Workshop

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The talk is based on joint work with Fabian Lehmann. The main thrust of that work was to develop a deformation theory for torsion-free SL(3,C) structures on 6-manifolds with boundary, partially extending the well-known Torelli theory in the closed case. We will outline this and a number of related constructions and questions which arise. These include a local differential geometric invariant of closed 3-forms on 5-manifolds, an obstruction space related to CR theory and a possible generalisation  of the affine isoperimetric inequality.

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Special Lagrangian submanifolds are a distinguished class of minimal surfaces inside Calabi-Yau manifolds. I will discuss the construction of the special Lagrangian analogue of the pair of pants surfaces in all dimensions, using a combination of PDE and geometric measure theory methods.

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Using an approach developed by Melrose, we will explain how to show that quiver varieties are quasi-asymptotically conical (QAC) provided some genericity assumption holds.  Relying on  this fine description of the geometry at infinity and a spectral gap for singular 3-Sasakian manifolds,  we will then explain how the reduced L^2 cohomology of a quiver variety can be computed, confirming a prediction made by Vafa and Witten in 1994.  This is a joint work with Panagiotis Dimakis.

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There are numerous speculative proposals for constructing enumerative invariants of Calabi-Yau 3-folds by counting special Lagrangian submanifolds. One of the many difficulties in defining such invariants is that special Lagrangians may collapse to multiple covers as parameters vary. For two-fold covers specifically, work of Siqi He shows that the local deformation theory is governed by the existence of Z_2-harmonic 1-forms. On the other hand, Taubes originally defined Z_2-harmonic 1-forms to describe limits of diverging sequences of flat SL(2,C) connections. In this talk I will discuss the connection between these two theories, and discuss progress toward and obstructions to gluing results providing the obverse of He's result in the realm of flat SL(2,C) connections.

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Allard’s regularity theorem provides regularity for stationary integral varifolds near a plane of multiplicity one: they must be smooth minimal graphs. What can be said when the plane instead has (integer) multiplicity larger than 1 has remained an open question and is crucial to the global regularity of stationary integral varifolds, as it currently prevents us concluding almost everywhere regularity. Examples such as the catenoid and complex varieties illustrate the complication. Even uniqueness of a multiplicity 2 plane as a tangent cone is not known in general, and those cases where there are results have additional variational assumptions, such as being area minimising.

In this talk, I will discuss a new regularity result for stationary integral varifolds near planes of multiplicity 2. This holds in arbitrary dimension and codimension, providing full regularity as a Lipschitz 2-valued graph (with certain improved estimates). The only additional assumption one needs is of a topological nature on the support of the varifold: no other variational assumptions are needed other than stationarity. This assumption can be directly checked in several cases of interest. I will also discuss hopes for the future. This is all based on joint work with Spencer Becker-Kahn and Neshan Wickramasekera.

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In his monumental work in the 1980s, Almgren showed that for $n$-dimensional locally area minimizing rectifiable currents of codimension $> 1$, the singular set has Hausdorff dimension at most $n-2$. In contrast with codimension one, it is well-known that area minimizers of higher codimension can admit branch point singularities, at which at least one tangent cone is an $n$-dimensional plane with integer multiplicity $\geq 2$. Almgren’s lengthy proof consisted of first using an elementary argument based on tangent cone type to show that the set of non-branch point singularities has Hausdorff dimension at most $n-2$, and then using a powerful array of new ideas to bound the dimension of the set of all branch point singularities. In this strategy, the exceeding complexity of the argument stems largely from the lack of an estimate giving decay of $T$ towards a unique tangent plane at a branch point.

We will discuss a new approach (joint work with Neshan Wickramasekera) in which we first focus on showing uniqueness of tangent cones at $\mathcal{H}^{n-2}$-a.e. singular point, and then we study the size and fine structure of the singular set. As part of our method, we introduce an approximate monotonicity formula for a new intrinsic planar frequency function which quantifies the rate at which $T$ decays towards an $n$-dimensional plane. We decompose the singular set of $T$ into the set of branch points $\mathcal{B}$ at which $T$ decays to a unique tangent plane faster than a fixed rate (of radius to a power) and the set $\mathcal{S}$ of all remaining singular points, which a priori might contain a large set of branch points. Using the monotonicity of planar frequency functions together with the relatively elementary parts of Almgren’s proof (namely Lipschitz and harmonic approximation), we obtain locally uniform estimates indicating that near each point of $\mathcal{S}$ and at each sufficiently small scale, $T$ is significantly closer to some non-planar cone than it is to any plane. An analysis of singularities using the planar frequency function and locally uniform estimates recovers Almgren’s dimension bound for the singular set of $T$ in a simpler way. Using a blow-up methods of L. Simon and Wickramaskera and the locally uniform estimates for $\mathcal{S}$, we show that $T$ has a unique non-planar tangent cone at $\mathcal{H}^{n-2}$-a.e. point of $\mathcal{S}$, and thus $T$ has a unique tangent cone at $\mathcal{H}^{n-2}$-a.e. singular point. Again using the blow-up method together with the approximate monotonicity of the planar frequency function, we show that the entire singular set, including $\mathcal{B}$, is countably $(n-2)$-rectifiable and that $T$ has a unique homogeneous harmonic blow-up relative to its tangent plane at each point of $\mathcal{B}$.