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Author Archives: Victoria Hain

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 R^+ 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 G_2 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 & 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.

Matei Toma: Lectures

January 10, 2018
TITLE: Moduli spaces of semistable sheaves with respect to Kähler polarizations

ABSTRACT: For a compact Kähler manifold (X,\omega) the Kobayashi-Hitchin correspondence gives homeomorphisms between moduli spaces of irreducible Hermite-Einstein connections and moduli spaces of stable vector bundles on X. 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 X is projective and \omega 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 L^2 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.

Gauge Theory and Special Holonomy, Imperial College, January 8-12, 2018

Arrival date: Sunday, January 7, 2018. Departure date: Saturday, January 13, 2018.

Talks will take place at Huxley Building   [ click for map ] —
→ Monday-Wednesday: Huxley Building room 140
→ Thursday-Friday: Huxley Building room 340 & 139 (see chart below)


Schedule

MON 8 JAN

TUE 9 JAN

WED 10 JAN

THU 11 JAN

FRI 12 JAN

9.00 -9.30

Registration

9.30-10.30

Walpuski  Joyce Haydys Weiß
(Huxley 340)
Li
(Huxley 340)

10.30-11.00

Q+A Q+A Q+A Q+A Q+A

11.00-11.30

Coffee Coffee Coffee Coffee Coffee

11.30-12.30

Thomas Walpuski Thomas He
(Huxley 340)
Acharya
(Huxley 340)

12.30-2.00

Lunch Lunch Lunch Lunch Lunch

2.00-3.00

Sibley Doan Toma Cherkis
(Huxley 139)
Nordström
(Huxley 340)

3.00-3.30

Coffee Coffee Coffee Coffee Coffee

3.30-4.30

de la Ossa Discussion Svanes Discussion
(Huxley 139)
Chen
(Huxley 340)

7.00-9.00

Social dinner

Speakers:

 

Yuguang Zhang: Lectures

September 14, 2017
TITLE: Collapsing of hyperkahler manifolds

ABSTRACT: In this talk, we study the collapsing of hyperkahler metrics on projective holomorphic symplectic manifolds along holomorphic Lagrangian fibrations. We prove that the Gromov-Hausdorff limits are compact metric spaces, which are half-dimensional special Kahler manifolds outside singular sets of real Hausdorff codimension 2.

Eirik Svanes: Lectures

January 10, 2018
TITLE: On marginal deformations of heterotic G2 geometries

ABSTRACT: A seven dimensional supersymmetric heterotic string compactification is a G_2 structure manifold Y equipped with an instanton bundle V, for which the geometry and bundle satisfy several coupled differential constraints. Ignoring higher curvature corrections, this includes G_2 holonomy manifolds with instanton bundles, but can also be more generic. Recently, the infinitesimal moduli of such compactifications was derived and identified with the first cohomology of a particular differential complex. I will review this result, and proceed to re-derive it from the two-dimensional sigma model point of view, whose target space is the above mentioned heterotic G_2 geometry. In particular, I will identify the worldsheet BRST operator whose cohomology is isomorphic to the infinitesimal deformations of the G_2 geometry. I will explain new concepts as they are introduced, in an effort to make the talk accessible to non-experts.

Slides of Lecture

 

September 14, 2017
TITLE: On the Coupled Moduli Space of Exceptional Holonomy Manifolds with Instanton Bundles

ABSTRACT: Recent years have seen a renaissance in the construction and study of new examples of manifolds with exceptional holonomy, instanton bundles over these spaces and their applications in physics and string theory. Due to anomalies and alpha’ corrections, the bundle often has a non-trivial back-reaction on the base geometry, and it can be important to keep this in mind when studying aspects of the solutions such as the moduli problem. These corrections are of particular importance in the context of the heterotic string, and I will review some recent work that highlights this and discuss the heterotic moduli problem in particular.

Slides of Lecture