The Geometry and Geometric Analysis Seminar at Purdue in Fall 2023 is usually on Mondays 11:30am-12:30pm Eastern Time in MATH 731 if we meet in person. Some talks will be on zoom and the link will be included in the email announcement.
Lvzhou Chen and Nicholas McCleerey are organizing this seminar in Fall 2023. If you have any questions or would like to suggest speakers, please contact one of us. If you would like to be on the mailing list, please email Lvzhou.
Title: The equicritical stratification and stratified braid groups
Abstract: One of the many guises of the braid group is as the fundamental group of the space of monic squarefree polynomials. From this point of view, there is a natural “equicritical stratification" according to the multiplicities of the critical points. These equicritical strata form a natural and rich class of spaces at the intersection of algebraic geometry, topology, and geometric group theory, and can be studied from many different points of view; their fundamental groups (“stratified braid groups”) look to be interesting cousins of the classical braid groups. I will describe some of my work on this topic thus far, which includes a partial description of the relationship between stratified and classical braid groups, and some progress towards showing that the equicritical strata are K(\pi,1) spaces.
Title: Deformation and rigidity of the holomorphic tangent bundle of a complex two ball quotient
Abstract: It is known from the work of Calabi-Vesentini in the 60's that a smooth locally Hermitian symmetric space is locally rigid in the space of deformation of the manifold. A natural question is whether the holomorphic tangent bundle of such spaces can be deformed among holomorphic vector bundles of the same determinant bundle. All the examples of locally Hermitian symmetric spaces for which the question has a known answer have a rigid tangent bundle in the above sense. In particular, for complex hyperbolic $n$ spaces, it is known from a work of Siu in 80's that there is no deformation for $n>2$ in the above sense. The case of $n=2$ is subtle and has been open. We will explain the problem from different perspectives, and show that there are examples with deformations in some complex two ball quotients.
Title: Fully nonlinear parabolic equations of real forms on Hermitian manifolds.
Abstract: Over the last many decades, fully nonlinear equations have played an important role in the development of complex differential geometry. This started with the solution to the complex Monge-Ampere equation by Yau in 1978, proving the Calabi conjecture. Soon after this, equations with more complicated structures were being studied for different applications, especially in Hermitian geometry. Some examples of this include equations involving $(n-1)$ forms as in the Gauduchon conjecture proved by Szekelyhidi, Tosatti, and Weinkove, or the Fu-Yau Hessian equation from string theory studied extensively by Phong, Picard, and Zhang.
Although having many similarities with the Monge-Ampere equation, such equations are often much more complicated. In this talk, we introduce a general class of such equations on Hermitian manifolds and discuss a new technique using the structure of the PDE that is effective in dealing with many of these difficulties. We take a parabolic approach and prove the long-time existence of solutions by deriving first and second-order estimates. Further, we prove the convergence of the solution to the elliptic case by deriving a Li-Yau-type Harnack inequality.
Title: Deformative magnetic marked length spectrum rigidity
Abstract: Given a closed surface with negative curvature, any closed curve on the surface which is not null-homotopic can be continuously deformed to a unique closed geodesic. The marked length spectrum is a function which takes a closed curve and returns the length of the corresponding geodesic. It was shown by Guillemin and Kazhdan that if two negatively curved metrics on a closed surface can be connected by a path of negatively curved metrics along which the marked length spectrum is constant, then the metrics are isometric. In this talk, I will present a generalization of this theorem to the setting of magnetic flows on a closed oriented surface with negative magnetic curvature.
Title: Surface Subgroups in Cocompact Kleinian Groups
Abstract: Kahn and Markovic proved the Surface Subgroup conjecture for closed hyperbolic 3-manifolds more than ten years ago. The surface subgroup they constructed can be as close as possible to Fuchsian. However, a closed hyperbolic 3-manifold can also have surface subgroups far away from being Fuchsian. Actually, provided any genus-2 quasi-Fuchsian group Γ and cocompact Kleinian group G, then for any K>1, we can find a surface subgroup H of G that is K-quasiconformally conjugate to a finite index subgroup F<Γ. We will point out the difference between my theorem and the original Surface Subgroup Theorem, discuss the proof idea, and introduce some applications. For instance, we can use this theorem to prove that the set of Hausdorff dimensions of limit sets of surface subgroups of G is dense in [1,2].
Title: Coherence, non-positive immersions and L^2-Betti numbers
Abstract: A group is said to be coherent if all of its finitely generated subgroups are finitely presented. Proving a group or a class of groups is coherent can often be a delicate task since many seemingly innocuous groups turn out to not be coherent, the prototypical example being the direct product of two non-abelian free groups. In this talk, I will discuss connections between coherence, the non-positive immersions property and the vanishing of the second L^2-Betti number. I will then show how these connections lead to a proof that all one-relator groups are coherent, solving an old problem of Baumslag's. Joint work with Andrei Jaikin-Zapirain.
Title: Special Lagrangian spheres in K3-fibred Calabi-Yau 3-folds
Abstract: Defined by Harvey and Lawson in a seminal paper, special Lagrangian (SL) submanifolds now play an important role in string theory and mirror symmetry. As calibrated submanifolds in manifolds with special holonomy, SL submanifolds are also an important source of high codimensional minimal submanifolds. In this talk, we discuss a new construction of SL 3-spheres in Calabi-Yau 3-folds equipped with a Lefschetz K3 fibration, where the Calabi-Yau metrics under consideration are near an adiabatic limit. The SL 3-spheres can be viewed as "thickenings" of segments in the base P^1 connecting two nodal values. This is joint work with Yu-Shen Lin.
Title: A Le Potier-type Isomorphism with Multiplier Submodule Sheaves
Title: Berndtsson's direct image theorem
Berndtsson's theorem, on a basic level, says that certain operations involving holomorphic and plurisubharmonic functions result in plurisubharmonic functions. After introducing the necessary notions, I will formulate an instance of the theorem, and discuss an application.
Title: On some new Monge-Ampere functionals
Title: An analog of a theorem of Stallings