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There is a broad body of work devoted to proving theorems of the following form: spaces with infinitely many special sub-spaces are either nonexistent or rare. Such finiteness statements are important in algebraic geometry, number theory, and the theory of moduli space and locally symmetric spaces. I will talk about joint work with Simion Filip and David Fisher proving a finiteness statement of this kind in a differential geometry setting. Our main theorem is that a closed negatively curved analytic Riemannian manifold with infinitely many closed totally geodesic hypersurfaces must be isometric to an arithmetic hyperbolic manifold. The talk will be more focused on providing background and context than details of proofs and should be accessible to a general audience.

CMC (constant mean curvature) surfaces are critical with respect to area as long as we restrict to competitor surfaces which 'enclose the same volume'. The index of a CMC surface is the number of ways we can locally deform it to reduce its area whilst enclosing the same amount of volume. We will show that the index of a CMC surface is bounded linearly from above in terms of its genus and a Willmore-type energy i.e. the more unstable a surface is the more genus and/or ‘Willmore energy’ it must have. Joint work with Luca Seemungal. 

For a closed Riemann surface X, and an irreducible representation R from the fundamental group of X to PSL(2,C), a seminal theorem of Donaldson proves the existence of an R-equivariant harmonic map from the universal cover of X into hyperbolic 3-space. We shall discuss a generalization of this result to the case when X is a punctured Riemann surface arising from an element of the enhanced Teichmüller space, and R is a framed PSL(2,C)-representation. Our technique involves the harmonic map heat flow, and the relation between the asymptotic behaviour of a harmonic map and the singular-flat geometry of its Hopf differential. This is joint work with Gobinda Sau. 

-In this talk I will describe recent advances in our understanding of the behaviour of Lagrangian mean curvature flow when the evolving submanifolds are 2-dimensional surfaces. In this case, we now have improved understanding of singularity formation and important examples which validate aspects of the well-known Thomas-Yau and Joyce conjectures.

Harmonic maps to symmetric spaces are used in the non-Abelian Hodge correspondence to bridge surface group representations with Higgs bundles. In special cases, these harmonic maps are conformal and hence give minimal surfaces in a symmetric space. In the first lecture, we look at the case of SL(3,R) and describe some asymptotics via Blaschke metrics.

In the second lecture, we will look at higher genus minimal Lagrangians in CP^2. There will be objects reminiscent of Higgs bundles, but which are not Higgs bundles. This will involve loop group methods and satisfying a closing condition.

In these two lectures I will explain how loop groups and loop algebras can be used to express the equations for a harmonic map of a Riemann surface into a symmetric space by meromorphic data---a generalized Weierstrass representation. I will discuss how to apply this method to special situations such as the construction of constant mean curvature surfaces in the 3-sphere. The lectures are intended as an introduction into this topic.

In this second talk on Lagrangian mean curvature flow, I will focus on key examples which illustrate the theory from the previous talk. In particular, we will see important objects arising in symplectic and Riemannian geometry, such as the Clifford torus, Whitney sphere and Lawlor necks.

Maximal surfaces in 3-dimensional Lorentz-Minkowski space arise as solutions to the variational problem of local area maximizing among the spacelike surfaces. These surfaces are zero mean curvature surfaces, and maximal surfaces with singularities are called generalized maximal surfaces. Maxfaces are a special class of these generalized maximal surfaces where singularities appear at points where the tangent plane contains a light-like vector. I will present the construction of a new family of maxfaces of high genus that are embedded outside a compact set and have arbitrarily many catenoid or planar ends using the node opening technique. The surfaces look like spacelike planes connected by small necks. Among the examples are maxfaces of the Costa-Hoffman-Meeks type. More specifically, the singular set form curves around the waists of the necks. In generic and some symmetric cases, all but finitely many singularities are cuspidal edges, and the non-cuspidal singularities are swallowtails evenly distributed along the singular curves. This work is conducted in collaboration with Dr. Hao Chen, Dr. Anu Dhochak, and Dr. Pradip Kumar.

Lagrangian mean curvature flow is potentially a powerful tool for tackling problems in symplectic geometry via geometric analysis, by studying the existence problem for minimal Lagrangian submanifolds. In this first talk I will give an overview of Lagrangian mean curvature flow and describe some of the key foundational results.

Harmonic maps to symmetric spaces are used in the non-Abelian Hodge correspondence to bridge surface group representations with Higgs bundles. In special cases, these harmonic maps are conformal and hence give minimal surfaces in a symmetric space. In the first lecture, we look at the case of SL(3,R) and describe some asymptotics via Blaschke metrics.

In the second lecture, we will look at higher genus minimal Lagrangians in CP^2. There will be objects reminiscent of Higgs bundles, but which are not Higgs bundles. This will involve loop group methods and satisfying a closing condition.