Home » Lectures

Category Archives: Lectures

Nicos Kapouleas: Lectures

May 27, 2021
TITLE: Gluing Eguchi-Hanson Metrics and a Question of Page

ABSTRACT:
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 t\to-\infty.

Panagiota Daskalopoulos: Lectures

May 26, 2021
TITLE: Ancient compact solutions to Ricci flow and Mean curvature flow

ABSTRACT:
Riemannian manifolds with special holonomy are ideal spaces on which to compactify M-theory, since the covariantly-constant spinors typical of such spaces give rise to supersymmetry in the effective, lower-dimensional theory.

However, it has long been recognized that other important features of the effective theory (including nonabelian gauge symmetry and massless chiral matter in four dimensions) cannot be realized by compactification on manifolds, and so it has been proposed to compactly M-theory on singular spaces as well. The effective theory in such a case is not derived exclusively from supergravity, but must contain other massless fields such as nonabelian gauge fields or massless matter fields corresponding to physics localized at singularities. One of the challenges is identify the new massless fields representing the new physics, based on the geometry of the singularities.

The case of gauge fields is very well studied and is understood to be derived from ADE singularities in (real) codimension four. The other cases (codimensions six and seven for compactifications to four dimension as well as codimension eight for compactifications to three dimension) are less well understood. Many examples are known, but in examples it is often assumed that the singularity is asymptotically a metric cone, which seems to have been justified on the basis of simplicity rather than on physical grounds.

We will propose a mathematical framework for studying singular spaces which contain a manifold with a metric of special holonomy as a dense open set, and we hope that this framework will capture all the relevant physical phenomena. We also hope that this is a reasonable framework mathematically — for example, one might hope that limits of metrics which are at finite distance from the bulk in a Weil-Peterson type metric would always fall into this class (as is already known for K3 surfaces).

Our main burden will be finding a good formulation for singularities of codimension seven, but we shall also discuss other relevant codimensions (four, six, and eight).

Richard Bamler: Lectures

May 28, 2021
TITLE: U(2)-invariant Ricci flows in dimension 4 and partial regularity theory for Ricci flows

ABSTRACT:
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 S^3 / \mathbb{Z}_k, k \geq 2, and whose Ricci flow develops a Type II singularity. This singularity is caused by the contraction of a 2-sphere of self-intersection -k. If k=2, 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 k \geq 3 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.

Dan Knopf: Lectures

May 28, 2021
TITLE: Ricci Solitons, Conical Singularities, and Nonuniqueness

ABSTRACT:
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.

Tristan Ozuch: Lectures

May 27, 2021
TITLE: Noncollapsed degeneration and desingularization of Einstein 4-manifolds

ABSTRACT:
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.

Simon Brendle: Lectures

May 26, 2021
TITLE: Ancient solutions to the Ricci flow in dimension 3

ABSTRACT:
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 \kappa-solutions: By definition, these are solutions to the Ricci flow which are defined for t \in (-\infty,T] and satisfy certain extra conditions (most importantly, a noncollapsing condition). Moreover, Perelman achieved a qualitative understanding of ancient \kappa-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 \kappa-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 \kappa-solutions in dimension 3.

Michael Douglas: Lectures

May 24, 2021
TITLE: Numerical Calabi-Yau metrics from holomorphic networks

ABSTRACT:
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.

Greg Moore: Lectures

January 13, 2021
Informal Remarks Complementary To, And Preparatory For,
Fei Yan’s Talk:

ABSTRACT:
Slides of Lecture, part 1
Slides of Lecture, part 2

Fei Yan: Lectures

January 13, 2021
TITLE: Line defects, UV-IR map and exact WKB

ABSTRACT:
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.

Slides of Lecture

Ivan Smith: Lectures

January 11, 2021
TITLE: Quadratic differentials as stability conditions

ABSTRACT:
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.

Slides of Lecture