RESEARCH TALKS

Da Rong Cheng (University of Miami)

A mapping approach to associative submanifolds

Abstract

Based on analogy with J-holomorphic maps, Aaron Smith proposed in 2011 a class of first-order PDEs whose solutions are mappings that conformally parametrize calibrated submanifolds in ambient spaces equipped with a vector cross product. I will report on previous and recent joint works with Spiro Karigiannis (Waterloo) and Jesse Madnick (Oregon) on the analytical properties and variational characterization of these mappings, focusing on the case of associative submanifolds in ambient spaces equipped with a G2-structure.

Tristan Collins (MIT)

Complete Calabi-Yau metrics and Monge-Ampere equations

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I will describe progress towards the construction of complete Calabi-Yau metrics on the complement of ample, simple normal crossings anti-canonical divisors. This talk will discuss joint works with Y. Li, and F. Tong and S.-T. Yau.

Aleksander Doan (Trinity College & UCL)

Cauchy-Riemann operators and quaternionic vortex equations

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It is well-known that the number of J-holomorphic curves in an almost manifold can change when we vary the almost complex structure J. I will talk about an analogous phenomenon in gauge theory on surfaces, where the number of solutions to certain partial differential equations can change when the equations are perturbed. These two phenomena are closely related, as they are both caused by a wall structure in the space of Cauchy-Riemann operators on a surface. This talk is based on joint work with Thomas Walpuski.

Shubham Dwivedi (Humboldt Universität)

Geometric flows of G2 and Spin(7)-structures

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We will discuss a family of flows of G2-structures on seven dimensional Riemannian manifolds. These flows are negative gradient flows of natural energy functionals involving various torsion components of G2-structures. We will prove short-time existence and uniqueness of solutions to the flows and a priori estimates for some specific flows in the family. We will discuss analogous flows of Spin(7)-structures. This talk is based on a joint work with Panagiotis Gianniotis and Spiro Karigiannis (arXiv:2311.05516) and another work in progress.

Teng Fei (Rutgers University Newark)

The geometry of 3-forms on symplectic 6-manifolds

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We study the geometries associated to various 3-forms on a symplectic 6-manifold of different orbital type. As an application, we show that how this can be used to find Lagrangian fibrations and other geometric structures of interest on symplectic manifolds.

Anna Fino (Università di Torino)

Strong G2-structures with torsion

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A $7$-manifold with a $G_2$-structure $\varphi$ admits a $G_2$-connection with totally skew-symmetric torsion if and only if $d * \varphi = \theta \wedge * \varphi$, where $\theta$ is the Lee form of the $G_2$-structure.
In the talk I will present recent results on $7$-manifolds admitting a $G_2$-connection with closed totally skew-symmetric torsion. In particular, I will discuss the twisted $G_2$ equation, which represents the $G_2$-analogue of the twisted Calabi-Yau equation for $SU(n)$-structures introduced by Garcia-Fernández, Rubio , Shahbazi and Tipler. The talk is based on a joint work with Lucia Merchan and Alberto Raffero.

Mario Garcia-Fernandez (Universidad Autónoma de Madrid and ICMA)

Pluriclosed flow and generalized geometry

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Pluriclosed flow, as introduced by Streets and Tian, is a parabolic evolution equation for hermitian metrics on a complex manifold which plays a distinguished role in the geometrization programme in complex non-Kähler geometry. In this talk I will overview the interplay between pluriclosed flow and generalized geometry and their salient implications to the existence theory. In particular, we will consider an extension of the flow for transitive courant algebroids with potential applications to Calabi-Yau threefolds. Based on joint work with J. Streets and R. Gonzalez Molina.

Andriy Haydys  (Université Libre de Bruxelles)

An index theorem for Z/2 harmonic spinors

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Multi-valued functions or sections appear in various areas of geometry and analysis, for example in the theory of minimal surfaces and in gauge theory. In this talk, I shall discuss mostly 2-valued harmonic spinors, in particular an index theorem for such objects in the setting where the branching locus is allowed to be singular. This is a joint project with R. Mazzeo and R. Takahashi.

Magdalena Larfors  (Uppsala University)

Strings and G2 structure

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Seven-dimensional, real, Riemannian manifolds with G2 structure have geometric properties that make them interesting in string theory. In this talk, I will discuss some of these aspects, and how they govern supersymmetric solutions of heterotic string theory.

Yang Li (MIT)

On the Donaldson Scaduto conjecture

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Part of the Donaldson-Scaduto conjecture is concerned with constructing certain special Lagrangians inside the Calabi-Yau 3-folds obtained from products of C with an An type hyperk¨ahler 4-manifold. We prove this conjecture by solving a real Monge-Amp`ere equation with a singular right-hand side, which produces a potentially singular special Lagrangian. Then, we prove the smoothness and asymptotic properties for the special Lagrangian using inputs from geometric measure theory. The method produces many other asymptotically cylindrical special Lagrangians. This is joint work with Saman Esfahani.

Jesse Madnick (University of Oregon)

Gauge Theory and Calibrated Geometry in Hyperkahler Manifolds

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In recent years, calibrated geometry and gauge theory in the Calabi-Yau, G2, and Spin(7) settings have seen significant advances. By contrast, the hyperkahler setting — the other Ricci-flat case on Berger’s list — has received less attention. In this talk, we explore various calibrated geometries and gauge-theoretic objects on hyperkahler manifolds, with a particular emphasis on conical singularities, Sp(n)-instantons, and deformed Sp(n)-instantons. This is joint work with Emily Windes, Ben Aslan, and Spiro Karigiannis.

Lucia Martín-Merchán (University of Waterloo)

Kähler-type identities for Nearly Kähler manifolds

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Verbitsky proved that Nearly Kähler 6-dimensional manifolds satisfy Kähler-type identities. These lead to a Hodge decomposition in the compact case, and restrictions on their Hodge numbers. In this talk, we present a new proof for most of these results that is independent of the dimension. This is work in progress with M. Albanese, S. Karigiannis and A. Milivojevic.

Goncalo Oliveira (Instituto Superior Tecnico IST)

Deformed Hermitian-Yang-Mills: an example

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Mirror symmetry is a somewhat mysterious phenomenon that relates the geometry of two distinct Calabi-Yau manifolds. In the realm of trying to understand this relationship an equation for a connection on a line bundle in a Kahler manifold appeared. This is commonly called the deformed Hermitian-Yang-Mills equation and I will explain what it is and some current joint work with Benoit Charbonneau and Rosa Sena-Dias which explicitly solves this equation on a specific setting. This helps in understanding the problem of the existence of solutions and explore (or rule out) possible stability conditions.

Daniel Platt (King’s College London)

New Spin(7)-instantons on compact manifolds

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Spin(7)-instantons are certain interesting principal bundle connections on 8-dimensional manifolds with a Spin(7)-structure. Not many examples of such instantons are known, which holds back the development numerical invariants using them. In the talk I will explain a new construction method for Spin(7)-instantons generating more than 20,000 examples. The construction takes place on Joyce’s first examples of compact Spin(7)-manifolds. In the talk, I will briefly review the manifold construction, which glues together an orbifold, an ALE space (Eguchi-Hanson space), and a product of two ALE spaces, which is a QALE space. I will then explain the instanton construction. It makes use of weighted Hölder norms that are known from other gluing constructions, but the presence of a QALE piece makes the analysis more interesting in our case. Time permitting, I will explain how we obtained a large number of examples. This is joint work with Mateo Galdeano, Yuuji Tanaka, and Luya Wang. (arXiv:2310.03451)

Ragini Singhal (Université Libre de Bruxelles)

Cohomogeneity-one nearly parallel G2 manifolds

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The metric cone over a nearly parallel G2 manifold has holonomy contained in Spin(7) hence they play an important role in the theory of special geometric structures. Although we get many examples of nearly parallel G2 manifolds coming from the 3-Sasakian geometry but there are no direct examples apart from the homogeneous ones. In this talk we will discuss the construction of cohomogeneity-one examples of nearly parallel G2 manifolds for which the principal orbits are SU(2)XSU(2) or its finite quotients. We will first parameterize the SU(3)-structure on the 6-dimensional principal orbit known as the nearly half-flat SU(3)-structure and then we will use this parameterization to discuss the SO(4)-invariant cohomogoneity-one nearly pa