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Author Archives: Victoria Hain

Ilyas Khan: Lectures

June 7, 2020 (Jointly with Alec Payne)
TITLE: Uniqueness of Asymptotically Conical Gradient Shrinking Solitons in G2-Laplacian Flow

ABSTRACT: The Laplacian flow is a natural geometric flow which deforms closed G2-structures on 7-manifolds. This flow could be a major avenue to insight into manifolds with G2-holonomy. However as with many geometric flows, singularities are expected to form in finite time. Self-similar soliton solutions to the flow are expected to play a significant role in the analysis of these singularities. In this talk, we consider self-shrinking solitons (these are necessarily noncompact) with prescribed asymptotics on their ends. In particular, we consider the important class of asymptotically conical (AC) shrinkers, the first examples of which were recently constructed by Haskins-Nordstrom. We describe our proof of the uniqueness of AC gradient shrinking solitons for the Laplacian flow of closed G2-structures: If two gradient shrinking solitons to Laplacian flow are asymptotic to the same closed G2-cone, then their G2-structures are equivalent, and in particular, the two solitons are isometric. This is joint work of Mark Haskins, Ilyas Khan and Alec Payne.

Alec Payne: Lectures

June 7, 2020 (Jointly with Ilyas Khan)
TITLE: Uniqueness of Asymptotically Conical Gradient Shrinking Solitons in G2-Laplacian Flow

ABSTRACT: The Laplacian flow is a natural geometric flow which deforms closed G2-structures on 7-manifolds. This flow could be a major avenue to insight into manifolds with G2-holonomy. However as with many geometric flows, singularities are expected to form in finite time. Self-similar soliton solutions to the flow are expected to play a significant role in the analysis of these singularities. In this talk, we consider self-shrinking solitons (these are necessarily noncompact) with prescribed asymptotics on their ends. In particular, we consider the important class of asymptotically conical (AC) shrinkers, the first examples of which were recently constructed by Haskins-Nordstrom. We describe our proof of the uniqueness of AC gradient shrinking solitons for the Laplacian flow of closed G2-structures: If two gradient shrinking solitons to Laplacian flow are asymptotic to the same closed G2-cone, then their G2-structures are equivalent, and in particular, the two solitons are isometric. This is joint work of Mark Haskins, Ilyas Khan and Alec Payne.

Chris Scaduto: Lectures

June 6, 2022
TITLE: Computing \nu-invariants of Joyce’s G_2-manifolds

ABSTRACT: Crowley and Nordström introduced the \nu-invariant of a G_2-structure on a 7-manifold, taking values in the integers modulo 48. This invariant, and its spectral refinement due to Crowley, Goette and Nordström, has been important in understanding the connected components of G_2 metric moduli spaces. In this talk I explain how to compute the \nu-invariant for many of the closed torsion-free G_2-manifolds defined by Joyce’s generalized Kummer construction.

Diarmuid Crowley: Lectures

June 10, 2022
TITLE: A prolegomena to finding higher homotopy in G_2-moduli spaces

ABSTRACT: I will begin this talk by reviewing topological approach to studying
the connected components
of G_2-moduli spaces via the \nu-invariant. Then I will look at the
possibility of detecting higher homotopy
groups in G_2-moduli spaces.

The idea is to define higher-dimensional \nu-invariants, which can detect
homotopy classes in \pi_i(G_2(M)) (where G_2(M) is the space of G_2-structures
on M) and even on the quotient \pi_i(G_2(M)) / \pi_i(\Diff(M)).

At the time of writing, passing to \pi_i of the actual moduli space G_2(M) /
Diff(M) or finding essential maps to the moduli space of torsion free
G_2-structures are open problems but we hope the methods may none the less be
of interest.

I will also discuss related results on the mapping class group of M and some
higher homotopy groups of Diff(M).

This is an early report on work in progress with Johannes Nordström and
Sebastian Goette.

Hongyi Liu: Lectures

June 7, 2022
TITLE: A compactness theorem for hyperkahler 4 manifolds with boundary

ABSTRACT:
A hyperkahler triple on a 4-manifold with boundary is a triple of symplectic
2-forms that are pointwise orthonormal with respect to the wedge product, so
its restriction to the boundary 3-manifold is a closed framing. Motivated by
previous work of Bryant, Fine-Lotay-Singer on the hyperkahler triple filling
problem, and a dimension 4 reduction of Donaldson’s elliptic boundary value
problem for G2 structures, we discuss some recent progress towards the
compactness part of these problems on compact 4-manifolds with boundary.

Ethan Torres: Lectures

June 8, 2022
TITLE: Getting High on Gluing Orbifolds

ABSTRACT:
Quantum field theories (QFTs) engineered from M-theory on singular non-compact manifolds often enjoy a rich dictionary between physical data and geometric quantities. For instance, when real codimension-4 ADE orbifold singularities extend out to the asymptotic boundary, the local operators of the QFT form representations of a (so-called) flavor group whose Lie algebra is specified by the ADE-types. We present a novel geometric procedure to calculate the global structure of this group. Additionally, taking into account line operators of the QFT, this structure may generalize to a 2-group and we give a geometric picture for this as well. For part I of this talk series, we will discuss these ideas in the context of 5d superconformal field theories engineered from M-theory on quotients of C^3.

Jeffrey Streets: Lectures

January 14, 2022
TITLE: Singular sets of generalized Einstein metrics

ABSTRACT: Various considerations in geometry and physics lead to natural generalizations of Einstein metrics which are coupled to differential forms.  In this talk I will describe recent joint work with X. Fu and A. Naber on the singularity formation of sequences of such structures.  In particular we show that the limit spaces are regular outside of a set of codimension 4, and satisfy certain sharp integral estimates, leading to geometric applications.

Lecture Notes

Jihwan Oh: Lectures

January 12, 2022
TITLE: G2 instantons in twisted M-theory

ABSTRACT: I will discuss a string theory way to study G2 instanton moduli space and explain how to compute the instanton partition function for a certain G2 manifold. An important insight comes from the twisted M-theory on the G2 manifold. Building on the example, I will explain a possibility to extend the story to a large set of conjectural G2 manifolds and a possible connection to 4d N=1 SCFT via geometric engineering. This talk is based on a work with Michele Del Zotto and Yehao Zhou.

Slides of Lecture

Hiraku Nakajima: Lectures

January 10, 2022
TITLE: Symmetric bow varieties

ABSTRACT: In my joint work with Takayama, I showed that Coulomb branches of quiver gauge theories of affine type A, as defined in earlier joint work with Braverman and Finkelberg, are isomorphic to Cherkis’ bow varieties. For quiver gauge theories of affine type D, or more generally of classical affine types, we introduce symmetric bow varieties, which are fixed point loci of involution on bow varieties.

Slides of Lecture

Francis Kirwan: Lectures

January 10, 2022
TITLE: Hyperkahler implosion

ABSTRACT: The hyperkahler quotient construction, which allows us to construct new hyperkahler spaces from suitable group actions on hyperkahler manifolds, is an analogue of symplectic reduction (introduced by Marsden and Weinstein in the 1970s), and both are closely related to the quotient construction for complex reductive group actions in algebraic geometry provided by Mumford’s geometric invariant theory (GIT). Symplectic implosion was introduced twenty years ago by Guillemin, Jeffrey and Sjamaar; in some sense it abelianises symplectic reduction. Hyperkahler implosion is in turn an analogue of symplectic implosion; both are related to a generalised version of GIT providing quotients for non-reductive group actions in algebraic geometry.

Slide of Lecture