Rutgers Geometry/Topology seminar: Fall 2019 - Spring 2020

Semi-local simply connectedness of non-collapsing Ricci limit spaces

We prove that any non-collapsing Ricci limit space is semi-locally simply connected. This is joint work with Guofang Wei.

Ideal Hyperbolic Polyhedra and Discrete Uniformization

Two seemingly unrelated problems turn out to be equivalent. The first is a problem of 3-dimensional hyperbolic geometry: Given a complete hyperbolic surface of finite area that is homeomorphic to a sphere with punctures, find a realization as convex ideal polyhedron in hyperbolic space. The second is a problem of discrete complex analysis: Given a closed triangle mesh of genus zero, find a discretely conformally equivalent convex triangle mesh inscribed in a sphere. The existence and uniqueness of a solution of the first (hence also the second) problem was shown by I. Rivin. His proof is not constructive. A variational principle leads to a new constructive proof.

Symplectic fillings of asymptotically dynamically convex manifolds

One natural question in symplectic topology is understanding symplectic fillings of a given contact manifold. In this talk, I will introduce a Floer theoretic method of studying a class of contact manifolds admitting only the trivial augmentation, called asymptotically dynamically convex (ADC) manifolds. I will construct various structure maps and explain why they are independent of fillings for ADC manifolds. Such invariance can be used to answer both existence and uniqueness questions of symplectic fillings.

Braid invariant relating knot Floer homology and Khovanov homology

Khovanov homology and knot Floer homology are two knot invariants that were defined around the same time, and despite their different constructions, share many formal similarities. After reviewing the construction of Khovanov homology and some of these similarities, we will discuss an algebraic braid invariant which is closely related to both Khovanov homology and the refinement of knot Floer homology into tangle invariants. This is a joint work with Nathan Dowlin.

A Generalization of the Tristram-Levine Knot Signatures as a Singular Furuta-Ohta Invariant for Tori

Given a knot K inside an integer homology sphere Y , the Casson-Lin-Herald invariant can be interpreted as a signed count of conjugacy classes of irreducible representations of the knot complement into SU(2) which map the meridian of the knot to a fixed conjugacy class. It has the interesting feature that it determines the Tristram-Levine signature of the knot associated to the conjugacy class chosen.

Turning things around, given a 4-manifold X with the integral homology of S1 × S3, and an embedded torus which is homologically non trivial, we define a signed count of conjugacy classes of irreducible representations of the torus complement into SU(2) which satisfy an analogous fixed conjugacy class condition to the one mentioned above for the knot case. Our count recovers the Casson-Lin-Herald invariant of the knot in the product case, thus it can be regarded as implicitly defining a Tristram-Levine signature for tori.

This count can also be considered as a singular Furuta-Ohta invariant, and it is a special case of a larger family of Donaldson invariants which we also define. In particular, when (X, T) is obtained from a self-concordance of a knot (Y, K) satisfying an admissibility condition, these Donaldson invariants are related to the Lefschetz number of an Instanton Floer homology for knots which we construct. Moreover, from these Floer groups we obtain Frøyshov invariants for knots which allows us to assign a Frøyshov invariant to an embedded torus whenever it arises from such a self-concordance.

Lattice point counting and saddle connections

Various questions concerning translation surfaces depend on counting saddle connections. For a certain class of translation surfaces, this reduces to the more general, yet more tractable problem of counting points in discrete orbits for the linear action of a lattice of SL(2,R) on the Euclidean plane. This can be done effectively, using either methods from ergodic theory or from number theory. We will discuss the latter aspect, based on recent joint work with Amos Nevo, René Rühr, and Barak Weiss.

Constructing knot Floer Homology via sutures

Sutured monopole Floer homology (SHM) and sutured Instanton Floer homology (SHI) were introduced by Kronheimer and Mrowka. They also constructed knot Floer homologies by taking the SHM or SHI of the knot complements with meridional sutures. These knot Floer homologies are the counterpart in monopole and instanton theories to HFK-hat in Heegaard Floer theory. In this talk, we will make use of other sutures on the boundary of knot complements and introduce new versions of knot Floer Homology in monopole and instanton theories, called KHM-minus and KHI-minus, which closely resemble HFK-minus in Heegaard Floer theory. We will present the construction as well as proving some basic properties. Based on this minus version, we can also define a tau invariant in monopole and instanton theory and prove its concordance invariance. If time permits, we will also present the computations for some special knots.

Free products and random walks in acylindrically hyperbolic groups

The properties of a random walk on a group which acts on a hyperbolic metric space have been well-studied in recent years. In this talk, I will focus on random walks on acylindrically hyperbolic groups, a class of groups which includes mapping class groups, Out(F_n), and right-angled Artin and Coxeter groups, among many others. I will discuss how a random element of such a group interacts with fixed subgroups that have certain nice properties. In particular, I will discuss when the subgroup generated by a random element and a fixed subgroup is a free product, and I will also describe some of the geometric properties of that free product. This is joint work with Michael Hull.

Long-time behavior of immortal Ricci flows

The limit behavior of immortal Ricci flows with appropriate curvature conditions are important in the application of Ricci flows to hunt for canonical metrics. In this talk, we will discuss how to simplify the singularity structure of the limit space as a result of the evolution of immortal Ricci flows with bounded or decaying curvature. To deal with the major difficulty of possible collapsing along the Ricci flow in the long time, we will present some new tools in understanding the metric measure structure of the collapsing geometry.

Compactifications of manifolds with boundary

In 1966, Larry Siebenmann once mused that his work (PhD thesis) was initiated at a time "when 'respectable' geometric topology was necessarily compact". That attitude has long since faded; today's topological landscape is filled with research in which noncompact spaces are primary objects. However, major successes in understanding and compactifiying certain noncompact spaces included here are fundamental to manifold topology and geometric group theory: Whitehead and Davis manifolds, Stalling's characterization of Euclidean spaces, Siebenmann's thesis and our recent Gu-Guilbault's manifold completion theorem --- to name just a few. My goal is to provide a quick access to some of those results by weaving them together with common interpretations, motivating examples and my current research. I hope this talk will give the audience with various interests a brief appreciation of some of that work.

Connected Heegaard Floer homology and homology cobordism

We study applications of Heegaard Floer homology to homology cobordism. In particular, to a homology sphere Y, we define a module HF_conn(Y), called the connected Heegaard Floer homology of Y, and show that this module is invariant under homology cobordism and isomorphic to a summand of HF_red(Y). The definition of this invariant relies on involutive Heegaard Floer homology. We use this to define a new filtration on the homology cobordism group, and to give a reproof of Furuta's theorem. This is joint work with Jen Hom and Tye Lidman.

Quantum representations and geometry of mapping class groups

The generalization of the Jones polynomial for links and 3-manifolds, due to Witten-Reshetiking-Turaev in the late 90’s, led to constructions of Topological Quantum Field Theory in dimensions (2+1). These theories also include representations of surface mapping class groups. The question of how much of the Thurston geometric picture of 3-manifolds is reflected in these theories is open. I will report on recent work in this direction with emphasis on the corresponding mapping class group representations. The talk is based on joint work with R. Detcherry and G. Belletti, R. Detcherry, T. Yang.

Coisotropic submanifolds of symplectic manifolds, leafwise fixed points, and spherical nonsqueezing

My talk is partly about joint work with Dusan Joksimovic, and with Jan Swoboda.

Consider a symplectic manifold (M,\omega), a closed coisotropic submanifold N of M, and a Hamiltonian diffeomorphism \phi on M. A leafwise fixed point for \phi is a point x\in N that under \phi is mapped to its isotropic leaf. These points generalize fixed points and Lagrangian intersection points. In classical mechanics leafwise fixed points correspond to trajectories that are changed only by a time-shift, when an autonomous mechanical system is perturbed in a time-dependent way.

J. Moser posed the following problem: Find conditions under which leafwise fixed points exist. A special case of this problem is V.I. Arnold's conjecture about fixed points of Hamiltonian diffeomorphisms.

The main result presented in this talk is that leafwise fixed points exist if the Hamiltonian diffeomorphism is the time-1-map of a Hamiltonian flow whose restriction to N stays C_0-close to the inclusion N \to M. I will also mention a version of this result that is locally uniform in the symplectic form and the coisotropic submanifold.

As an application of a related result, no neighbourhood of the unit sphere symplectically embeds into the unit symplectic cylinder. This sharpens Gromov's nonsqueezing result.

The number of surfaces of fixed genus embedded in a 3-manifold

It was noticed before that presence of embedded essential surfaces in a 3-manifold can give information about that manifold. However to construct, classify or count such surfaces is a non-trivial task. If 3-manifold is complement of an alternating link with n crossings in a 3-sphere, we previously showed that the number of genus-g closed surfaces is bounded by a polynomial in n. This was the first polynomial bound. We then extended the result to spanning surfaces, and to 3-manifolds obtained as Dehn fillings of alternating links. This was joint work with Joel Hass and Abigail Thompson. In the talk, I will discuss its generalizations. One generalization is a joint work with Jessica Purcell. It concerns any cusped 3-manifold that is complement of a link alternating on some embedded surface in an arbitrary 3-manifold. Another one is joint work with Marc Lackenby. There, we prove that for any closed hyperbolic 3-manifold, there are polynomially many genus-g surfaces in terms of hyperbolic volume of the manifold.

Light bulbs in 4-manifolds

In 2017, Gabai proved the light bulb theorem, showing that if $R$ and $R'$ are 2-spheres homotopically embedded in a 4-manifold with a common dual, then with some condition on 2-torsion in $\pi_1(X)$ one can conclude that $R$ and $R'$ are smoothly isotopic. Schwartz later showed that this 2-torsion condition is necessary, and Schneiderman and Teichner then obstructed the isotopy whenever this condition fails. I showed that when $R'$ does not have a dual, we may still conclude the spheres are smoothly concordant.

I will talk about these various definitions and theorems as well as new joint work with Michael Klug generalizing the result on concordance to the situation where $R$ has an immersed dual (and $R'$ may have none), which is a common condition in 4-dimensional topology.

Flat and hyperbolic geometry of surfaces

The uniformization theorem tells us that the deformation space of constant curvature metrics on a surface also describes the moduli space of its complex structures. We will review some constructions coming from hyperbolic geometry and some coming from complex analysis. Using Bonahon and Thurston’s shear coordinates for Teichmüller space, Mirzakhani proved that two flows, namely earthquake flow from hyperbolic geometry and Teichmüller horocycle flow from complex analysis, are measurably isomorphic. Mirzakhani’s correspondence does not factor through uniformization and it is defined only on a measurably generic set. We will introduce a new coordinate system for Teichmüller space that allows us to extend Mirzakhani’s conjugacy everywhere in a natural way relating the hyperbolic geometry of a surface to certain singular flat metrics induced by quadratic differentials. This is joint work (in progress) with Aaron Calderon.

Taut sutured handlebodies as twisted homology products

A basic problem in the study of 3-manifolds is to determine when geometric objects are of ‘minimal complexity’. We are interested in this question in the setting of sutured manifolds, where minimal complexity is called ‘tautness’.

One method for certifying that a sutured manifold is taut is to show that it is homologically simple - a so-called ‘rational homology product’. Most sutured manifolds do not have this form, but do always take the more general form of a ‘twisted homology product’, which incorporates a representation of the fundamental group. The question then becomes, how complicated of a representation is needed to realize a given sutured manifold as such?

We explore the case of sutured handlebodies, and see even among the simplest class of these, twisting is required. We give examples that, when restricted to solvable representations, the twisting representation cannot be ‘too simple’.

Statistics for random curves

For S a finite type orientable surface, we study random walks on the Cayley graph of the fundamental group associated to a finite generating set. In particular we show that generically, the self-intersection number of a curve is bounded from above and below by the square of its word length, and we obtain bounds on the lifting degree (the minimum degree cover to which the curve admits a lift with no self-intersection) and the minimum geodesic length of the curve in any complete hyperbolic metric on the surface. We apply this to study bounds due to Dowdall, relating the dilatation of a point-pushing pseudo-Anosov to the self-intersection number of the defining curve, and we prove that generically, these bounds can be dramatically improved. This represents joint work with Jonah Gaster.

Involutive Floer homology of surgeries

We recall the definition of involutive Heegaard Floer homology of three-manifolds and then give a generalization of Ozsvath-Szabo's mapping cone formula to involutive Floer homology. From this, we'll give some applications to the homology cobordism of integer homology three-spheres. This is joint work with Kristen Hendricks, Jen Hom, and Ian Zemke.

Partial hyperbolicity and pseudo-Anosov dynamics

We study partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds. We classify these diffeomorphisms and obtain results concerning the global structure of these diffeomorphisms in these manifolds. We announce preliminaries results showing the following: up to finite iterates, if a partially hyperbolic diffeomorphism of a hyperbolic 3-manifold is not leaf conjugate to the time one map of a topological Anosov flow, then it can be obtained as a blow down of a power of a "step up" map of an R-covered topological Anosov flow. In particular this shows that a hyperbolic 3-manifold that admits a partially hyperbolic diffeomorphism also admits a topological Anosov flow. Very similar techniques can be used to prove that if a partially hyperbolic diffeomorphism of a Seifert fibered spaces induces a pseudo-Anosov diffeomorphism in the base surface, then it is not dynamically coherent, and it is also obtained by an appropriate blow down of a smooth R-covered Anosov flow.