Rutgers Geometry/Topology seminar: Fall 2018 - Spring 2019

The Inequalities of Hitchin-Thorpe and Thorpe

J. A. Thorpe and N. Hitchin independently discovered that the Euler characteristic and signature of a compact, oriented, 4-dimensional Einstein manifold must satisfy a remarkable inequality. As a consequence there are infinitely many simply-connected topological manifolds that cannot support an Einstein metric. An underemphasized aspect of the story of this so-called Hitchin-Thorpe inequality is that it is only a special case of Thorpe's original and more general inequality. In this talk, we give some background on the classical Hitchin-Thorpe inequality as well as Thorpe's inequality before discussing our recent Generalized Thorpe Inequality, which identifies the classical inequalities as resulting from a pure result in Chern-Weil theory.

Normal generation and fibered cones

This is joint work with Harry Baik, Eiko Kin and Hyunshik Shin. We showed that for any 2d slice in the Thurston's fibered cone, all but finitely many primitive points correspond to pseudo anosov maps that normally generate the mapping class group.

The Price twist and trisections

Let S be an RP^2 embedded in a smooth 4-manifold X^4. With some mild conditions, the Price twist is a surgery operation on S that yields a 4-manifold homeomorphic (but not necessarily diffeomorphic) to X^4. In particular, for every RP^2 embedded in S^4, this operation yields a homotopy 4-sphere.

In this talk, we will understand the Price twist via the theory of trisections. In particular, I will show how to produce an explicit trisection diagram of the surgered 4-manifold.

This is joint work with Seungwon Kim.

Continuous sections of families of complex algebraic varieties

Families of algebraic varieties exhibit a wide range of fascinating topological phenomena. Even families of zero-dimensional varieties (configurations of points on the Riemann sphere) and one-dimensional varieties (Riemann surfaces) have a rich theory closely related to the theory of braid groups and mapping class groups. In this talk, I will survey some recent work aimed at understanding one aspect of the topology of such families: the problem of (non)existence of continuous sections of "universal" families. Informally, these results give answers to the following sorts of questions: is it possible to choose a distinguished point on every Riemann surface of genus g in a continuous way? What if some extra data (e.g. a level structure) is specified? Can one instead specify a collection of n distinct points for some larger n? Or, in a different direction, if one is given a collection of n distinct points on CP^1, is there a rule to continuously assign an additional m distinct points? In this last case there is a remarkable relationship between n and m. For instance, we will see that there is a rule which produces 6 new points given 4 distinct points on CP^1, but there is no rule that produces 5 or 7, and when n is at least 6, m must be divisible by n(n-1)(n-2). These results are joint work with Lei Chen.

Schwartz functions on sub-analytic manifolds

Schwartz functions are classically defined on R^n as smooth functions such that they, and all their (partial) derivatives, decay at infinity faster than the inverse of any polynomial. The space of Schwartz functions is a Frechet space, and its continuous dual space is called the space of tempered distributions. A third space that plays a key role in the Schwartz theory is the space of tempered functions – a function is said to be tempered if point-wise multiplication by it preserves the space of Schwartz functions. This theory was formulated on R^n by Laurent Schwartz, later on Nash manifolds (smooth semi-algebraic varieties) by Fokko du Cloux and by Avraham Aizenbud and Dmitry Gourevitch, and on singular algebraic varieties by Boaz Elazar and myself.

The goal of this talk is to present the recently developed Schwartz theory on sub-analytic manifolds. I will first explain how one can attach a Schwartz space to an arbitrary open subset of R^n. Then, I will define (globally) sub-analytic manifolds – loosely speaking these are manifolds that locally look like sub-analytic open sub-sets of R^n (I will explain what are these too) and have some ”finiteness” property. Model theorists may think of definable manifolds in R^{an}. I will prove that one can intrinsically define the space of Schwartz functions (as well as the spaces of tempered functions and of tempered distributions) on these manifolds, and prove that these spaces are ”well behaved” (in the sense that they form sheaves and co-sheaves on the Grothendieck sub-analytic topology). Along the way we will see where sub-analyticity is used, and why this theory is ill-defined in the category of smooth (not necessarily sub-analytic) manifolds. Mainly, some ”polynomially bounded behaviour” (that holds in the sub-analytic case thanks to Lojasiewicz’s inequality) is required. As time permits I will describe some possible applications.

Geometry of biperiodic alternating links

In this talk we will study the hyperbolic geometry of alternating link complements in the thickened torus. We will give conditions which imply the hyperbolicity of alternating link complements in the thickened torus. We show that these complements admit a positively oriented, unimodular geometric ideal triangulation, and determine sharp upper and lower volume bounds. For links which arise from semi-regular Euclidean tilings, called semi-regular links, we explicitly determine the complete hyperbolic structure on their complements and will discuss nice consequences like determination of exact volumes, arithmeticity and commensurability for this class of links. We will also discuss the Volume Density Conjecture and examples.

The Seiberg-Witten equations and the length spectrum of hyperbolic three-manifolds

While both hyperbolic geometry and Floer homology have both been tremendously successful tools when studying three-dimensional topology, their relationship is still very mysterious. In this talk, we provide sufficient conditions for a hyperbolic rational homology sphere not to admit irreducible solutions to the Seiberg-Witten equations in terms of its volume and the length spectrum (i.e. the set of lengths of closed geodesics). We discus explicit examples in which this criterion can be applied. This is joint work with Michael Lipnowski.

Mahler measure and the Vol-Det Conjecture

A basic open problem is to understand how the hyperbolic volume of knots and links is related to diagrammatic knot invariants. The Vol-Det Conjecture relates the volume and determinant of alternating links. We prove the Vol-Det Conjecture for infinite families of alternating links using the dimer model, the Mahler measure of 2-variable polynomials, and the hyperbolic geometry of biperiodic alternating links. This is joint work with Abhijit Champanerkar and Matilde Lalin.

Compactness for generalized Seiberg-Witten equations

The Seiberg-Witten equation can be generalized to higher-rank vector bundles with a hyperKahler moment map. Examples among these are the Kapustin-Witten equations, Vafa-Witten equations, Seiberg-Witten equations with multiple spinors, and the ADHM equations. Witten, Haydys, and Doan-Walpuski have proposed several conjectures about the gauge theories from these equations, and the conjectures suggest that they will reflect topological information of the base manifold beyond the Yang-Mills and Seiberg-Witten equations. The analytic difficulty of establishing gauge theories based on generalized Seiberg-Witten equations is the lack of compactness. It was proved by Taubes and Haydys-Walpuski that solutions to many of these equations satisfy certain compactness properties described by Z/2-harmonic spinors. In this talk, we will discuss some recent progress on the compactness problems for generalized Seiberg-Witten equations. Some of the works presented in the talk are in collaboration with Thomas Walpuski.

Application of extremal length in elliptic estimates

Extremal length is a conformal invariant which is useful in a wide variety of areas. Discrete extremal length on graphs is also useful in discrete potential theory and circle packing problems. In this talk we will report their new applications in gradient estimates for 2nd order (discrete) elliptic PDEs in divergence form. We will also give one motivating example, which is about synchronization conditions for coupled oscillators on lattices.

Circle packings and Delaunay circle patterns for complex projective structures

At the interface of discrete conformal geometry and the study of Riemann surfaces lies the Koebe-Andreev-Thurston theorem. Given a triangulation of a surface S, this theorem produces a unique hyperbolic structure on S and a geometric circle packing whose dual is the given triangulation. In this talk, we explore an extension of this theorem to the space of complex projective structures - the family of maximal CP^1-atlases on S up to Möbius equivalence. Our goal is to understand the space of all circle packings on complex projective structures with a fixed dual triangulation. As it turns out, this space is no longer a unique point and evidence suggests that it is homeomorphic to Teichmüller space via uniformization - a conjecture by Kojima, Mizushima, and Tan. In joint work with Jean-Marc Schlenker, we show that this projection is proper, giving partial support for the conjectured result. Our proof relies on geometric arguments in hyperbolic ends and allows us to work with the more general notion of Delaunay circle patterns, which may be of separate interest. I will give an introductory overview of the definitions and results and demonstrate some software used to motive the conjecture.

A characterization of separable subgroups of 3-manifold groups

The subgroup separability is a property in group theory that is closely related to low dimensional topology, especially lifting \pi_1-injective immersed objects in a space to be embedded in some finite cover and the virtual Haken conjecture of 3-manifolds resolved by Agol. We give a complete characterization on which finitely generated subgroups of finitely generated 3-manifold groups are separable. Our characterization generalizes Liu's spirality character on \pi_1-injective immersed surface subgroups of closed 3-manifold groups. A consequence of our characterization is that, for any compact, orientable, irreducible and boundary-irreducible 3-manifold M with nontrivial torus decomposition, \pi_1(M) is LERF if and only if for any two adjacent pieces in the torus decomposition of M, at least one of them has a boundary component with genus at least 2.

Koebe circle domain conjecture and the Weyl problem in the hyperbolic 3-space

In 1908, Paul Koebe conjectured that every open connected set in the plane is conformally diffeomorphic to an open connected set whose boundary components are either round circles or points. The Weyl problem, in the hyperbolic setting, asks for isometric embedding of surfaces of curvature at least -1 in to the hyperbolic 3-space. We show that there are close relationships among the Koebe conjecture, the Weyl problem and the work of Alexandrov and Thurston on convex surfaces. This is a joint work with Tianqi Wu.

When can you twist out exponentially growing outer automorphisms?

A theme in the study of mapping class groups is the construction of partial pseudo-Anosov mapping classes by various means. The oldest construction, which dates to Thurston, is to take the product of squares of two Dehn twists about curves that intersect. The resulting mapping class will be pseudo-Anosov when restricted to the subsurface filled by the curves. The study of the outer automorphism group of a free group has progressed analogously to the study of mapping class groups. In Out(F_r), the analog of a partial pseudo-Anosov mapping class is an exponentially growing outer automorphism. Using surfaces with free fundamental group, one can define Dehn twists in Out(F_r). In this talk I will present an algorithmic criterion to decide when a subgroup of Out(F_r) generated by two Dehn twists contains an exponentially growing outer automorphism, giving a complete characterization of the appropriate Out(F_r) analog of "intersection" for the analog of Thurston's construction.

Index-energy estimates for Yang--Mills connections and Einstein metrics

We prove a conformally invariant estimate for the index of Schrodinger operators acting on vector bundles over four-manifolds, related to the classical Cwikel-Lieb-Rozenblum estimate. Applied to Yang-Mills connections we obtain a bound for the index in terms of its energy which is conformally invariant, and captures the sharp growth rate. Furthermore we derive an index estimate for Einstein metrics in terms of the topology and the Einstein-Hilbert energy. Lastly we derive conformally invariant estimates for the Betti numbers of an oriented four-manifold with positive scalar curvature. This is joint work with Matthew Gursky (University of Notre Dame) and Jeffrey Streets (UC Irvine).

Index-energy estimates for Yang--Mills connections and Einstein metrics

The abstract commensurator (aka “virtual automorphisms”) of a group encodes “hidden symmetries”, and is a natural generalization of the automorphism group. In this talk, I will give an introduction to these mysterious and classical groups and then discuss their residual finiteness. Recall that residual finiteness is a property enjoyed by linear groups (by A. I. Malcev), mapping class groups of closed oriented surfaces (by EK Grossman), and branch groups (by definition!). Moreover, by work of Armand Borel, Gregory Margulis, G. D. Mostow, and Gopal Prasad, the abstract commensurator of any irreducible lattice in any “nice enough” semisimple Lie group is locally residually finite (a property is termed “local” if it is satisfied by every finitely generated subgroup of the group). “Nice enough” is sufficiently broad that the only remaining unknown case is PSL_2(R). Are abstract commensurators of lattices in PSL_2(R) locally residually finite? Lattices here are commensurable with either a free group of rank 2 or the fundamental group of an oriented surface of genus 2. I will present a complete answer to this question with a proof that is computer-assisted. Our answer and methods open up new questions and research directions, so graduate students are especially encouraged to attend. This talk covers joint work with Daniel Studenmund.

A 2-category of hamiltonian manifolds

We will construct a strict 2-category whose objects are Lie groups, 1-morphism spaces are generated by hamiltonian manifolds, and 2-morphism spaces are generated by equivariant lagrangian correspondences. Our construction involves the completion of a partial 2-category, which we will define. This is an attempt of extending Wehrheim and Woodward's "Floer Field Theory" down to dimension 1. Ultimately, our category should be promoted to a 3-category, and one should be able to define an extended field theory in dimensions 1+1+1+1 that should incorporate Donaldson's polynomials.

Tiling billiards and interval exchange transformations

Tiling billiards is a new dynamical system where a beam of light refracts through a planar tiling. It turns out that, for a regular tiling of the plane by congruent triangles, the light trajectories can be described by interval exchange transformations. I will explain this surprising correspondence, and will also discuss the behavior of the system for other interesting tilings.

Introduction to geometric aspects of Ricci flow

In this talk, aimed for topologists and geometers, we will survey some works of others on understanding the qualitative behavior of Ricci flow. We will start with some low-dimensional results and then discuss some higher dimensional results.

A central limit theorem for random closed geodesics on surfaces

In 2013, Chas, Li, and Maskit produced numerical experiments on random closed geodesics on a hyperbolic pair of pants. Namely, they drew uniformly at random conjugacy classes of a given word length, and considered the hyperbolic lengths of the corresponding closed geodesic on the pair of pants. Their experiments lead to the conjecture that the length of these closed geodesics satisfies a central limit theorem. I will discuss a proof of this conjecture obtained in joint work with I. Gekhtman and G. Tiozzo, and its generalizations to all negative curved surfaces.

Rigidity of sphere packing on triangulated 3-manifolds

Sphere packing on triangulated 3-manifolds is an analogue of circle packing on triangulated surfaces. In this talk, we will describe some rigidity results of sphere packings on 3-manifolds. Part of the talk is based on recent joint work with Xiaokai He.

Embedded Delaunay triangulations for point clouds of surfaces in R^3

The use of geometric methods for shape comparison of real life objects has seen success in finding algorithms to do so. Hence efficiency and effectiveness of the algorithms inspired by theses ideas should have geometric meaning. In this talk we will see sufficient conditions on the point cloud to show that the diagonal switch algorithm finds a embedded Delaunay triangulation, which is a desired preliminary property for a initial data set for a surface.

Exploring algebraic rigidity in mapping class groups

A classical theorem of Powell (with roots in the work of Mumford and Birman) states that the pure mapping class group of a connected, orientable, finite-type surface of genus at least 3 is perfect, that is, it has trivial abelianization. We will discuss how this fails for infinite-genus surfaces and give a complete characterization of all homomorphisms from pure mapping class groups of infinite-genus surfaces to the integers. This characterization yields a direct connection between algebraic invariants of pure mapping class groups and topological invariants of the underlying surface. This is joint work with Javier Aramayona and Priyam Patel.

Ribbon concordances and knot Floer homology

A ribbon concordance between two knots is a concordance with only index 0 and 1 handles. In this talk, we will show that the map induced by a ribbon concordance on knot Floer homology is an injection. A corollary is the monotonicity of the Seifert genus under ribbon concordance. If time permits, we will describe some additional results concerning ribbon concordances which are joint work with Maggie Miller and Andras Juhasz.

Partially hyperbolic diffeomorphisms in dimension 3

These diffeomorphisms exhibit weaker forms of hyperbolicity and are extremely common. We study these in dimension 3 and prove some rigidity or classification results. We assume that the diffeomorphism is homotopic to the identity, and show that certain invariant foliations associated with the diffeomorphism have a structure that is well determined. This has some important consequences when the manifold is either hyperbolic or Seifert: under certain conditions we prove the diffeomorphism is up to iterates and finite covers, leaf conjugate to the time map of a topological Anosov flow. This is joint work with Thomas Barthelme, Steven Frankel, and Rafael Potrie.

Torus representations with connected isotropy groups and a conjecture of Hopf

A conjecture of Hopf from the 1930s states the following: A closed, even-dimensional Riemannian manifold with positive sectional curvature has positive Euler characteristic. In joint work with Michael Wiemeler and Burkhard Wilking, this conjecture is confirmed under the additional assumption that the isometry group has rank at least five. Similar previous results required bounds on the rank that grew to infinity in the manifold dimension. The main new tool is a structural result for representations of tori with the special property that all isotropy groups are connected. Such representations are surprisingly rigid, and we analyze them using only elementary techniques. A full classification of such representations remains an open problem.

Tame Quasiconformal Motions and Teichmuller Spaces

The concept of “quasiconformal motion” was first introduced by Sullivan and Thurston. They asserted that any quasiconformal motion of a set in the sphere over an interval can be extended to the sphere. However, in our recent work, we gave a counterexample to that assertion. Based on this counterexample, we introduced a new concept called “tame quasiconformal motion” and show that their assertion is true for tame quasiconformal motions. Actually, we proved a much more general result that, any tame quasiconformal motion of a closed set in the sphere, over a simply connected Hausdorff space, can be extended as a quasiconformal motion of the sphere. Furthermore, we showed that this extension can be done in a conformally natural way. The fundamental idea is to show that the Teichmuller space of a closed set in the sphere is a “universal parameter space” for tame quasiconformal motions of that set over a simply connected Hausdorff space. This talk is based on a joint work with Sudeb Mitra, Hiroshige Shiga, and Zhe Wang.