# Rutgers Geometry/Topology Seminar -- Spring 2015

### Tuesdays 3:30-4:30 in room 525

 Date Speaker Title Jan. 27 No seminar Snow storm!!! Feb. 3 Ara Basmajian, CUNY Arcs, orthogeodesics, and hyperbolic surface identities Feb. 10 Sergio Fenley, FSU and Princeton Counting closed orbits of Anosov flows in free homotopy classes Feb. 17 Ilya Kofman, CUNY Geometrically and diagrammatically maximal knots Feb. 24 Chris Woodward, Rutgers Floer theory and minimal model transitions Mar. 3 Max Lipyanskiy Gromov-Uhlenbeck Compactness Mar. 10 Tian Yang, Stanford On type-preserving representations of the four-punctured sphere group Mar. 17 No Seminar Spring Break Mar. 24 Sucharit Sarkar, Princeton A Khovanov stable homotopy type Mar. 31 Parlier Hugo, Fribourg Switzerland Chromatic numbers for hyperbolic surfaces Apr. 7 Moira Chas, Stony Brook Computer driven theorems and questions in geometry Apr. 14 Tax Holiday! Apr. 21 Tony Bahri, Rider and Princeton Some familiar spaces from a different point of view Apr. 28 Priyam Patel, Purdue Separability Properties of Right-Angled Artin Groups May 410-11amHill 124 Quanshen Jiu, Capital Normal University and Princeton U On Liouville Type of Theorems to the Three-dimensional Incompressible Axisymmetric Navier-Stokes Equations

# Fall 2014

 Date Speaker Title Sept. 2 No Seminar Sept. 9 Christian Millichap, Temple U Geometrically Similar Hyperbolic Pretzel Knots Sept. 16 Xiaochun Rong, Rutgers Comparison geometry on singular spaces Sept. 23 Vladimir Retakh, Rutgers Non-commutative triangulations Sept. 30 Yi Hu, U of Arizona Moduli Spaces of Stable Polygons Oct. 7 Alexander Dranishnikov, U of Florida On the LS-category of the product of Lens spaces Oct. 14 Zhenlei Zhang, Capital Normal University/Princeton Univeristy Regularity of Kahler-Ricci flow on Fanothreefolds Oct. 21 Murat Savas, Gazi University, Turkey Orthogonal Projection to k-plane in Hyperbolic and Spherical n-Spaces Oct. 28 Ran Ji, CUNY The Martin boundary for manifolds with nonpositive curvature Nov. 4 Lingling Kong. Northeasten Normal Universiity and Rutgers University Singularities of the lightlike surfaces in anti de Sitter 3-space Nov. 11 Shmuel Weinberger, U of Chicago Equivariant Classification Problems Nov. 18 Ed Chien, Rutgers Square Tiling of Closed Surfaces Nov. 25 Glen Wilson, Rutgers Motivic Adams spectral sequence over finite fields Dec. 2 Don Davis, Lehigh On the topological complexity of 2-torsion lens spaces Dec. 9 Lee Kennard, UC at Santa Barbara On the Chern problem with symmetry

# Abstracts

### Geometrically Similar Hyperbolic Pretzel Knots

Given a hyperbolic 3-manifold M, there are a number of geometric invariants of interest. Two such invariants are the volume of M and the length spectrum of M, that is, the set of all lengths of closed geodesics in M counted with multiplicities. It is natural to ask how often can hyperbolic manifolds have the same volume, the same length spectrum, or perhaps even both. In this talk, we shall construct large families of hyperbolic pretzel knots whose complements have both the same volume and the same initial length spectrum, but are pairwise incommensurable, i.e., they do not share a common finite sheeted cover. In particular, we shall show that the number of hyperbolic knot complements with the same volume and initial length spectrum grows at least factorially fast with the volume and the number of twist regions. This proof relies on Ruberman's work on mutations along Conway spheres in least area form that preserve volume, and expanding this analysis to see when these Conway spheres could intersect short geodesics in a hyperbolic 3-manifold.

### Comparison geometry on singular spaces

We will present an introduction talk on comparison geometry, which provides tools in understanding the geometry of curvature of Riemannian metrics. We will give a brief history, and we will disscuss some recent development of comparison geometry on singular spaces which are Gromov-Hausdorff limit spaces of Riemannian manifolds.

### Noncommutative triangulations

Ptolemy identity plays a major role in the theory of triangulations. I will discuss a noncommutative version of Ptolemy identity and decribe some related invariants. The talk is based on a joint paper with A. Berenstein (U. of Oregon)

### Moduli Spaces of Stable Polygons

I will describe the moduli spaces of stable spatial polygons. This may be considered as the symplectic construction of the Deligne-Mumford moduli spaces of stable pointed rational curves. It is also a manifestation of a general principle that predicts a correspondence between symplectic and Geometric Invariant Theory quotients. The talk will be elementary.

### On the LS-category of the product of Lens spaces

In all known examples of degree one maps between manifolds the image is simpler than the domain. Yuli Rudyak conjectured that for the Lusternik-Schnirelmann category cat(M)\ge cat (N) for a map f:M-->N with deg(f)=1. Since Rudyak's partial results from the late 90s there was no progress on the problem. In this talk I'll give an introduction to the LS-category and show a reduction of Rudyak's problem to computation of the LS-category of the product of lens spaces.

### Regularity of Kahler-Ricci flow on Fanothreefolds

In this talk I will discuss a joint work with Professor Tian on the regularity of Kahler-Ricci flow on three dimensionalFano manifolds. We will show that the Kahler-Ricci flow converge in the Cheeger-Gromov topology to a Kahler-Ricci soliton with codimension four singularities. By establishing the Partial C^0 estimate under the Ricci flow it will be seen that the limit is also an algebraic object.

### Motivic Adams spectral sequence over finite fields

To any field $k$, we consider the motivic stable homotopy category over $k$ constructed by Morel and Voevodsky. In this setting, one can construct a motivic Adams spectral sequence (MASS) which converges to the 2-complete stable homotopy groups of the motivic sphere spectrum. By using computer calculations of the E2 page of the MASS, we are able to compute the 2-complete stable homotopy groups of the motivic sphere spectrum $\pi_{n,m}$ for $n \leq 12$ over finite fields of odd characteristic. This talk will introduce the motivic stable homotopy category and present the results of our computations. This work is joint with Paul Ostvaer and Knight Fu.

### Orthogonal Projection to k-plane in Hyperbolic and Spherical n-Spaces

We will describe the orthogonal projection taking a point in hyperbolic, or spherical n-space and mapping it along a geodesic to the point, where geodesic meets orthogonally the chosen k-plane of projection. Via such projection, we obtain the distance formula between a point and a k-plane in the hyperbolic and spherical n-spaces. For a given n-simplex, we also obtain the exact formula for the altitude and the perpendicular foot from a given vertex to its opposite k-face. These results are proved by using the Schur complement of a sub-matrix in Gram and Edge matrices.

### The Martin boundary for manifolds with nonpositive curvature

Anderson and Schoen proved that for a complete, simply connected manifold with pinched negative curvature, the Martin boundary can be identified with the geometric boundary. In this talk, I will first introduce the Martin compactification for Cartan-Hadamard manifolds. I will then relax the lower bound on the curvature assumption and generalize Anderson and Schoen’s result. Time permitting, I will also discuss the Martin boundary for manifolds admiting some zero sectional curvature.

### On the Chern problem with symmetry

Manifolds admitting positive sectional curvature have been of interest since the origin of global Riemannian geometry, but their classification is open. In fact, we do not have a classification of the possible fundamental groups. I will discuss some of what is known about this problem. Along the way, we will discuss a question of S.-S. Chern posed in the 1960s, important examples by R. Shankar in the 1990s, and more recent classification results in the presence of symmetry by X. Rong, myself, and others.

### On the topological complexity of 2-torsion lens spaces

The topological complexity of a topological space is the minimum number of rules required to specify how to move between any two points of the space. A rule'' must satisfy the requirement that the path varies continuously with the choice of end points. We use connective complex K-theory to obtain new lower bounds for the topological complexity of 2-torsion lens spaces. We follow a program set up by Jesus Gonzalez, and answer a question posed by him.

### Singularities of the lightlike surfaces in anti de Sitter 3-space

In this talk, I will introduce singularities of lightlike surfaces and focal surfacesof spacelike curves in anti de Sitter space times sphere. To do that, we construct an anti de Sitter height function and a Lightcone height function, and then show the relation between singularities of the lightlike surfaces (respectively, the focal surfaces) and that of the anti de Sitter height functions (respectively, the Lightcone height functions).

### Equivariant Classification Problems

(Based on work with Cappell and Yan, and conversations with Klein.) The framework for classifying high dimensional manifolds, surgery theory, does not work for manifolds with group actions in the absence of of a condition called the gap hypothesis. When this fails, the usual h-principles and surgery theory (which is a slightly perturbed h-principle) fail. I will explain a method that, in principle, solves this under a much less restrictive hypothesis -- using "nonlinear functors" and explain what it means in some concrete cases. In particular I will use this to explain some (previously known) examples of "exotic" group actions on tori.

### Square Tiling of Closed Surfaces

In a paper from the 1940s, Brooks, Smith, Stone, and Tutte proved a classical theorem which produces a tilling of a rectangle by squares associated to any connected planar graph. We generalize this result to high genus surfaces. Given a closed surface S, a non-zero first homology class and a graph G on S so that each component of S-G is simply connected, we show that exists a singular flat metric and a square tiling on S associated to the graph and the homology class. The proof uses analogues of Kirchoff's circuit laws and discrete harmonic forms.

### Geometrically and diagrammatically maximal knots

The ratio of volume to crossing number of a hyperbolic knot is bounded above by the volume of a regular ideal octahedron, and a similar bound is conjectured for the knot determinant per crossing. We show that many families of alternating knots and links simultaneously maximize both ratios. This is joint work with Abhijit Champanerkar and Jessica Purcell.

### Arcs, orthogeodesics, and hyperbolic surface identities

Let X be a compact hyperbolic surface with either geodesic or horocyclic boundary. The homotopy class (rel the boundary) of a non-trivial arc from the boundary to itself can be realized by an orthogeodesic- a geodesic segment perpendicular to the boundary at its initial and terminal points. This talk is about a special subclass of orthogeodesics called primitive orthogeodesics. In work with Hugo Parlier and Ser Peow Tan we show that the primitive orthogeodesics arise naturally in the study of maximal immersed pairs of pants in X and are intimately connected to regions of X in the complement of the natural collars. These considerations lead to continuous families of new identities- equations that remain constant on the space of hyperbolic structures.

### Counting closed orbits of Anosov flows in free homotopy classes

There are Anosov and pseudo-Anosov flows so that some orbits are freely homotopic to infinitely many other orbits. An Anosov flow is R-covered if either the stable or unstable foliations lift to foliations in the universal cover with leaf space homeomorphic to the reals. These are extremely common. A free homotopy class is a maximal collection of closed orbits of the flow that are pairwise freely homotopic to each other. The first result is that if an R-covered Anosov flow has all free homotopy classes that are finite, then up to a finite cover the flow is topologically conjugate to either a suspension or a geodesic flow. This is a strong rigidity result that says that infinite free homotopy classes are extremely common amongst Anosov flows in 3-manifolds. We also mention other examples with infinite free homotopy classes.

The second part of the talk is about analysing growth of length of orbits in a fixed infinite free homotopy class. We analyse the interaction of such a free homotopy class with the torus decomposition of the manifold: for examples whether all orbits in the infinite free homotopy classes are contained in a Seifert piece or atoroidal piece. There is a natural ordering of an infinite subset of such a collection, indexed as (gamma_i). We analyse the growth of the length of gamma_i as a function of i. We obtain several inequalities: for example if the manifold is hyperbolic then the growth of length of gamma_i is exponential. These inequalities have consequences for the ergodic theory of the Anosov flow.

(This is joint work with Thomas Barthelme of Penn State University)

### Gromov-Uhlenbeck Compactness

Let M be a symplectic manifold with a hamiltonian group action by G. We introduce an analytic framework that relates holomorphic curves in the symplectic quotient of M to gauge theory on M. As an application of these ideas, we discuss the relation between instanton Floer homology and Lagrangian Floer homology of representation varieties.

### Floer theory and minimal model transitions

I will describe a method for finding Floer-non-trivial Lagrangians in a symplectic manifold, by running the minimal model program backwards. This is work in progress (partly with Charest and Schultz). Examples in moduli spaces of polygons will be discussed.

### On type-preserving representations of the four-punctured sphere group

We give counterexamples to a conjecture of Bowditch that if a non-elementary type-preserving representation of a punctured surface group into PSL(2,R) sends every non-peripheral simple closed curve to a hyperbolic element, then the representation must be discrete faithful. The counterexamples come from relative Euler class of representations of the four-punctured sphere group. As a related result, we show that the mapping class group action on each non-extremal component of the character space of type-preserving representations of the four-punctured sphere group is ergodic. The main tool we use is Penner’s lengths coordinates of the decorated character spaces defined by Kashaev.

### A Khovanov stable homotopy type

We will start our story with the Jones polynomial, a revolutionary knot invariant introduced by V. Jones in 1984. We will then talk about Khovanov homology of knots, which is a "categorification" of the Jones polynomial constructed by M. Khovanov. Finally, we will discuss a recent stable homotopy level refinement of Khovanov homology, which is joint work with R. Lipshitz, and a more algebraic topological reformulation of this invariant using the Burnside category, which is joint work with T. Lawson and R. Lipshitz. Along the way, we will mention topological applications of these three knot invariants.

### Chromatic numbers for hyperbolic surfaces

Given a metric space X and a positive real number d, the chromatic number of X,d is the minimum number of colors needed to color points of the metric space such that any two points at distance d are colored differently. When X is a metric graph (and d is 1) this is the usual chromatic number of a graph. When X is the Euclidean plane (the d is irrelevant) the chromatic number is known to be between 4 and 7 (finding the exact value is known as the Hadwiger-Nelson problem). For the hyperbolic plane even less is known and it is not even known whether or not it is bounded by a quantity independent of d.

This talk is about finding different bounds on the chromatic number of hyperbolic surfaces and is based on joint work with Camille Petit.

### Computer driven theorems and questions in geometry

Given an orientable surface S with negative Euler characteristic, a minimal set of generators of the fundamental group of S, and a hyperbolic metric on S, each free homotopy class C of closed oriented curves on S, determines three numbers: the word length (that is, the minimal number of generators and inverses necessary to express C as a cyclically reduced word), the minimal self-intersection and the geometric length.

These three numbers, can be explicitly computed (or approximated) by the help of computer. These computations lead us find counterexamples to existing conjectures and to establish new conjectures.

For instance, we conjectured that the distribution of self-intersection of classes of closed directed curves on a surface with boundary, sampling by word length, appropriately normalized, tends to a Gaussian when the word length goes to infinity. Later on, jointly with Lalley, we proved this result. Recently, Wroten extended this result to closed surfaces.

In another direction, the computer allowed to us to study the relation between self-intersection of curves and length-equivalence. (Two classes a and b of curves are length equivalent if for every hyperbolic metric m on S, m(a)=m(b).)

### Separability Properties of Right-Angled Artin Groups

Right-Angled Artin groups (RAAGs) and their separability properties played an important role in the recent resolutions of some outstanding conjectures in low-dimensional topology and geometry. We begin this talk by defining two separability properties of RAAGs, residual finiteness and subgroup separability, and provide a topological reformulation of each. We then discuss joint work with K. Bou-Rabee and M.F. Hagen regarding quantifications of these properties for RAAGs and the implications of our results for the class of virtually special groups.

### Some familiar spaces from a different point of view

Polyhedral products arise naturally in a variety of mathematical contexts including toric geometry/topology, complements of subspace arrangements, intersections of quadrics, arachnid mechanisms, homotopy theory, and lately, number theory. After a brief survey, I shall describe geometric and algebraic approaches to the computation of their cohomology. A report of joint work with Martin Bendersky, Fred Cohen and the late Sam Gitler.

### Some familiar spaces from a different point of view

The Liouville type of theorem plays a key role in the blow-up approach to study the global regularity of the three-dimensional Navier-Stokes equations. In this talk, we will present Liouville type of theorems to the 3-D axisymmetric Navier-Stokes equations with swirls under some suitable assumptions on swirl component velocity $u_\theta$, which are scaling invariant. It is known that $ru_\theta$ satisfies the maximum principle. The assumptions on $u_\theta$ will be natural and useful to make further studies on the global regularity to the three-dimensional incompressible axisymmetric Navier-Stokes equations.