomposes into four operators. In this talk, I will introduce these operators and discuss the structure of the (graded) associative algebra generated by these four components of d, subject to relations deduced from d squaring to zero. Then I will compare this algebra to the corresponding one in the complex (i.e. integrable) case, we shall see they are very different strictly speaking but similar in a weak sense (quasi-isomorphic). This is based on joint work with Shamuel Aueyung and Jin-Cheng Guu.

9/16: Yury Ustinovskiy (Lehigh U.)
Title: Scalar Curvature in Generalized Kahler Geometry

Abstract: Construction and classification of the constant scalar curvature metrics in Kahler geometry has been a central problem in complex geometry. It is related to deep notions of stability in algebraic geometry and represents a challenging analytic problem. In this talk we turn our attention to the generalized geometry introduced by Hitchin and its Kahler counterpart formulated by Gualtieri. Recently, several notions of scalar curvature in this context have emerged, mirroring the fundamental results in the classical Kahler geometry. We will review various approaches to the definition of generalized scalar curvature, prove their equivalence, and set up a formal GIT picture for the search of the constant generalized scalar curvature metrics. This talk is based on a joint work with Vestislav Apostolov and Jeffrey Streets.

9/23: Prof. Paul Feehan (U. Rutgers, New Brunswick)
Title: Virtual Morse-Bott Index, Moduli Spaces of Pairs, and Applications to Topology of Smooth Four-Manifolds

Abstract: In joint work with Tom Leness (arXiv:2010.15789), we compute a version of the classical Morse-Bott index, called a virtual Morse-Bott index, for each critical point of a Hamiltonian function for a circle action on a certain complex analytic moduli space. In the case of Hamiltonian functions of circle actions on smooth, almost Kaehler (symplectic) manifolds, the virtual Morse-Bott index coincides with the classical Morse-Bott index considered by Bott (1954) and Frankel (1959). Positivity of virtual Morse-Bott indices implies downward gradient flow in the top stratum of smooth points in the complex analytic space. We apply our method to the moduli space of SO(3) monopoles over a complex, Kaehler surface, we use the Hirzebruch-Riemann-Rich Theorem to compute virtual Morse-Bott indices of all critical strata (Seiberg-Witten moduli subspaces), and we prove that these indices are positive in a setting motivated by the conjecture that all closed, smooth four-manifolds of Seiberg-Witten simple type obey the Bogomolov-Miyaoka-Yau inequality.

9/30: Prof. Xiaowei Wang (U. Rutgers, Newark)
Title: Moment map and convex function

Abstract: We will present a theory of the moment map coupled with a convex function and study the property of the corresponding critical point of composing functional. As an application, we interpret the Kahler-Ricci solitons as a special case of the generalized extremal metric. This is a joint work with King-Leung Lee and Jacob Sturm.

10/7: Prof. Mehdi Lejmi (Bronx, CUNY)
Title: Second-Chern-Einstein metrics in dimension 4

Abstract: Thanks to the work of Gauduchon and Ivanov, it is known that the only 4-dimensional compact Hermitian non-Kahler second-Chern-Einstein manifold is the Hopf surface. In this talk, we investigate the existence of second-Chern-Einstein metrics on 4-dimensional compact almost-Hermitian manifolds. For instance, we remark that under a certain hypothesis the dual of the Lee form is a Killing vector field. This leads to a description of such manifolds in some cases. This is a joint work with Giuseppe Barbaro.

10/14: Prof. Scott Wilson (Queens, CUNY)
Title: Rational homotopy obstructions to the existence of certain complex structures on smooth manifolds.

Abstract: I will explain an argument for ruling out certain complex structures on manifolds in real dimension greater than or equal to 4. The obstructions come from the underlying rational homotopy type (and cohomology ring) of the manifold, which measure the failure of the deRham dga to be formal. In the situation I'll discuss, there is a topological lower bound on analytic invariants that measure the wildness of the bi-complex of differential forms on a complex manifold. Several explicit examples will be given in dimension 6, using nilmanifolds (some of which have complex structures), showing there is no complex structure which is ``ddc in degree 1 and ddc+3 in degree 2''. I'll give all the necessary background and details to make this statement precise. This is joint work with Jonas Stelzig.

10/21: Prof. Xumin Jiang (Fordham U.)
Title: Kahler-Einstein metrics on complex hyperbolic cusps and a continuous cusp closing process.

Abstract: Let D be a complex torus together with a negative holomorphic line bundle L over D. Let V be any closed tubular neighborhood of the zero section of L. Datar-Fu-Song proved the existence of a unique complete Kahler-Einstein metric on V \ D if a boundary condition is assumed. We study the asymptotics of such Kahler-Einstein metrics and prove a sharp exponentially decaying estimate. Then we introduce the Tian-Yau construction of Ricci-flat Kahler metrics and glue a Tian-Yau model with the cusp model. We show that the glued metric is close to the unique Kahler-Einstein metric on the glued manifold. This is a joint work with Hans-Joachim Hein and Xin Fu.

10/28: Prof. Aleksandar Milivojevic (Max Planck Inst., Bonn)
Title: Formality, Massey products, and non-zero degree maps

Abstract: I will discuss results related to the following basic question in rational homotopy theory: given a non-zero degree map of rational Poincare duality spaces (e.g. closed orientable manifolds), does formality of the domain imply formality of the target? As a helpful tool, which might be of independent utility, I will describe a construction that takes in a cohomologically finite dimensional and connected commutative differential graded algebra and extends it to one that satisfies Poincare duality. This is joint work with Jonas Stelzig and Leopold Zoller.

11/4: Prof. Nicoletta Tardini (U. Parma)
Title: HKT manifolds and Hodge theory

Abstract: Hyperkahler with torsion (HKT for short) manifolds are defined as smooth manifolds endowed with an hypercomplex structure (I,J,K) and a real and positive (in the quaternionic sense) del-closed (2,0)-form (here the bidegree and del are taken with respect to the complex strucure I). Examples of these manifolds are hyperkahaler manifolds. We will show that compact HKT manifolds, under some natural assumptions, behave very similarly to Kahler manifolds, both from the complex and "symplectic" points of view. In particular, we will discuss their cohomological and metric properties. This is joint work with Giovanni Gentili.

11/11: Prof. Amir Babak Aazami (Clark U.)
Title: On the geometry of pp-wave spacetimes

Abstract: Gravitational plane waves were discovered by A. Einstein in 1916 as solutions of his linearized equations, modeling gravitational radiation propagating at the speed of light. Almost a century later, their existence was confirmed by LIGO. But it turns out that these spaces are as remarkable mathematically as they are physically. In this talk, we explore some of the rich geometric properties they exhibit, not just in Lorentzian geometry, wherein they hold a distinguished place, but also in Riemannian geometry, in symplectic and complex geometry, and in dynamical systems.

11/18: Kevin Smith (Columbia U.)
Title: The continuity equation on Hopf and Inoue surfaces

Abstract: We study the continuity equation of La Nave-Tian, extended to the Hermitian setting by Sherman-Weinkove, on Hopf and Inoue surfaces. We prove a priori estimates for solutions in both cases, and Gromov-Hausdorff convergence of Inoue surfaces to a circle.

11/25: no meeting (Thanksgiving)

12/2: Informal meeting
Title: Faculty and student discussion (Q&A)

12/9: Informal meeting
Title: Faculty and student discussion (Q&A)