Focusing on the Lie group SU(n) and associated symmetric spaces,
I investigate two related topics. The main result is that the bi-invariant
Einstein metric on SU(2n+1) is isolated in the moduli space of Einstein
metrics, even though it admits infinitesimal deformations. This gives a
non-Kaehler, non-product example of this phenomenon adding to the famous
example of Koiso from the eighties. I also explore the relationship between
the question of rigidity and instability (under the Ricci flow) of Einstein
metrics, and present results in this direction for complex Grassmannians.
Asymptotically conical Calabi-Yau manifolds are non-compact Ricci-flat Kähler manifolds that are modeled on a Ricci-flat Kähler cone at infinity. I will present a classification result for such manifolds. This is joint work with Hans-Joachim Hein (Fordham/Muenster).
On symplectic manifolds, there are intrinsincally symplectic cohomologies of differential forms that are analogous to the Dolbeault cohomology on complex manifolds. These cohomologies are isomorphic to the de Rham cohomologies on odd-dimensional sphere bundles over the symplectic manifold. In this talk, I will describe how we can use this sphere bundle perspective to define a novel Morse-type theory on symplectic manifolds associated with the symplectic cohomologies. This is joint work with Xiang Tang and Li-Sheng Tseng.
Understanding moduli spaces is one of the central questions in
algebraic geometry. This talk will survey one of the main aspects of
research in moduli theory — the compactification problem. Roughly speaking,
most naturally occurring moduli spaces are not compact and so the goal is to
come up with geometrically meaningful compactifications. We will begin by
looking at the case of algebraic curves (i.e. Riemann surfaces) and progress
to higher dimensions, where the theory is usually divided into three main
categories: general type (i.e. negatively curved), Calabi-Yau (i.e. flat),
and Fano (i.e. positively curved, where the theory is connected to the
existence of KE metrics). Time permitting, we will use this motivation to
discuss moduli spaces of K3 surfaces (simply connected compact complex
surfaces with a no-where vanishing holomorphic 2-form).
(A joint seminar with the Geometry & Topology Seminar series.)
The generalized Ricci flow is first studied by Streets and Tian.
It can be viewed as the Ricci flow of connections with torsion which has
many applications in non-Kähler geometry. In this talk, we will study the
fixed points of the generalized Ricci flow which are called the steady
generalized Ricci solitons. Similar to Ricci flow, generalized Ricci flow is
also a gradient flow so we can further compute the variation formulas and
see some linear stable examples. Our main result is to show that the concept
of dynamical stability and linear stability are equivalent on the steady
gradient generalized Ricci solitons.
The Monge-Ampere equation det(D^2 u) = 1 arises in prescribed
curvature problems and in optimal transport. An interesting feature of the
equation is that it admits singular solutions. We will discuss new examples
of convex functions on R^n that solve the Monge-Ampere equation away from
finitely many points, but contain polyhedral and Y-shaped singular
structures. Along the way we will discuss geometric and applied motivations
for constructing such examples, as well as their connection to a certain
Typical comparison results in Riemannian geometry, such as for
volume or for spectrum of the Laplacian, require Ricci curvature lower
bounds. In dimension three, we can prove several sharp comparison estimates
assuming only a scalar curvature bound. The talk will present these results,
their applications, and describe how dimension three is used in the proofs.
Joint work with Jiaping Wang.
In this talk, I will present parts of a recent paper written
jointly with Mario Garcia-Fernandez and Jeff Streets ("NonKahler Calabi-Yau
Geometry and Pluriclosed Flow" arxiv:2106.13716). In particular, I intend to
discuss long-time existence and convergence of solutions to pluriclosed flow
on manifolds satisfying a flatness condition. As the pluriclosed flow does
not admit a scalar reduction in general, I will introduce the language of
holomorphic Courant algebroids and a result of J.M. Bismut that will allow
us to derive apriori estimates for an equivalent coupled