# Rutgers Geometry/Topology seminar: Fall 2017 - Spring 2018

## Fall 2017

 Date Speaker Title (click for abstract) Sep. 5th No Seminar Sep. 12th Hongbin Sun (Rutgers) Geometric finite amalgamations of hyperbolic 3-manifold groups are not LERF Sep. 19th Chenxi Wu (Rutgers) An upper bound on the translation length in the curve complex Sep. 26th Semeon Artamonov (Rutgers) Genus two analogue of A_1 spherical DAHA Oct. 3rd Xuwen Zhu (Stanford) Deformation theory of constant curvature conical metrics Oct. 6th (Friday) 1-2pm Hill 005 Priyam Patel (UC Santa Barbara) Lifting curves simply in finite covers Oct. 10th Sajjad Lakzian (Fordham University) Compactness theory for harmonic maps into locally CAT(1) spaces Oct. 17th Jiayin Pan (Rutgers) A proof of Milnor conjecture in dimension 3 Oct. 24th Lizhi Chen (Lanzhou University, China) Homology systole over mod 2 coefficients and systolic freedom Oct. 31st Asilya Suleymanova (Max Planck Institute, Germany) On the spectral geometry of manifolds with conic singularities Nov. 7th Nadav Dym (Weizmann Institute of Science, Israel) Provably good convex methods for mapping problems Nov. 14th Artem Kotelskiy (Princeton) Bordered theory for pillowcase homology Nov. 21st No Seminar Happy Thanksgiving! Nov. 28th Kyle Hayden (Boston College) Complex curves through a contact lens Dec. 5th Dec. 12th Sam Taylor (Temple University)

## Spring 2018

 Date Speaker Title (click for abstract) Jan. 16th No Seminar Jan. 23rd Jan. 30th Joseph Maher (CUNY) Feb. 6th Feb. 13th Feb. 20th Feb. 27th March 6th March 13th No Seminar Spring break March 20th March 27th April 3rd April 10th April 17th April 24th

## Abstracts

### 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.)