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Author Archives: Victoria Hain

Yuri Tschinkel: Lectures

September 8, 2022
TITLE: Equivariant Birational Geometry

Slides of Lecture

Camillo De Lellis: Lectures

September 8, 2022
TITLE: Area-minimizing integral currents: singularities and structure

ABSTRACT: Almgren’s famous Big Regularity Paper proves that the interior singular set of any m-dimensional area-minimizing integral current T in any smooth Riemannian manifold M has (Hausdorff) dimension at most m-2. Except for the case m=2, when it was proved that interior singularities are isolated, little is known about the structure of the singular set. Moreover, a recent theorem by Liu proves that we cannot expect it to be a C1 (m-2)-dimensional submanifold (unless the ambient M is real-analytic) as in fact, it can be a fractal set of any Hausdorff dimension α ≤ m-2. On the other hand, it seems likely that it is an (m-2)-rectifiable set, i.e., that it can be covered by countably many C1 submanifolds.

In this talk, I will explain why the problem is challenging and how it can be broken down into easier pieces following a recent joint work with Anna Skorobogatova.

Lecture postponed.

Jiahua Tian: Lectures

September 14, 2022
TITLE: 5D SCFTs from C3 Orbifolds

ABSTRACT: In this talk, we will consider the orbifold singularities X = C3/Γ where Γ is a finite subgroup of SU(3). M-theory on X gives rise to a rich class of 5D SCFTs. We will study these SCFTs via 3D McKay correspondence which relates the group theoretical data of \Gamma to the physical properties of the 5D SCFT. We find some theories in this class can be obtained by gauging certain discrete symmetries in other 5D SCFTs. I will report on recent progress on relating 5D SCFTs to 4D electric-magnetic duality in a class of G2 compactification.

Slides of Lecture

Ling Lin: Lectures

September 13, 2022
TITLE: Gravity, geometry and generalized symmetries

ABSTRACT: I will discuss how geometric constraints in string compactifications inspire consistency conditions of quantum theories of gravity, which can be formulated in the framework of generalized symmetries.

Slides of Lecture

Greg Parker: Lectures

September 13, 2022
TITLE: Gluing Z2-Harmonic Spinors

ABSTRACT: Z2-harmonic spinors are singular generalizations of classical harmonic spinors and appear in two contexts in gauge-theory. First, they arise as limits of sequences of solutions to equations of Seiberg-Witten type on low-dimensional manifolds; second, they are the simplest type of (singular) Fueter section–objects which arise naturally on calibrated submanifolds in the study of gauge theory on manifolds with special holonomy. These two pictures are related by proposals for defining invariants of manifolds with special holonomy.

In this talk, after giving an introduction to these ideas, I will discuss a gluing result that shows a generic Z2-harmonic spinor on a 3-manifold necessarily arises as the limit of a family of two-spinor Seiberg-Witten monopoles. Due to the singularities of the Z2-harmonic spinor, the relevant operators in the gluing problem are only semi-Fredholm and possess an infinite-dimensional cokernel. To deal with this, the proof requires the analysis of families of elliptic operators degenerating to a singular limit, and the study of deformations of the singularities which are used to cancel the infinite-dimensional cokernel. At then end, I will discuss related problems in gauge theory and calibrated geometry.

Gorapada Bera: Lectures

September 13, 2022
TITLE: Deformations and gluing of asymptotically cylindrical associatives

ABSTRACT: Given a matching pair of asymptotically cylindrical (Acyl) G_2 manifolds the twisted connected sum construction produces a one parameter family of closed G_2 manifolds. We describe when we can construct closed rigid associatives in these closed G_2 manifolds by gluing suitable pairs of Acyl associatives in the matching pair of Acyl G_2 manifolds. The hypothesis and analysis in the gluing theorem requires some understanding of the deformation theory of Acyl associatives which will also be discussed. At the end we will describe examples of closed associatives coming from Acyl holomorphic curves or special Lagrangians.

Federico Trinca: Lectures

September 12, 2022
TITLE: T^2-invariant associatives in G_2 manifolds with cohomogeneity-two symmetry

ABSTRACT: A classical way to construct calibrated submanifolds is via symmetry reduction. In this talk, we will consider G_2 manifolds with a T^2\times SU(2) structure-preserving action of cohomogeneity-two. For each of these manifolds, we describe the geometry of the T^2-invariant associative submanifolds using moment type maps for the group action. As an application, we describe an associative fibration, in the Karigiannis–Lotay sense, on the Bryant–Salamon manifolds S^3\times \mathbb{R}^4.
This is joint work in progress with B. Aslan.

Izar Alonso: Lectures

September 12, 2022
TITLE: Heterotic systems, balanced SU(3)-structures and coclosed G_2-structures in cohomogeneity one manifolds

ABSTRACT: When considering compactifications of heterotic string theory down to 4D, the Hull–Strominger system arises over a six-dimensional manifold endowed with an invariant nowhere-vanishing holomorphic (3,0)-form. When compactifying down to 3D, we get the heterotic G_2 system over a manifold with a G_2-structure. In this talk, we describe these systems and then study the existence of some of the geometric structures required by them in the cohomogeneity one setting.

For the former one, we provide a non-existence result for balanced non-Kähler SU(3)-structures which are invariant under a cohomogeneity one action on a simply connected six-manifold. For the later one, we find a family of coclosed G_2-structures on certain seven-dimensional cohomogeneity one manifolds. Part of this talk is based on a joint work with F. Salvatore.

Slides of Lecture

Henry Liu: Lectures

September 12, 2022
TITLE: Multiplicative vertex algebras and wall-crossing in equivariant K-theory

ABSTRACT: K-theory is an interesting multiplicative refinement of
cohomology, and many cohomological objects arising in enumerative
geometry have K-theoretic analogues — modular forms become Jacobi
forms, Yangians become quantum affine algebras, etc. I will explain
how this sort of refinement goes for vertex algebras. As an
application, Joyce’s recent “universal wall-crossing” machine, which
operates by making the homology of certain moduli stacks into vertex
algebras, can be lifted to equivariant K-theory, e.g. thereby proving
the main conjecture on semistable invariants in refined Vafa-Witten
theory. In a different direction, I expect there to be some hidden
multiplicative vertex algebra structure on the aforementioned quantum
affine algebras, which can be viewed as symmetry algebras controlling
various enumerative and physical theories.

Aaron Kennon: Lectures

September 11, 2022
TITLE: Geometric Flows of 3-Sasakian Structures

ABSTRACT: Geometric flows of G_2-Structures are expected to be valuable tools for determining when a G_2-Structure with torsion may be deformed to one which is torsion-free. Several flows of G_2-Structures have been proposed to provide insight into this question, including the Laplacian flow and the Laplacian coflow. Here we consider an alternative application of these geometric flows to the study of Nearly Parallel G_2-Structures, specifically those originating from 3-Sasakian geometry. We write down an ansatz for co-closed G_2-Structures given in terms of the 3-Sasakian data and consider how scaled versions of the Laplacian flow and coflow behave when we start the flows at one of these structures. These results provide us with insight into the stability/instability of the Nearly Parallel G_2-Structures which are special co-closed G_2-Structures in this ansatz. We then can compare these stability results with the analogous conclusions for the scaled Ricci flow starting at a G_2-metric corresponding to our ansatz for the G_2-Structure. This is joint work with Jason Lotay.