Trimester Program: "Metric Analysis"

Title: Analytically one dimensional planes and CLP

Abstract: I will discuss joint work with Guy C. David that is related to conformal dimension and analytically one dimensional planes. We aimed to solve two quite different problems. Failing to solve either of them, we instead proved a somewhat surprising implication between them. The first problem is if a certain two dimensional version of the Laakso diamond space attains its conformal dimension. The second problem is if a metric and measure exists on the two dimensional sphere, which satisfies the Poincaré inequality, and whose Cheeger co-tangent bundle is one dimensional. Our result shows that a positive answer to the first implies a positive answer to the second. Depending on your perspective, the first of these seems possible, while the second seems impossible. In the talk, I will explain these problems, and an interesting connection to certain three dimensional hyperbolic buildings.

The seminar will be followed by coffee/tea and cake in the usual room.

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Title: Sublinear bilipschitz equivalence and symmetric spaces

Abstract: Sublinear bilipschitz equivalence is a weakening of quasi-isometry arising from considerations about asymptotic cones of metric spaces. It occurs between certain pairs of homogeneous Riemannian manifolds which are not quasiisometric. I will present the classification of Riemannian globally symmetric spaces up to sublinear bilipschitz equivalence, and some of the tools it involves. This is joint work with Ido Grayevsky.

Title: A counterexample to Kleiner's conjecture

Abstract: For Q > 1, a Q-Loewner space is a metric space satisfying arguably one of the strongest pair of analytic conditions: Q-Ahlfors regularity and Q-Poincare inequality. For such metric spaces, there is a rich theory of quasiconformal mappings developed by Heinonen and Koskela. This theory is also useful whenever a metric space is quasisymmetric to a Loewner space.

Every metric space that is quasisymmetric to a Loewner space satisfies a natural discrete variant of the Loewner property, the combinatorial Loewner property (CLP).  In 2006, Kleiner conjectured that the converse holds for self-similar metric spaces. CLP is a very generic property among self-similar fractals and boundaries of hyperbolic groups, and it is often fairly easy to verify. Thus, a positive answer to the conjecture would have led to many new Loewner structures. In my recent joint work with Sylvester Eriksson-Bique, we disproved Kleiner’s conjecture by constructing a self-similar CLP-space that is not quasisymmetric to any Loewner space.

The seminar will be followed by coffee/tea and cake in the usual room.

Title: Minimal surfaces in symmetric spaces and Labourie's Conjecture

Abstract: For S a closed surface of genus at least 2, Hitchin representations from π₁(S) to PSL(n,R) naturally generalize Fuchsian representations to PSL(2,R) (which are essentially equivalent to hyperbolic metrics on S). Labourie proved that every Hitchin representation comes with an invariant minimal surface in the corresponding Riemannian symmetric space. Motivated by questions in higher Teichmüller theory, he conjectured that uniqueness holds as well. He then proved the conjecture for n=3.

In this talk, we'll explain how we produced large area minimal surfaces that give counterexamples to Labourie's conjecture for all n at least 4, and along the way we'll survey some aspects of the theory of harmonic maps and minimal immersions to symmetric spaces. Time permitting, we will discuss some more recent and related results that make contact with the theory of harmonic maps to buildings.

This is all joint work with Peter Smillie.

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Title: The Quantitative Geometry of Geodesics

Abstract: The goal of quantitative geometry is to provide effective versions of known existence theorems for geometric objects. For example, following Serre's proof of the existence of infinitely many geodesics connecting any two points on a closed Riemannian manifold, one may attempt to prove a length bound for these geodesics. In this talk, we will provide a survey of current quantitative theorems concerning geodesics and explore how such results can be proven. In particular, we discuss recent work on the existence of "short" geodesics that meet a given submanifold orthogonally.

• Video

Title: Characterization of metric spaces with a metric fundamental class

Abstract: We study geometric and analytic aspects of metric manifolds (metric spaces homeomorphic to a closed, oriented manifold). We discuss different properties of such spaces that guarantee the existence of a non-trivial integral current without boundary. Such an object should be thought of as an analytic analog of the fundamental class of the space and can also be interpreted as giving a way to make sense of Stokes' theorem in this setting.

• Video