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May 27, 2021
TITLE: Gluing Eguchi-Hanson Metrics and a Question of Page
I will discuss a paper of Simon Brendle and myself (Comm. Pure Appl. Math. 70 (2017), no. 7, 1366-1401) motivated by a question Page asked in 1981. Page’s question was based on a physical picture for the Ricci-flat Kähler metrics on the K3 surface proposed by Gibbons-Pope and Page in 1978. In this picture the K3 metrics are viewed as desingularizations by Eguchi-Hanson manifolds of the 16 orbifold points of the quotient of a torus by the antipodal map. Such a construction was carried out rigorously by Topiwala, LeBrun-Singer, and Donaldson. Page’s question was whether some of the Eguchi-Hanson metrics can be attached with the opposite orientation resulting in a manifold of different topology carrying a non-K\”ahler Ricci-flat metric.
We imposed a discrete group of symmetries on such a construction to simplify the situation while still capturing the essential features. We then studied the obstructions to such a construction which arise from the interaction of the Eguchi-Hanson manifolds being attached, because the obstructions arising from the interaction with the flat background all vanish. It turns out that these obstructions cannot be overcome and the construction fails. Finally we made use of the non-vanishing obstruction to construct and study ancient solutions of the Ricci flow where the Eguchi-Hanson manifolds attached shrink to orbifold points as .
May 26, 2021
TITLE: Ancient compact solutions to Ricci flow and Mean curvature flow
Some of the most important problems in partial differential equations are related to the understanding of singularities. This usually happens through a blow up procedure near the potential singularity which uses the scaling properties of the equation. In the case of a parabolic equation the blow up analysis often leads to special solutions which are defined for all time , for some . We refer to them as ancient solutions. The classification of such solutions often sheds new insight to the singularity analysis.
In this lecture we will discuss Uniqueness Theorems for ancient compact solutions to the Ricci flow and Mean curvature flow. Emphasis will be given to the complete classification of compact κ-noncollapsed solutions to the 3- dim Ricci flow. We will also discuss the 2-dim case where the κ-noncollapsed condition is not necessary for uniqueness.
May 28, 2021
TITLE: U(2)-invariant Ricci flows in dimension 4 and partial regularity theory for Ricci flows
In this talk I will present work of my former graduate student, Alexander Appleton, on cohomogeneity-1 Ricci flows in dimension 4 that are invariant under an isometric U(2)-action.
I will first show that there are certain U(2)-invariant, metrics that are asymptotic to , , and whose Ricci flow develops a Type II singularity. This singularity is caused by the contraction of a 2-sphere of self-intersection . If , then the singularity is modeled on the Eguchi-Hanson metric. This is the first example of a finite-time singularity in Ricci flow whose blow-up limit is Ricci flat. Numerical evidence suggests that the singularities in the case are modeled on a new family of non-collapsed, U(2)-invariant, steady solitons, which were also constructed by Appleton.
In the last part of the talk, I will relate Appleton’s results to a new compactness and partial regularity theory for Ricci flows. In the context of this theory, Appleton’s examples are optimal, as they exhibit a singular set with the lowest possible codimension.
May 28, 2021
TITLE: Ricci Solitons, Conical Singularities, and Nonuniqueness
In dimension n=3, there is a complete and well-posed theory of weak solutions of Ricci flow: existence in the form of “Ricci Flow Spacetimes” was proved by Kleiner and Lott, and uniqueness was proved by Bamler and Kleiner. I will describe joint work with Angenent in which we show that uniqueness should not be expected to hold for Ricci flow weak solutions in dimensions n≥5. Specifically, we produce a discrete family of asymptotically conical gradient shrinking solitons, each of which encounters a finite-time local singularity and thereafter admits non-unique forward continuations by gradient expanding solitons. Moreover, we exhibit these evolutions as Ricci Flow Spacetimes and show that topological nonuniqueness after the first singularity time is possible for the solutions we construct.
May 27, 2021
TITLE: Noncollapsed degeneration and desingularization of Einstein 4-manifolds
We study the noncollapsed singularity formation of Einstein 4-manifolds. We prove that any smooth Einstein 4-manifold close to a singular one in a mere Gromov-Hausdorff (GH) sense is the result of a gluing-perturbation procedure that we develop and which handles the presence of multiple trees of singularities at arbitrary scales.
This sheds light on the structure of the moduli space of Einstein 4-manifolds near its boundary and lets us show that spherical and hyperbolic orbifolds cannot be GH-approximated by smooth Einstein metrics. New obstructions specific to the compact situation moreover raise the question of whether or not a sequence of Einstein 4-manifolds degenerating while bubbling out gravitational instantons has to be Kähler-Einstein. These obstructions are even more restrictive in the Ricci-flat situation.
May 26, 2021
TITLE: Ancient solutions to the Ricci flow in dimension 3
The Ricci flow is a natural evolution equation for Riemannian metrics on a manifold. The central problem is to understand singularity formation. In other words, what does the geometry look like at points where the curvature is large? In his spectacular 2002 breakthrough, Perelman showed that, for a solution to the Ricci flow in dimension 3, the high curvature regions are modeled on so-called ancient -solutions: By definition, these are solutions to the Ricci flow which are defined for and satisfy certain extra conditions (most importantly, a noncollapsing condition). Moreover, Perelman achieved a qualitative understanding of ancient -solutions in dimension 3; this is sufficient for topological conclusions.
In this lecture, I will discuss recent work which gives a complete classification of all noncompact ancient -solutions in dimension 3, thereby confirming a conjecture of Perelman. Time permitting, I will mention joint work with Panagiota Daskalopoulos and Natasa Sesum which gives a complete classification of all compact ancient -solutions in dimension 3.
May 24, 2021
TITLE: Numerical Calabi-Yau metrics from holomorphic networks
We propose machine learning inspired methods for computing numerical Calabi-Yau (Ricci flat Ka ̈hler) metrics. In joint work with Yidi Qi and Subramanian Lakshminarasimhan, we implemented them using Tensorflow/Keras and compared with previous work. We will also discuss ideas for numerical study of other Ricci flat metrics and of SYZ fibrations.
Slides of Lecture
January 13, 2021
TITLE: Line defects, UV-IR map and exact WKB
In this talk I’ll give an overview of the relations between class S theories and Hitchin systems, focusing on roles played by line defects in class S theories. Deforming onto the Coulomb branch triggers a UV-IR map for line defects, corresponding to a trace map for certain flat nonabelian connections over a Riemann surface. The UV-IR map admits a q-deformation, which corresponds to a quantum trace map embedding certain skein algebra into a quantum torus algebra. I’ll also briefly describe connections to exact WKB and a potential q-deformation thereof.
January 11, 2021
TITLE: Quadratic differentials as stability conditions
Consider a quasi-projective Calabi-Yau 3-fold which is an affine conic fibration over a two-dimensional surface. I will explain why the space of stability conditions on (a subcategory of) its Fukaya category can be understood in terms of meromorphic quadratic differentials on the surface. This talk reports on old joint work with Tom Bridgeland.