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# Category Archives: Lectures

## 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.

## Xenia de la Ossa: Lectures

### January 8, 2018

TITLE: The geometry and moduli of heterotic G2 structures

ABSTRACT: A heterotic system is a quadrupole , where is a seven dimensional manifold with an integrable structure and is the corresponding associative three form, is a bundle on with an instanton connection , and is an instanton connection on the tangent bundle . is a three form given in terms of the field and the Chern-Simons forms of and (the anomaly cancelation condition) which is further constrained so that it is equal to a natural three form uniquely determined by the structure on . This constraint mixes up the geometry of with that of the bundles.

I will describe the tangent space of the moduli space of these systems. We first prove that a heterotic system is equivalent to an exterior covariant derivative acting on forms with values on the bundle which satisfies , for some appropriately defined projection of the operator . Remarkably, this equivalence implies the (Bianchi identity of) the anomaly cancellation condition. We show that the infinitesimal moduli space is given by the cohomology group and that it is finite dimensional. Our analysis leads to results that are of relevance to all orders in . Time permitting, I will comment on work in progress about the finite deformations of heterotic systems and the relation to differential graded Lie algebras.

From the physics perspective these structures give rise to very interesting three dimensional gauge supergravity theories (on Minkowski or AdS3) with only N=1 supersymmetry. Very little is known about these theories, as opposed to those with or , however we seem to have just enough supersymmetry to be able to deduce some interesting features of the effective field theories.

## 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.

## Hartmut Weiß: Lectures

### January 11, 2018

TITLE: On the asymptotic geometry of the Hitchin metric

ABSTRACT: I will report on recent joint work with Rafe Mazzeo, Jan Swoboda and Frederik Witt on the asymptotic geometry of the Hitchin metric. This is the natural metric on the moduli space of Higgs bundles. We describe the difference to a more elementary semiflat metric, thus confirming part of a more general proposal of Gaiotto, Moore and Neitzke.

## Benjamin Sibley: Lectures

### January 8, 2018

TITLE: A complex analytic structure on the compactification of Hermitian-Yang-Mills moduli space

ABSTRACT: A key aspect of gauge theory is finding a suitable compactification for the moduli space instantons. For instantons on higher dimensional manifolds, a rough compactification has been defined by Tian, analogous to Uhlenbeck’s compactification of the moduli space of anti-self-dual connections on a four-manifold.

In the case when the base manifold is Kähler, and the bundle in question is hermitian, instantons which are unitary and give rise to a holomorphic structures are Hermitian-Yang-Mills connections. A sequence of such connections is known to bubble at most along a codimension 2 analytic subvariety, and so one might hope that the resulting compactification has the structure of a complex analytic space. I will attempt to explain why this true in the case when the base is projective. This gives a higher dimensional analogue of a theorem of Jun Li for algebraic surfaces. This is joint work in progress with Daniel Greb, Matei Toma, and Richard Wentworth.