| Date | Speaker | Title |
May 28, 2019: Melkana Brakalova "p-integrable Teichmüller spaces from the quasi-conformal mappings point of view"
Abstract: There are different approaches to investigating properties of the universal Teichmüller space \(\mathcal{T} \) and its many subspaces. We consider \(\mathcal{T} \) to be the set of equivalent classes of quasisymmetric automorphisms of the real line or of the unit circle. We are interested in the p-integrable subspaces \(\mathcal{T}_p \), p>0, of \(\mathcal{T} \), which we define as the equivalent classes of quasisymmetric automorphisms that admit a q.c. extension (to the upper half-plane or the unit disk, resp.) with p-integrable complex dilatation w.r.t. the Poincaré metric (finite hyperbolic \(L^p \) norm). We show that such spaces belong to the little Teichmüller space and that they are continuously differentiable for \( 0 < p \leq 1 \) using local properties of quasiconformal maps. We discuss the length space property of the \( T^p \) spaces and some open questions. This is work in progress joint with V. Alberge.
May 14, 2019: Samuel Lin "From Pointwise to Global"
Abstract: We explore the following pointwise geometric condition on Riemannian manifolds. Let R be a curvature like tensor on a Riemannian manifold. The nullity of R at a point is defined as the dimension of the kernel of the tensor R. An n-dimensional Riemannian manifold is said to have R-conullity at most k if the nullity of R is at least n-k at every point. The main goal of this talk is to introduce the global theorems of Florit-Ziller and Schmidt-Wolfson for manifolds of conullity at most two. Ideas for possible generalizations will also be discussed.
April 30, 2019: Adam Epstein "Transversality principles in holomorphic dynamics"
Abstract:
The moduli space of all degree D rational maps is an orbifold of dimension 2D−2. We
present a language for describing dynamically natural subspaces, for example, the loci of
maps having
• specified critical orbit relations,
• cycles of specified period and multiplier,
• parabolic cycles of specified degeneracy and index,
• Herman ring cycles of specified rotation number,
or some combination thereof. We present a methodology for proving the smoothness and
transversality of such loci. The natural setting for the discussion is a family of deformation
spaces arising functorially from first principles in Teichmuller theory. Transversality
flows from an infinitesimal rigidity principle (following Thurston), in the corresponding
variational theory viewed cohomologically (following Kodaira-Spencer). Results for deformation
spaces may then be transferred to moduli space. Moreover, the deformation
space formalism and associated transversality principles apply more generally to finite
type transcendental maps.
April 16, 2019 in KH 008: Glen Whitney "It Matters How You Say It"
Abstract: Math is in part a quest to see similarities between things that are not obviously related. The desire to make mathematical principles apply in as wide a range of contexts as possible is one facet of that quest: the more general a statement, the more specific instances it ties together. One stumbling block in our way, however, is that when you try to generalize certain principles, you discover they can't possibly be true in a wider setting; but if you simply restate these principles in an equivalent but different formulation, then suddenly they become obviously applicable in the wider setting! This talk will explore this confounding phenomenon in the setting of trying to move from two dimensions to three in ordinary geometry, specifically relating to a simple way of computing areas and volumes.
April 9, 2019: Lien-Yung Kao "Pressure Metric for Teichmueller Spaces of Punctured Surfaces"
Abstract: Thurston pointed out that one can use variations of lengths of closed geodesics on hyperbolic surfaces to construct a Riemannian metric on the Teichmueller space. For closed surfaces cases, Wolpert proved this Riemannian metric is indeed the Weil-Petersson metric, and McMullen proposed a dynamics (more precisely, thermodynamic formalism) approach to recover this Riemannian metric and called it the pressure metric. In this talk, I will discuss how to extend this dynamics construction to non-compact finite area hyperbolic surfaces. If time permitted, I will also discuss several related (entropy) rigidity results.
April 2, 2019: Martin Bridgeman "Epstein Surfaces, W-Volume, and the Osgood-Stowe Differential"
Abstract: We show that Epstein surfaces and W-volume can be described simply in terms of the Osgood-Stowe differential, a generalization of the Schwarzian derivative. Using this, we obtain short proofs of many of the properties of Epstein surfaces and W-volume derived by Epstein and Krasnov-Schlenker as well as new results describing the variation of the W-volume in terms of thr Osgood-Stowe differential and a description of the W-volume in terms of curvature forms.
March 26, 2019: Ara Basmajian "The type problem and the geometry of Riemann surfaces"
Abstract: Riemann surfaces have a dual existence both as complex analytic objects and as (hyperbolic) geometric ones. While the complex analytic and geometric theory of finite type Riemann surfaces is well developed, such a study for infinite type (that is, infinitely generated fundamental group) Riemann surfaces is still in its infancy. This talk is about continuing work in our investigation of the relationship between the extremal lengths of arc families going out the ends of the surface, ergodicity of the geodesic flow, and growth rates of simple closed geodesics that go out the end. This is joint work with Hrant Hakobyan and Dragomir Saric.
November 6, 2018: Sema Salur "Calibrated Geometries and Applications"
Abstract: Calibrated submanifolds are distinguished classes of minimal submanifolds and their moduli spaces are expected to play an important role in geometry, low dimensional topology and theoretical physics. Examples of these submanifolds are special Lagrangian 3-folds for Calabi-Yau, associative 3-folds and coassociative 4-folds for G2 and Cayley 4-folds for Spin(7) manifolds. In this talk we first give an introduction to calibrated geometries and a survey of recent research on the deformation theory of calibrated submanifolds inside Ricci-flat manifolds. We then study the deformations of Lagrangian submanifolds and extend the theory to “Lagrangian Type” submanifolds inside G2 manifolds. If time permits, we will also discuss relations between G2 and contact structures.
October 30, 2018: Otis Chodosh "A splitting theorem for scalar curvature"
Abstract: This is joint work with Michael Eichmair and Vlad Moraru. I will discuss a scalar curvature generalization of the classical splitting theorem (due to Cheeger--Gromoll) for Ricci curvature.
October 23, 2018: Baris Coskunzer "Minimal Surfaces in Hyperbolic 3-Manifolds"
Abstract: In this talk, we will discuss the existence question for closed embedded minimal surfaces in 3-manifolds. After reviewing the classical results on the subject, we will show the existence of smoothly embedded closed minimal surfaces in infinite volume hyperbolic 3-manifolds.
October 9, 2018: Peter McGrath "Existence and Uniqueness for Free Boundary Minimal Surfaces"
Abstract: Let $\mathbb{B}^3$ be the unit ball in \( \mathbb{R}^3\) and consider the family of surfaces contained in \(\mathbb{B}^3 \) with boundary on the unit sphere \( \mathbb{S}^2 \). The critical points of the area functional amongst this class are called Free Boundary Minimal Surfaces. The latter surfaces are physically realized by soap films in equilibrium and have been the subject of intense study. In the 1980s, it was proved that flat equatorial disks are the only free boundary minimal surfaces with the topology of a disk. It is conjectured that a surface called the critical catenoid is the unique (up to ambient rotations) embedded free boundary minimal annulus. I will discuss some recent progress towards resolving this conjecture. I will also discuss some sharp bounds for the areas of free boundary minimal surfaces in positively curved geodesic balls which extend works of Fraser-Schoen and Brendle in the Euclidean setting.
September 25, 2018: Chen-Yun Lin "An embedding theorem: geometric analysis behind data analysis"
Abstract: High-dimensional data can be difficult to analyze. Assume data are distributed on a low-dimensional manifold. The Vector Diffusion Mapping (VDM), introduced by Singer-Wu, is a non-linear dimension reduction technique and is shown robust to noise. It has applications in cryo-electron microscopy and image denoising and has potential application in time-frequency analysis. In this talk, I will present a theoretical analysis of the VDM for its mathematical foundation. Specifically, I will discuss parametrisation of the manifold and an embedding which is equivalent to the truncated VDM. In the differential geometry language, I use eigen-vector fields of the connection Laplacian operator to construct local coordinate charts that depend only on geometric properties of the manifold. Next, I use the coordinate charts to embed the entire manifold into a finite-dimensional Euclidean space. The proof of the results relies on solving the elliptic system and provide estimates for eigenvector fields and the heat kernel and their gradients.