8/26: no meeting
9/2: College closed (Labor day)
9/9: Jiahao Hu (Stony Brook University)
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.
Title: The four components of d on almost complex manifolds
9/16: Yury Ustinovskiy (Lehigh U.)
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.
Title: Scalar Curvature in Generalized Kahler Geometry
9/23: Prof. Paul Feehan (U. Rutgers, New Brunswick)
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.
Title: Virtual Morse-Bott Index, Moduli Spaces of Pairs, and Applications to Topology of Smooth Four-Manifolds
9/30: Prof. Xiaowei Wang (U. Rutgers, Newark)
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.
Title: Moment map and convex function
10/7: Prof. Mehdi Lejmi (Bronx, CUNY)
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.
Title: Second-Chern-Einstein metrics in dimension 4
10/14: Prof. Scott Wilson (Queens, CUNY)
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.
Title: Rational homotopy obstructions to the existence
of certain complex structures on smooth manifolds.
10/21: Prof. Xumin Jiang (Fordham U.)
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.
Title: Kahler-Einstein metrics on complex hyperbolic cusps and a continuous cusp closing process.
10/28: Prof. Aleksandar Milivojevic (Max Planck Inst., Bonn)
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.
Title: Formality, Massey products, and non-zero degree maps
11/4: Prof. Nicoletta Tardini (U. Parma)
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.
Title: HKT manifolds and Hodge theory
11/11: Prof. Amir Babak Aazami (Clark U.)
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.
Title: On the geometry of pp-wave spacetimes
11/18: Kevin Smith (Columbia U.)
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.
Title: The continuity equation on Hopf and Inoue surfaces
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)