Rutgers Geometry/Topology seminar: Fall 2017 - Spring 2018

Geometric finite amalgamations of hyperbolic 3-manifold groups are not LERF

We will show that, for any two finite volume hyperbolic 3-manifolds, the amalgamations of their fundamental groups along nontrivial geometrically finite subgroups are always not LERF. A consequence of this result is: all arithmetic hyperbolic manifolds with dimension at least 4, with possible exceptions in 7-dimensional manifolds defined by the octonion, their fundamental groups are not LERF.

An upper bound on the translation length in the curve complex

This is a collaboration with Hyungyul Bsik and Hyunshik Shin. We found an asymptotic upper bound for the translation length in the curve complex for primitive integer points in a fibered cone.

Genus two analogue of A_1 spherical DAHA

Double Affine Hecke Algebra can be viewed as a noncommutative (q,t)-deformation of the SL(N,C) character variety of the fundamental group of a torus. This deformation inherits major topological property from its commutative counterpart, namely Mapping Class Group of a torus SL(2,Z) acts by automorphisms of DAHA. In my talk I will define a similar algebra for a closed genus two surface and show that the corresponding Mapping Class Group acts by automorphisms of such algebra. (This talk is based on arXiv:1704.02947 joint with Sh. Shakirov)

Deformation theory of constant curvature conical metrics

In this joint work with Rafe Mazzeo, we would like to understand the deformation theory of constant curvature metrics with prescribed conical singularities on a compact Riemann surface. We construct a resolution of the configuration space, and prove a new regularity result that the family of constant curvature conical metrics has a nice compactification as the cone points coalesce. This is one key ingredient to understand the full moduli space of such metrics with positive curvature and cone angles bigger than 2π.

Lifting curves simply in finite covers

It is a well known result of Peter Scott that the fundamental groups of surfaces are subgroup separable. This algebraic property of surface groups also has important topological implications. One such implication is that every immersed (self-intersecting) closed curve on a surface lifts to an embedded (simple) one in a finite cover of the surface. A natural question that arises is: what is the minimal degree of a cover necessary to guarantee that a given closed curve lifts to be embedded? In this talk we will discuss various results answering the above question for hyperbolic surfaces, as well as several related questions regarding the relationship between geodesic length and geometric self-intersection number. Some of the work that will be presented is joint with T. Aougab, J. Gaster, and J. Sapir.

Compactness theory for harmonic maps into locally CAT(1) spaces

We determine the complete bubble tree picture for a sequence of harmonic maps, with uniform energy bounds, from a compact Riemann surface into a compact locally CAT(1) space. In the smooth setting, Parker established the bubble tree picture by exploiting now classical analytic results about harmonic maps. Our work demonstrates that these results can be reinterpreted geometrically. Indeed, in the absence of a PDE we prove analogous results by taking advantage of the local convexity properties of the target space. Included in this paper are an $\epsilon$-regularity theorem, an energy gap theorem, and a removable singularity theorem for harmonic maps. We also prove an isoperimetric inequality for conformal harmonic maps with small image (i.e. minimal surfaces in the non-smooth setting). This is a joint work with Christine Breiner.

A proof of Milnor conjecture in dimension 3

We present a proof of Milnor conjecture in dimension 3, which says that any open manifold of non-negative Ricci curvature has a finitely generated fundamental group. The proof is based on the structure of Ricci limit spaces.

Homology systole over mod 2 coefficients and systolic freedom

I am going to talk about problems around homology systole over mod 2 torsion coefficients. Given a Riemannian metric defined on a closed manifold, we define mod 2 homology systole to be the infimum of volumes of cycles representing nontrivial classes in homology group with mod 2 coefficients. Gromov conjectured that there would be systolic rigidity for mod 2 homology systoles, similar to homotopy 1-systolic inequalities on aspherical manifolds. However, later work shows that counterexample exists. In the talk, different aspects related to this conjecture will be explained. In particular, there are two types of motivations to study this problem. The first motivation is based on Gromov’s essential systolic inequality on aspherical manifolds. Another recent motivation is from quantum information theory.

On the spectral geometry of manifolds with conic singularities

In this talk we consider the heat kernel of the Laplace-Beltrami operator on a Riemannian manifold. On any closed smooth Riemannian manifold the heat trace expansion gives some geometrical information such as dimension, volume and total scalar curvature of the manifold. On a manifold with conic singularities we derive a detailed asymptotic expansion of the heat trace using the Singular Asymptotics Lemma of Jochen Brüning and Robert T. Seeley. Then we investigate how the terms in the expansion reflect the geometry of the manifold. Can one hear a singularity?

Provably good convex methods for mapping problems

Computing mappings or correspondences between surfaces is an important tool for many applications in computer graphics, computer vision, medical imaging, morphology and related fields. Mappings of least angle distortion (conformal) and distance distortion (isometric) are of particular interest. The problem of finding conformal/isometric mappings between surfaces is typically formulated as a difficult non-convex optimization problem. Convex methods relax the non-convex optimization problem to a convex problem which can then be solved globally. The main issue then is whether the global solution of the convex problem is a good approximation for the original global solution. In this talk we will discuss two families of convex relaxations.

Conformal: We relax the problem of computing planar conformal mappings (Riemann mappings) to a simple convex problem which can be solved by solving a system of linear equations. We show that in this case the relaxation is exact- the solution of the convex problem is guaranteed to be the Riemann mapping!

Discrete isometric: for perfectly isometric asymmetric surfaces, the well known doubly-stochastic (DS) relaxation is exact. We generalize this result to the more challenging and important case of symmetric surfaces, once exactness is correctly defined for such problems. For non-isometric surfaces it is difficult to achieve exactness. Several relaxations have been proposed for such problems, where the more accurate relaxations are generally also more time consuming. We will describe two algorithm which strike a good balance between accuracy and efficiency: The DS++ algorithm, which is provably better than several popular algorithms but does not compromise efficiency, and the Sinkhorn-JA algorithm, which gives a first-order algorithm for efficiently solving the strong but high-dimensional JA relaxation. We utilize this algorithmic improvement to achieve state of the art results for shape matching and image arrangement.

Bordered theory for pillowcase homology

Pillowcase homology is a geometric construction, which was developed in order to better understand and compute a knot invariant I(K) called singular instanton knot homology. Motivated by the problem of extending pillowcase homology to tangles, we will introduce the following construction. The pillowcase P is a torus factorized by hyperelliptic involution, and after removing 4 singular points one obtains a 4-punctured 2-sphere P*. First, we will associate an algebra A to the pillowcase P*. Second, to an immersed curve L inside P* we will associate an A∞ module M(L) over A. Then we will show how, using these modules, one can recover and compute Lagrangian Floer homology (i.e. geometric intersection number) for immersed curves.

Complex curves through a contact lens

Every four-dimensional Stein domain has a height function whose regular level sets are contact three-manifolds. This allows us to study complex curves via their intersection with these contact level sets, where we can comfortably apply three-dimensional tools. We use this perspective to characterize the links in Stein-fillable contact manifolds that bound complex curves in their Stein fillings. (Some of this is joint work with Baykur, Etnyre, Hedden, Kawamuro, and Van Horn-Morris.)

Veering triangulations and fibered faces of 3-manifolds

Agol’s veering triangulation for 3-manifolds that fiber over the circle can be obtained very explicitly, via a construction of Gueritaud, from the stable and unstable laminations of the monodromy. We study the way in which these triangulations interact with the curve complexes of the surface and its subsurfaces. This allows us to examine the “profile” of subsurface projections associated to each fiber in a fibered face of the Thurston norm ball, obtaining some bounds that do not depend on the complexity of the fibers. This is joint work with Yair Minsky.

An intuitive glimpse at Ricci flow

Ricci flow was created and developed over many years by highly original works of Richard Hamilton. By revolutionary works, Grisha Perelman carried out Hamilton's program and proved Thurston's Geometrization Conjecture. In this talk we discuss Ricci flow from an elementary and intuitive perspective, with an emphasis on taking a glimpse at a few things one might hope for in higher dimensions.

Veering triangulations: theory and experiment

Every fibered hyperbolic 3-manifold M has a canonically associated veering triangulation. This triangulation (technically, an ideal triangulation of a certain surgery parent of M) was introduced by Agol, and has nice combinatorial and dynamical properties. The question is: how much geometry does it encode? I will describe the results of a large-scale computational experiment that provides some intriguing answers. Then, I will promote one of the experimental results to a theorem, outlining a proof that generic mapping classes give rise to non-geometric veering triangulations. This is joint work with Sam Taylor and Will Worden.

Random mapping class group elements have generic foliations

A pseudo-Anosov element of the mapping class group determines a quadratic differential, which lies in the principal stratum if all zeroes are simple, equivalently, if the corresponding foliations have trivalent singularities. We show that this occurs with asymptotic probability one for random walks on the mapping class group, and furthermore, the hitting measure on the boundary gives weight zero to foliations with saddle connections. This is joint work with Vaibhav Gadre.

Representations of Kauffman bracket skein algebras of a surface

The definition of the Kauffman bracket skein algebra of an oriented surface was originally motivated by Witten's topological quantum field theory interpretation of the Jones polynomial for knots. But the skein algebra is also closely related to the SL(2,C)-character variety of the surface. We'll discuss recent methods for constructing finite-dimensional representations of the skein algebra, and their role in bridging quantum topology.

Partially hyperbolic diffeomorphisms in dimension 3

These diffeomorphisms exhibit weaker forms of hyperbolicity and are extremely common. Such a diffeomorphism f has stable, unstable and center bundles invariant under df. This is an very intense area of research currently. We review basic examples, conjectures. We also talk about dynamical coherence - this has to do with the integrability of some of the bundles above - unlike the strictly hyperbolic case, there are many non integrable recent examples. This leads to some recent counterexamples of a main conjecture.

Finiteness of maximal geodesic submanifolds of hyperbolic hybrids

Reid and McMullen have both asked whether or not the presence of infinitely many finite-volume totally geodesic surfaces in a hyperbolic 3-manifold implies arithmeticity of its fundamental group. I will explain why large classes of non-arithmetic hyperbolic n-manifolds, including the hybrids introduced by Gromov and Piatetski-Shapiro and many of their generalizations, have only finitely many finite-volume immersed totally geodesic hypersurfaces. These are the first examples of finite-volume n-hyperbolic manifolds, n>2, for which the collection of all finite-volume totally geodesic hypersurfaces is finite but nonempty. In this talk, I will focus mostly on dimension 3, where one can even construct link complements with this property. This is joint work with David Fisher, Jean-François Lafont, and Nicholas Miller.

The shape of Kahler-Einstein metric on the universal Teichmuller space

The universal Teichmuller space is an infinite dimensional generalization of the Teichmuller space. In early 2000s, Takhtajan-Teo gave a Hilbert structure to the Universal Teichmuller space. Within this Hilbert structure, they proved several profound theorems about the geometry of the Kahler-Einstein metric (which coincides with the L^2-pairing). We consider the curvature operator of this metric and prove that it's bounded, nonpositive definite and noncompact. This is based on joint work with Y. Wu (Tsinghua university).

Non-negative Ricci curvature, stability at infinity, and finite generation of fundamental groups

In 1968, Milnor conjectured that any open n-manifold M of non-negative Ricci curvature has a finitely generated fundamental group. This conjecture remains open today. In this talk, we show that if there is an integer k such that any tangent cone at infinity of the Riemannian universal cover of M is a metric cone, whose maximal Euclidean factor has dimension k, then \pi_1(M) is finitely generated. In particular, this confirms the Milnor conjecture for a manifold whose universal cover has Euclidean volume growth and unique tangent cone at infinity.

Circle patterns with obtuse exterior intersection angles

Koebe-Andreev-Thurston theorem studies the existence and rigidity of circle patterns of a given combinatorial type and the given non-obtuse exterior intersection angles. By using topological degree theory and variational principle, in this talk I will generalize the Koebe-Andreev-Thurston theorem to the case of obtuse exterior intersection angles.

Sphere packings and Arithmetic Lattices

It has been known for sometime that the Apollonian packing, as well as certain other infinite circle/sphere packings, are "integral" packings, i.e. they can be realized so that the bends (the reciprocal of radii) of constituent circles/spheres are integers. Most of the known integral packings exhibit a stronger integral property, and we refer to them as "super-integral" packings. Relating them to the theory of arithmetic reflection lattices, we prove that super-integral packings exists only in finitely many dimensions, and only in finitely many commensurability classes.

Hidden Symmetries and Commensurability of 2-bridge link complements

The canonical triangulations and symmetry groups of 2-bridge link complements are well understood and relatively easy to describe. We leverage this fact to show that non-arithmetic 2-bridge link complements have no hidden symmetries (i.e., symmetries of a finite cover that do not descend to symmetries of the link complement itself), and are pairwise incommensurable. I will discuss the relevant background material, and give a rough sketch of the proof. This is joint work with Christian Millichap.

Kahler-Ricci flow with cusp singularities and some applications

We generalized Lott-Zhang's maximal time existence of cusp Kahler-Ricci flow with superspatial initial data to arbitrary initial data with zero Lelong number. First we construct the solution with smooth and bounded initial by perturbations. Then by Demailly's decreasing approximation method which was developed by Guedj-Zeriahi, we can construct the solution with arbitrary zero lelong number data. If time permits, we will talk about the applications on the minimal model program, especially on log canonical varieties. This work is joint with Albert Chau and Ka-Fai Li.

Contrasting Various Notions of Convergence in Geometric Analysis

We explore the distinctions between L^p convergence of metric tensors on a fixed Riemannian manifold vs. GH, uniform, and intrinsic flat convergence of the the resulting sequence of metric spaces. We provide a number of examples which demonstrate these notions of convergence do not agree even for two dimensional warped product manifolds with warping functions converging in the L^p sense. We then prove a theorem which requires L^p bounds from above and C^0 bounds from below on the warping functions to obtain enough control for the limits to agree. This is joint work with Christina Sormani.