Abstracts Riemannian 2024

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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)

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 parallel G2-structures on the Berger space. This talk is based on arXiv2310.11233 and a joint work in progress with Simon Salamon.

 Conifold transitions are a mechanism in which a Calabi–Yau 3-fold is deformed into another by contracting (-1,-1)-curves and smoothing out the resulting conical singularities. It is fantasized that all Calabi–Yau 3-folds can be linked by a sequence of these transitions. Since this process changes the Hodge numbers, it can deform a K\”{a}hler Calabi–Yau 3-fold into a non-K\”{a}hler one, which has sparked the study of differential geometric structures that are preserved by such transitions. In this talk, we will briefly discuss string-theoretic generalizations of the K\”{a}hler condition. We will also outline the construction of balanced metrics and Hermitian Yang–Mills metrics through conifold transitions by Fu–Li–Yau and Collins–Picard–Yau respectively before showing that conifold transitions are continuous in the Gromov–Hausdorff topology. This is based on work-in-progress with Benji Friedman and Sébastien Picard.

This talk discusses joint work with Dwivedi and Platt on associative submanifold in G_2–manifolds arising from Joyce’s generalised Kummer construction. Time permitting, I will also touch upon the work of my student Dominik Gutwein on coassociative submanifolds.