Colloquium

Abundant existence of minimal hypersurfaces

This is the third talk in a series of three talks on the variational theory of minimal hypersurfaces. In this talk, I will discuss the following theorems. By the works of Marques-Neves and Song, every closed Riemannian manifold M^n, 3 \leq n \leq 7, contains infinitely many closed, minimal hypersurfaces. This was conjectured by Yau. For generic metrics, stronger results have been obtained. Irie, Marques and Neves proved that for a generic metric g on M, the union of all closed, minimal hypersurfaces is dense in (M, g). This theorem was later quantified by Marques, Neves and Song; they proved that for a generic metric there exists an equidistributed sequence of closed, minimal hypersurfaces in (M, g). In higher dimensions, Li proved that every closed Riemannian manifold, equipped with a generic metric, contains infinitely many closed minimal hypersurfaces. The Weyl law for the volume spectrum, proved by Liokumovich, Marques and Neves, played a major role in the proofs of these theorems. Inspired by the abundant existence of closed minimal hypersurfaces, we showed that the number of closed c-CMC hypersurfaces in a closed Riemannian manifold M^n, n \geq 3, tends to infinity as c \rightarrow 0^+.  

Morse theory for the area functional

This is the second talk in a series of three talks on the variational theory of minimal hypersurfaces. In this talk, I will discuss the following theorems. When the ambient dimension 3 \leq n \leq 7, Marques and Neves showed that the index of the min-max minimal hypersurface is bounded from above by the dimension of the parameter space. Zhou proved the multiplicity one property for the min-max minimal hypersurfaces, which was conjectured by Marques and Neves. In the Almgren-Pitts min-max theory, the min-max width is realized by the area of a closed minimal hypersurface, with the possibility that the connected components of the minimal hypersurface can have different multiplicities. The multiplicity one theorem says that for a generic metric, all the min-max minimal hypersurfaces have multiplicity one. Using the Morse index upper bound and multiplicity one theorem, Marques and Neves have proved the following theorem. For a generic metric g, there exists a sequence of closed, embedded, two-sided minimal hypersurfaces {S_p} in (M^n, g) such that the Morse index Ind(S_p) = p and area(S_p) \sim p^{1/n}. In higher dimensions (i.e. when n \geq 8), the Morse index upper bound has been proved by Li.

Min-max construction of minimal hypersurfaces

In the 1960s, Almgren developed a min-max theory to construct closed minimal submanifolds in an arbitrary closed Riemannian manifold. The regularity theory in the co-dimension 1 case was further developed by Pitts and Schoen-Simon. In particular, by the combined works of Almgren, Pitts and Schoen-Simon, in every closed Riemannian manifold M^n, n \geq 3, there exists at least one closed, minimal hypersurface. Recently, the Almgren-Pitts min-max theory has been further developed to show that minimal hypersurfaces exist in abundance.

In addition to the Almgren-Pitts min-max theory, there is an alternative PDE based approach for the min-max construction of minimal hypersurfaces. This approach was introduced by Guaraco and further developed by Gaspar and Guaraco. It is based on the study of the limiting behaviour of solutions to the Allen-Cahn equation. In my talk, I will briefly describe the Almgren-Pitts min-max theory and the Allen-Cahn min-max theory and discuss the question to what extent these two theories agree.

Gromov-Tischler theorem for symplectic stratified spaces

Singular symplectic spaces appear naturally as examples of reduced Hamiltonian phase spaces in physics as well as singular projective algebraic varieties in mathematics. We give a unified and geometric definition for these objects, and prove a singular variant of the Gromov-Tischler theorem: such a space with an integral symplectic form can always be embedded symplectically inside the complex projective space. On the way we discuss the topology of stratified spaces, symplectic reduction and h-principles. 

This is joint work with Mahan Mj.

The approximation by conjugation method in smooth ergodic theory

Limit sets of paths in Outer space

In analogy to the mapping class group acting on the Teichmuller space, we have the group of outer automorphisms of the free group acting on Culler-Vogtmann's `Outer space'. The limit sets of geodesics in Teichmuller space exhibit very interesting and varied phenomena with respect to the Teichmuller metric, Thurston metric and Weil Petersson metric. In this talk, we will look for similar results for `folding/unfolding' paths in Outer space.

Weil-Petersson geometry of Teichmuller space

The Weil-Petersson metric is a negatively curved, incomplete Riemannian metric on the Teichmuller space with connections to hyperbolic geometry. In this talk we present some results about the behavior of geodesics of the metric and its relation to subsurface coefficients in analogy with continued fraction expansions.

Dynamics on spaces of discrete sets

The Dimer Model in 3 dimensions

The dimer model, also referred to as domino tilings or perfect matching, are tilings of the Z^d lattice by boxes exactly one of whose sides has length 2 and the rest have length 1. This is a very well-studied statistical physics model in two dimensions with many tools like height functions and Kasteleyn determinant representation coming to its aid. The higher dimensional picture is a little daunting because most of these tools are limited to two dimensions. In this talk I will describe what techniques can be extended to higher dimensions and give a brief account of a large deviations principle for dimer tilings in three dimensions that we prove analogous to the results by Cohn, Kenyon and Propp (2000). 

This is joint work with Scott Sheffield and Catherine Wolfram.

A higher dimensional analog of Margulis' construction of expanders

The first explicit example of a family of expander graphs was quotients of the Cayley graph of a group G, having Property (T), by subgroups of finite index. This construction is due to Margulis, in a special case, and Alon-Milman in general.  We will discuss a higher dimensional analog of this result that can be obtained by replacing 'expander graphs' by 'higher spectral expanders', 'group having Property (T)'  by 'strongly n-Kazhdan group' and and 'Cayley graph' by 'n-skeleton of the universal cover of a K(G,1) simplicial complex'.  New examples of 2-dimensional spectral expanders are obtained using this construction.