Kyoto

I will introduce recent work on the two- and three-dimensional uniform spanning trees (USTs) that establish the laws of these random objects converge under rescaling in a space whose elements are measured, rooted real trees, continuously embedded into Euclidean space. (In the three-dimensional case, the scaling result is currently only known along a particular scaling sequence.) I will also discuss various properties of the intrinsic metrics and measures of the limiting spaces, including their Hausdorff dimension, as well as the scaling limits of the random walks on the two- and three-dimensional USTs. In the talk, I will attempt to emphasise where the differences lie between the two cases, and in particular the additional challenges that arise when it comes to the three-dimensional model.
    The two-dimensional results are joint with Martin Barlow (UBC) and Takashi Kumagai (Kyoto). The three-dimensional results are joint with Omer Angel (UBC) and Sarai Hernandez-Torres (UBC).

The quantum unipotent coordinate ring has a cluster algebra structure. On the other hand, this ring is isomorphic to the Grothendieck ring of the module category of quiver Hecke algebras (QHA). We can prove that cluster monomials of the quantum unipotent coordinate ring correspondi to real simple modules. This is a joint work with Seok-Jin Kang, Myungho Kim and Se-jin Oh.

Special Lagranigian submanifolds are area-minimizing Lagrangian submanifolds of Calabi--Yau manifolds. One can define the moduli space of compact special Lagrangian submanifolds of a (fixed) Calabi--Yau manifold. Mclean proves it has a structure of manifold (of dimension finite). It isn't compact in general, but one can compactify it by using geometric measure theory.

Kontsevich conjectured a mirror symmetry, and special Lagrangians should be "mirror" to holomorphic vector bundles. By using algebraic geometry one can compactify the moduli space of holomorphic vector bundles. By "counting" holomorphic vector bundles in Calabi--Yau 3-folds Richard Thomas defined holomorphic Casson invariants (Donaldson-Thomas invariants).

So far as I know it's an open question (probably very difficult) whether one can "count" special Lagrangians, or define a nice structure on the (compactified) moduli space of special Lagrangians.

To do it one has to study singularities of special Lagrangians.

One can smooth singularities in suitable situations: given a singular special Lagrangian, one can construct smooth special Lagrangians tending to it (by the gluing technique). I've proved a uniqueness theorem in a "symmetric" situation: given a symmetric singularity, there's only one way to smooth it (the point of the proof is that the symmetry reduces the problem to an ordinary differential equation).

More recently I've studied a non-symmetric situation together with Dominic Joyce and Joana Oliveira dos Santos Amorim. Our method is based on Lagrangian Floer theory, and is effective at least for pairs of two (special) Lagrangian planes intersecting transversely.

I'll give the details in the talk.

Special Lagrangian submanifolds are area minimizing Lagrangian submanifolds discovered by Harvey and Lawson. There is no obstruction to deforming compact special Lagrangian

submanifolds by a theorem of Mclean. It is however difficult to understand singularities of

special Lagrangian submanifolds (varifolds). Joyce has studied isolated singularities with multiplicity one smooth tangent cones. Suppose that there exists a compact special Lagrangian submanifold M of dimension three with one point singularity modelled on the Clliford torus cone. We may apply the gluing technique to M by a theorem of Joyce.

We obtain then a compact non-singular special Lagrangian submanifold sufficiently close to M as varifolds in Geometric Measure Theory. The main result of this talk is as follows: all special Lagrangian varifolds sufficiently close to M are obtained by the gluing technique.

The proof is similar to that of a theorem of Donaldson in the Yang-Mills theory.

One first proves an analogue of Uhlenbeck's removable singularities theorem in the Yang-Mills theory. One uses here an idea of a theorem of Simon, who proved the uniqueness of multiplicity one tangent cones of minimal surfaces. One proves next the uniqueness of local models for desingularizing M (see above) using symmetry of the Clifford torus cone.

These are the main part of the proof.