# The geometry of k-Ricci curvature and a Monge-Ampere equation

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A joint Geometry and Analysis seminar.

Abstract: The k-Ricci curvature interpolates between the Ricci curvature and holomorphic sectional curvature. For the positive case, a recent result asserts that the compact Kaehler manifolds with positive k-Ricci are projective and rationally connected. This generalizes the previous results of Campana, Kollar-Miyaoka-Mori for the Fano case and the Heirer-Wong and Yang for holomorphic sectional curvature case. For the negative case, all compact Kaehler manifolds with negative k-Ricci admit a Kaehler-Einstein metric with negative scalar curvature. I shall explain how to get this result by solving a complex Monge-Ampere equation.

# A nonlinear spectrum on closed manifolds

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Abstract: The p-widths of a closed Riemannian manifold are a nonlinear

analogue of the spectrum of its Laplace--Beltrami operator, which was

defined by Gromov in the 1980s and corresponds to areas of a certain

min-max sequence of hypersurfaces. By a recent theorem of

Liokumovich--Marques--Neves, the p-widths obey a Weyl law, just like

the eigenvalues do. However, even though eigenvalues are explicitly

computable for many manifolds, there had previously not been any >=

2-dimensional manifold for which all the p-widths are known. In recent

joint work with Otis Chodosh, we found all p-widths on the round

2-sphere and thus the previously unknown Liokumovich--Marques--Neves

Weyl law constant in dimension 2.

# Hamilton-Ivey estimates for gradient Ricci solitons

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One special feature for the Ricci flow in dimension 3 is the

Hamilton-Ivey estimate. The curvature pinching estimate provides a lot of

information about the ancient solution and plays a crucial role in the

singularity formation of the flow in dimension 3. We study the pinching

estimate on 3 dimensional expanding and 4 dimensional steady gradient Ricci

solitons. A sufficient condition for a 3-dimensional expanding soliton to

have positive curvature is established. This condition is satisfied by a

large class of conical expanders. As an application, we show that any

3-dimensional gradient Ricci expander C^2 asymptotic to certain cones is

rotationally symmetric. We also prove that the norm of the curvature tensor

is bounded by the scalar curvature on 4 dimensional non Ricci flat steady

soliton singularity model and derive a quantitative lower bound of the

curvature operator for 4-dimensional steady solitons with linear scalar

curvature decay and proper potential function. This talk is based on a joint

work with Zilu Ma and Yongjia Zhang.

# Geometric Flows of G2-Structures

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There are several natural flows of G2-Structures that may be useful for resolving important problems in the theory of G2-Holonomy Manifolds. Here we review what is known and discuss some open problems for two such flows: The Laplacian flow and the Laplacian co-flow. We will then present some new results for each of these flows.

# Degeneration of 7-dimensional minimal hypersurfaces with bounded index

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A 7D minimal and locally-stable hypersurface will in general have a discrete singular set, provided it has no singularities modeled on a union of half-planes. We show in this talk that the geometry/topology/singular set of these surfaces has uniform control, in the following sense: if $M_i$ is a sequence of 7D minimal hypersurfaces with uniformly bounded index and area, and discrete singular set, then up to a subsequence all the $M_i$ are bi-Lipschitz equivalent, with uniform Lipschitz bounds on the maps. As a consequence, we prove the space of $C^2$ embedded minimal hypersurfaces in a fixed $8$-manifold, having index $\leq I$, area $\leq \Lambda$, and discrete singular set, divides into finitely-many diffeomorphism types.

# Rigidity and instability of SU(n) symmetric spaces

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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

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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).

# Cohomology and Morse theory on symplectic manifolds

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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.

# Moduli spaces in algebraic geometry

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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.)