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# Category Archives: Lectures

## Fabian Lehmann: Lectures

### September 13, 2021

TITLE: Non-compact Spin(7)-manifolds

ABSTRACT:

In the non-compact setting, symmetry reduction methods can be used to simplify the condition for Spin(7)-holonomy, which in general is given by a large, non-linear, first order PDE system, to a system of ODEs. I will talk about a particular example with symmetry group SU(3). I will outline a rigorous proof for the existence of two families of complete Spin(7)-metrics, where all members are either asymptotically locally conical (ALC), or asymptotically conical (AC).

These families were conjectured to exist earlier and fit into the landscape of other known families of non-compact G2 and Spin(7) holonomy spaces. Time permitting, I will also discuss the deformation theory of AC Spin(7)-manifolds. The talk is based on arXiv:2012.11758 and arXiv:2101.10310.

## Max Hübner: Lectures

### September 13, 2021

TITLE: Higgs Bundles for G2 Manifolds and Brane/Particle Probes

ABSTRACT:

We consider M-theory on a local, ALE-fibered G2 manifold. At low energies the effective physics is described by a partially twisted 7d SYM theory. The BPS equations describe a Higgs bundle associated to the ALE fibration of the G2 manifold. We describe how M2-branes probing the G2 manifold descend to particles probing the Higgs bundle. Such probe particles attach a Morse-Witten complex to the geometry. This complex is generated by the singular subloci of the G2 manifold. Supersymmetric three-spheres are in correspondence with flow trees and give rise to boundary maps and cup products.

## Jingxiang Wu: Lectures

### September 13, 2021

TITLE: Kondo line defect and affine oper/Gaudin correspondence

ABSTRACT: It is well-known that the spectral data of the Gaudin model associated to a finite semisimple Lie algebra is encoded by the differential data of certain flat connections associated to the Langlands dual Lie algebra on the projective line with regular singularities, known as oper/Gaudin correspondence. Recently, some progress has been made in understanding the correspondence associated with affine Lie algebras.

I will present a physical perspective from Kondo line defects, physically describing a local impurity chirally coupled to the bulk conformal field theory. The Kondo line defects exhibit interesting integrability properties and wall-crossing behaviors, which are encoded by the generalized monodromy data of affine opers. In the trigonometric setting, this reproduces the known ODE/IM correspondence in the physics literature. I will explain how the recently proposed 4d Chern Simons theory provides a new perspective which suggests the possibility of a physicists’ proof. Along the way, I will also present new examples of ODE/IM correspondences. The talk is based on [2003.06694][2010.07325][2106.07792] in collaboration with D. Gaiotto, J Lee, B. Vicedo.

## Zhenhua Liu: Lectures

### September 14, 2021

TITLE: Every finite graph arises as the singular set of a compact 3-d calibrated area minimizing submanifold

ABSTRACT: Calibrated submanifolds play an important role in special holonomic geometries. Prime examples include associative and coassociative submanifolds, special Lagrangian submanifolds, and Cayley subamnifolds. However, currently, we do not know much about the moduli space of such submanifolds. Just recall that a nice sequence of smooth algebraic curves can converge to a curve with branch points. Thus, it’s both inevitable and important to understand the singularities of calibrated submanifolds and how the singularities are formed. Now, Almgren’s Big Regularity Theorem and De Lellis-Spadaro’s new proof show that n-dimensional area minimizing integral currents are smooth manifolds outside of a singular set of dimensions at most n-2. Since calibrated submanifolds are area minimizing, we have this optimal bound on the dimension of their singular sets. A more geometrically applicable question is the fine structure of the singular set. The problem has been settled in dimension 2 by Chang and De Lellis-Spadaro-Spolaor, and they prove that all 2-dimensional area minimizing currents are branched with minimal immersion in the interior. Starting from dimension 3, very little is known. In particular, it is not even known if a line segment can appear as the singular set of a 3-d calibrated submanifold. In this direction, we show that given any (not necessarily connected) finite graph in the combinatorial sense, we can construct a calibrated 3-dimensional calibrated submanifold on a 7-dimensional closed compact Riemannian manifolds, so that the singular set of the surfaces consist of precisely this finite graph. The calibration is a modification of the special Lagrangian form. Both the metric and the calibration form are smooth. The constructions are based on some unpublished ideas of Professor Camillo De Lellis and Professor Robert Bryant.

## Daniel Platt: Lectures

### September 15, 2021

TITLE: New estimates for G2-structures on resolutions of orbifolds

ABSTRACT: Joyce and Karigiannis extended the generalised Kummer construction and constructed torsion-free G2-structures on resolutions of G2-orbifolds. In the talk I will explain a different analytic setup to study the same problem, using weighted Hölder norms, which gives improved control over the torsion-free G2-structure and a slightly simpler proof compared to the original proof. This has applications in G2-instantons, and potential applications in associative submanifolds and resolutions of singularities at different length scales. This is the content of arXiv:2011.00482.

## 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 .

## Panagiota Daskalopoulos: Lectures

~~May 26, 2021~~

Lecture cancelled

TITLE: Ancient compact solutions to Ricci flow and Mean curvature flow

ABSTRACT:

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.

## 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 , , 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.

## 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.