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# Author Archives: Victoria Hain

## Rafe Mazzeo: Lectures

### April 11, 2018

TITLE: Analysis of elliptic operators on complete spaces with asymptotically regular collapsing geometry

ABSTRACT: Elliptic theory for asymptotically cylindrical and asymptotically conical spaces is now classical and can be approached in many ways. When dealing with slightly more intricate geometries at infinity it is often helpful or even necessary to use more sophisticated tools. This talk will discuss a general and systematic theory which leads to sharp mapping results for ALF/ALG type metrics and prospects for a similar theory for singular fibrations over QAC spaces.

### April 10, 2018

TITLE: The large-scale structure of the Hitchin moduli space

ABSTRACT: The moduli space of solutions to the Hitchin equations on a Riemann surface carries a natural hyperKaehler metric, and questions and conjectures about its asymptotic structure have emerged out of the physics literature. There has been a lot of progress on this recently. I will discuss recent results, showing why this space might reasonably be called QALG.

## Ruobing Zhang: Lectures

### April 12, 2018

TITLE: Gravitational collapsing of K3 surfaces II

ABSTRACT: We will exhibit some new examples of collapsed hyperkähler metrics on a K3 surface. This is my recent joint work with Hans-Joachim Hein, Song Sun and Jeff Viaclovsky. We will construct a family of hyperkähler metrics on a K3 surface which are collapsing to a closed interval. Geometrically, each regular fiber is a Heisenberg manifold and each singular fiber is a singular circle fibration over a torus. In our example, each bubble limit is either the Taub-NUT space or a complete hyperk\”ahler space constructed by Tian-Yau. The regularity estimates in this example in fact confirms a general picture given by the -regularity theorem we present in Lecture 2. Technically, our examples are achieved by a new gluing construction. Continuing Jeff Viaclovsky’s introduction to the construction of this example, we will go through the details of the proof. We will also discuss some variations of the main gluing construction and some possible developments.

### April 9, 2018

TITLE: Quantitative nilpotent structure and regularity theorems of collapsed Einstein manifolds

ABSTRACT: This talk is on the new developments of the structure theory for collapsed Einstein manifolds. We will start with some motivating examples of collapsed Ricci-flat manifolds. Our main focus is the -regularity and structure theorems for collapsed Einstein manifolds which is included in my joint work with Aaron Naber. First, in the context of manifolds with Ricci curvature uniformly bounded from below, we show that every point on such a manifold can be associated with a nilpotent rank which has a sharp upper bound. This follows from an effective version of the Generalized Margulis Lemma. The main part of the -regularity theorem gives the following dichotomy: either the curvatures are uniformly bounded or the nilpotent rank drops.

### April 9, 2018

TITLE: Introduction to Ricci curvature and the convergence theory

ABSTRACT: The first talk is an overview of the convergence and regularity theory of the manifolds with Ricci and sectional curvature bounds. Specifically, we will review some both classical and new structure theory such as the -regularity theorems, the fibration theorems, and the structure of the limit spaces. The main part is to introduce the analytic tools in studying the non-collapsing manifolds and we will see why most tools legitimately fail in the collapsed context. Another emphasis is the development of the Generalized Margulis Lemma which gives the local collapsing geometry at the level of the fundamental group.

## Jeff Viaclovsky: Lectures

### April 11, 2018

TITLE: Gravitational collapsing of K3 surfaces I

ABSTRACT: I will discuss a construction of collapsing sequences of Ricci-flat metrics on K3 surfaces with Tian-Yau and Taub-NUT metrics occurring as bubbles. This is joint work with Hans-Joachim Hein, Song Sun, and Ruobing Zhang. Lecture II will be given by Zhang.

## Guofang Wei: Lectures

### April 9, 2018

TITLE: Manifolds with integral curvature bound

ABSTRACT: We begin with a review of early joint work with P. Petersen on the Laplacian and volume comparison for manifolds with only integral Ricci curvature bounds. We then present recent joint work with X. Dai and Z. Zhang producing a local Sobolev constant estimate for such manifolds without assuming a lower bound on volume. We close with applications of this theorem to produce a maximum principle, a gradient estimate, and to extend the L_2 Hessian estimate of Cheeger-Colding and Colding-Naber to manifolds with only lower bounds on their integral Ricci curvature.

## Xiaochun Rong: Lectures

### April 11, 2018

TITLE: Collapsed manifolds with Ricci local bounded covering geometry

ABSTRACT: Collapsed manifolds with local bounded covering geometry (i.e., sectional curvature bounded in absolute value) has been well-studied; the basic discovery by Cheeger-Fukaya-Gromov is the existence of a compatible local nilpotent symmetry structures whose orbits point to all collapsed directions.

In this talk, we will report an on-going work in generalizing the structural result to collapsed manifolds with (partially) local Ricci bounded covering geometry; which may contain a large class of collapsed Calabi-Yau manifolds and Ricci flat manifolds with special holonomy. Our construction of local nilpotent symmetry structures does not reply on the work of Cheeger-Fukaya-Gromov; which gives alternative approach to the structural result.

## Xuemiao Chen: Lectures

### January 12, 2018

TITLE: Singularities of Hermitian Yang Mills connections and the Harder-Narasimhan-Seshadri filtration

ABSTRACT: I will talk about joint work with Song Sun on the tangent cones of Hermitian Yang Mills connections with point singularity.

## Siqi He: Lectures

### January 11, 2018

TITLE: The extended Bogomolny equations and generalized Nahm pole solutions

ABSTRACT: We will discuss Witten’s gauge theory approach to Jones polynomial and Khovanov homology by counting solutions to some gauge theory equations with singular boundary conditions. When we reduce these equations to 3-dimensional, we call them the extended Bogomolny equations. We develop a Donaldson-Uhlenbeck-Yau type correspondence for the moduli space of the extended Bogomolny equations on Riemann surface Σ times with Generalized Nahm pole singularity at Σ × {0} with the stable SL(2,R) Higgs bundle. This is joint work with Rafe Mazzeo.

## Sergey Cherkis: Lectures

### January 11, 2018

TITLE: Octonionic Monopoles and another look at the Twistor Transform

ABSTRACT: An octonionic monopole is a solution of an octonionic generalization of the Bogomolny equation. Conjecturally, it is dual to a solution of the Haydys-Witten equation and plays central role in using seven-dimentional gauge theory to provide invariants of knot and coassociative cycles in holonomy manifolds.

Motivated by the search for a model octopole solution, we present a twistorial view of the bow construction of instantons on the multi-Taub-NUT space. We emphasize its quaternionic formulation and its relation to the complex Ward construction, posing a question of similar octonionic-quaternionic relations for the octopole.

## Richard Thomas: Lectures

### January 8 and January 10, 2018

TITLE: Introduction to coherent sheaves

ABSTRACT: I will try to give a coherent introduction to sheaf theory.

Coherent sheaves can be thought of as singular holomorphic vector bundles on complex manifolds, and can be used to compactify moduli of bundles. They thus give a way to define higher dimensional gauge theory invariants on projective varieties, and give examples that demonstrate some of the phenomena that can arise on more general manifolds of special holonomy.

After an introductory first lecture I will focus on some of (depending on audience tastes): curve counting via sheaves, stable pairs, the relationship to GW theory (MNOP conjecture) and Gopaukmar-Vafa invariants, the Serre construction relating codimension two subvarieties to rank 2 bundles, smoothing of singularities of reflexive sheaves.

**January 10, 2018**:

**January 8, 2018**:

## Matei Toma: Lectures

### January 10, 2018

TITLE: Moduli spaces of semistable sheaves with respect to Kähler polarizations

ABSTRACT: For a compact Kähler manifold () the Kobayashi-Hitchin correspondence gives homeomorphisms between moduli spaces of irreducible Hermite-Einstein connections and moduli spaces of stable vector bundles on . Whereas gauge theoretical compactifications for these spaces are known to exist by work of Donaldson, Uhlenbeck and Tian, the question of constructing modular compactifications in complex geometry is still open in the above setting.

In this talk we report on some recent progress in this direction obtained by two different methods jointly with Daniel Greb and Julius Ross and with Daniel Greb and Peter Heinzner respectively. We deal with the case when is projective and is an arbitrary Kähler class, which arises in wall crossing phenomena in algebraic geometry. Unlike the first one, the second method is GIT-free and it is likely to extend to the general situation.