The Seminar usually takes place on Wednesdays at 2:30 PM in Room 617 on the sixth floor of Wachman Hall.
Vasily Dolgushev, Temple University
I will introduce a functor from a poset of certain finite index normal subgroups of the braid group on 4 strands to the category finite groupoids. The limit of this functor coincides with the profinite version of the Grothendieck-Teichmueller group introduced by Vladimir Drinfeld in 1990. My interest in this functor is motivated by the famous question posed by Yasutaka Ihara at the ICM of 1990. This talk is based on a joint work with Khanh Le and Aidan Lorenz.
Emily Stark, Technion - Israel Institute of Technology
Two far-reaching methods for studying the geometry of a finitely generated group with non-positive curvature are (1) to study the structure of the boundaries of the group, and (2) to study the structure of its finitely generated subgroups. Cannon--Thurston maps, named after foundational work of Cannon and Thurston in the setting of fibered hyperbolic 3-manifolds, allow one to combine these approaches. Mitra (Mj) generalized work of Cannon and Thurston to prove the existence of Cannon--Thurston maps for normal hyperbolic subgroups of a hyperbolic group. I will explain why a similar theorem fails for certain CAT(0) groups. I will also explain how we use Cannon--Thurston maps to obtain structure on the boundary of certain hyperbolic groups. This is joint work with Algom-Kfir--Hilion and Beeker--Cordes--Gardham--Gupta.
Nick Miller, Indiana University
Nick Salter, Columbia University
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.
Saul Schleimer, University of Warwick
Singular euclidean structures on surfaces are a key tool in the study of the mapping class group, of Teichmüller space, and of kleinian three-manifolds. François Guéritaud, while studying work of Ian Agol, gave a powerful technique for turning a singular euclidean structure (on a surface) into a triangulation (of a three-manifold). We will give an exposition of some of this work from the point of view of Delaunay triangulations for the L^\infty metric. We will review the definitions in a relaxed fashion, discuss the technique, and then present applications to the study of strata in the space of singular euclidean structures. If time permits, we will also discuss the naturally occurring algorithmic questions.
This is joint work with Mark Bell and Vaibhav Gadre. Some of our results are independently due to Ian Frankel, who has further applications.
Hannah Schwartz, Bryn Mawr College
Two knots in S^3 are ambiently isotopic if and only if there is an orientation preserving automorphism of S^3 carrying one knot to the other (this follows from the classical result that every orientation preserving automorphism of S^3 is isotopic to the identity). In this talk, we will examine a family of smooth 4-manifolds in which the analogue of this fact does not hold, i.e. each manifold contains a pair of smoothly embedded, homotopic 2-spheres that are related by a diffeomorphism, but not smoothly isotopic. In particular, the presence of 2-torsion in the fundamental groups of these 4-manifolds can be used to obstruct even a topological isotopy between the 2-spheres.
Ben Stucky, University of Oklahoma
In 2013, Joseph Lauer and Daniel Wise showed that a one-relator group whose defining relator has exponent at least 4 admits a proper, cocompact action on a CAT(0) cube complex, thus verifying a powerful non-positive curvature condition for these groups. To do this, they build a system of nicely-behaved codimension-1 subspaces (“walls”) in the universal cover and invoke a construction due to Sageev. I will describe a generalization of this result to one-relator products, namely, that a one-relator product of locally indicable groups whose defining relator has exponent at least 4 admits a geometric action on a CAT(0) cube complex if the factors do. The main tools are geometric small-cancellation results for van Kampen diagrams over these groups, which allow us to argue that walls are plentiful and geometrically well-behaved in the universal cover. Relative hyperbolicity of these one-relator products and relative quasiconvexity of wall stabilizers both play a central role.
Jeffrey Meyer, Cal State San Bernardino
The length of the shortest closed loop in a Riemannian manifold is called the systole. There are deep connections between the systole and the volume of a manifold. Recently there has been interest in how the systole grows as one goes up a tower of covers. Interestingly, this growth is deeply related to number theory. In this talk, I will go over some examples, these deep connections, and recent results. I will start by concretely looking at the systole growth up covers of flat tori. I will then discuss the celebrated result of Buser and Sarnak in which they showed that systolic growth is logarithmic in area up congruence covers of arithmetic hyperbolic surfaces. I will conclude by discussing my results from a recent paper with collaborators Sara Lapan and Benjamin Linowitz in which we show that the systolic growth up congruence p-towers is a least logarithmic in volume for all arithmetic simple locally symmetric spaces.
Jonah Gaster, McGill University
Abstract: In the context of proving that the mapping class group has finite asymptotic dimension, Bestvina-Bromberg-Fujiwara exhibited a finite coloring of the curve graph, i.e. a map from the vertices to a finite set so that vertices of distance one have distinct images. In joint work with Josh Greene and Nicholas Vlamis we give more attention to the minimum number of colors needed. We show: The separating curve graph has chromatic number coarsely equal to g log(g), and the subgraph spanned by vertices in a fixed non-zero homology class is uniquely g-1-colorable. Time permitting, we discuss related questions, including an intriguing relationship with the Johnson homomorphism of the Torelli group.
Final presentations from Math 9072 on the Poincaré homology sphere and related topics.
Presenters: Thomas Ng, Rebekah Palmer, Khánh Le, and Elham Matinpour.
Matthew Stover, Temple University
Reid and McMullen 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.
Feng Luo, Rutgers University
PATCH Seminar (joint with Bryn Mawr, Haverford, and Penn)
Abstract: We discuss some of the recent work on discrete conformalgeometry of polyhedral surfaces. The relationship among discrete conformal geometry, the work of Thurston and Alexandrov on convex surfaces in hyperbolic 3-space, and the Koebe circle domain conjecture will be addressed. We also show that the discrete uniformization maps converge to the conformal maps. This is joint work with D. Gu, J. Sun, and T. Wu.
In the morning background talk (at 11:30am), I will review geometric notions such as Delaunay triangulations.
Caitlin Leverson, Georgia Tech
PATCH Seminar (joint with Bryn Mawr, Haverford, and Penn)
Abstract: Legendrian knots are topological knots which satisfy extra geometric conditions. Two classes of invariants of Legendrian knots in \(S^3\) are ruling polynomials and representations of the Chekanov-Eliashberg differential graded algebra (DGA). Given a knot \(K\) and a positive permutation braid \(\beta\), we give a precise formula relating a specialization of the ruling polynomial of the satellite \(S(K,\beta)\) with certain counts of representations of the DGA of the original knot \(K\). We also introduce an \(n\)-colored ruling polynomial, defined analogously to the \(n\)-colored HOMFLY-PT polynomial, and show that the 2-graded version of it arises as a specialization of the \(n\)-colored HOMFLY-PT polynomial. This is joint work with Dan Rutherford.
In the morning background talk (at 10:00 AM), I will give an introduction to Legendrian satellite knots, ruling polynomials, and representations of the Chekanov-Eliashberg DGA.
Bena Tshishiku, Harvard University
Abstract: In the 1960s Atiyah and Kodaira constructed surface bundles over surfaces with many interesting properties (e.g. they're holomorphic with closed base and the total space has nonzero signature). Many questions remain about these examples, including a precise description of their monodromy, viewed as a subgroup of the symplectic group. In this talk I will discuss some recent progress toward this question. The main result is that the monodromy is arithmetic (as opposed to being thin). This is ongoing joint work with Nick Salter.
Kei Nakamura, Rutgers University
Abstract: It has been known for sometime that the Apollonian circle 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 show that super-integral packings exists only in finitely many dimensions, and only in finitely many commensurability classes.
Wouter van Limbeek, University of Michigan
Abstract: In 1893, Hurwitz showed that a Riemann surface of genus \(g \geq 2\) admits at most \(84(g-1)\) automorphisms; equivalently, any 2-dimensional hyperbolic orbifold \(X\) has $\(Area(X)\geq \pi / 42\). In contrast, such a lower bound on volume fails for the n-dimensional torus \(T^n\), which is closely related to the fact that \(T^n\) covers itself nontrivially. Which geometries admit bounds as above? Which manifolds cover themselves? In the last decade, more than 100 years after Hurwitz, powerful tools have been developed from the simultaneous study of symmetries of all covers of a given manifold, tying together Lie groups, their lattices, and their appearances in differential geometry. In this talk I will explain some of these recent ideas and how they lead to progress on the above (and other) questions.
Ara Basmajian, CUNY Graduate Center
Abstract: In this talk we first describe some of the known results on the geometry and topology of infinite (topological) type surfaces and then we investigate the relationship between Fenchel-Nielsen coordinates and when the geodesic flow on such a surface is ergodic. Ergodicity of the geodesic flow is equivalent to the surface being of so called parabolic type (the surface does not carry a Green's function), and hence this problem is intimately connected to a version of the classical type problem in the study of Riemann surfaces. Specifically, we study so called tight flute surfaces -- (possibly incomplete) hyperbolic surfaces constructed by linearly gluing infinitely many tight pairs of pants along their cuffs -- and the relationship between their type and geometric structure. This is joint work with Hrant Hakobyan and Dragomir Saric.
Jing Tao, University of Oklahoma
Abstract: In this talk, I will describe an elementary and topological argument that gives bounds for the stable commutator lengths in right-angled Artin groups.
Olga Plamenevskaya, SUNY Stony Brook
PATCH Seminar, joint with Bryn Mawr, Haverford, and Penn
Abstract: Due to work of Giroux, contact structures on 3-manifolds can be topologically described by their open books decompositions (which in turn can be encoded via fibered links). A contact structure is called planar if it admits an open book with fibers of genus 0. Symplectic fillings of such contact structures can be understood, by a theorem of Wendl, via Lefschetz fibrations with the same planar fiber. Using this together with topological considerations, we prove a new obstruction to planarity (in terms of intersection form of fillings) and obtain a few corollaries. In particular, we consider contact structures that arise in a canonical way on links of surface singularities, and show that the canonical contact structure on the link is planar only if the singularity is rational. (Joint work with P. Ghiggini and M. Golla.)
In the background talk (11:00 AM), I will discuss topological properties of Lefschetz fibrations over a disk, focusing on the case where fiber is a surface of genus 0. The boundary of the 4-manifold given by Lefschetz fibration has an induced open book and a contact structure. This will be the setting for my second talk.
Christine Lee, University of Texas
PATCH Seminar, joint with Bryn Mawr, Haverford, and Penn
Abstract: Quantum link invariants lie at the intersection of hyperbolic geometry, 3-dimensional manifolds, quantum physics, and representation theory, where a central goal is to understand its connection to other invariants of links and 3-manifolds. In this talk, we will introduce the colored Jones polynomial, an important example of quantum link invariants. We will discuss how studying properly embedded surfaces in a 3-manifold provides insight into the topological and geometric content of the polynomial. In particular, we will describe how relating the definition of the polynomial to surfaces in the complement of a link shows that it determines boundary slopes and bounds the hyperbolic volume of many links, and we will explore the implication of this approach on these classical invariants.
In the background talk (9:30, AM) I'll introduce the colored Jones polynomial and discuss the many conjectures/open problems surrounding the polynomial, to give the research talk more context.
Lee Mosher, Rutgers Newark
In the course of our theorem on the \(H^2_b\)-alternative for \(Out(F_n)\) — every finitely generated subgroup of \(Out(F_n)\) is either virtually abelian or has second bounded cohomology of uncountable dimension — the case of subgroups of natural embeddings of \(Aut(F_k)\) into \(Out(F_n)\) led us to subgroups of \(Aut(F_k)\) which have interesting new hyperbolic actions arising from “suspension” constructions, generalizing a thread of hyperbolic suspension constructions which goes back to a theorem of W. Thurston. In this talk we will describe these suspension constructions, and we will speculate on what may unify them.
This is joint work with Michael Handel.
Benjamin Collas, Bayreuth
The goal of Grothendieck-Teichmüller theory is to lead an arithmetic study of the moduli spaces of curves via their geometric fundamental group. Once identified to the profinite orbifold fundamental group, the latter provides a computational framework in terms of braid and mapping class groups.
While the classical GT theory, as developed by Drinfel'd, Lochak, Nakamura, Schneps et al., essentially deals with the schematic or topological properties of the spaces ``at infinity'', the moduli spaces of curves also admit a stack or orbifold structure that encodes the automorphisms of curves. The goal of this talk is to show how fundamental group theoretic properties of the mapping class groups and Hatcher-Thurston pants decomposition lead to orbifold arithmetic results, then to potential finer GT groups.
We will present in detail this analytic Teichmüller approach and indicate the essential obstacles encountered, before briefly explaining how they can be circumvent in terms of arithmetic geometry.
Sam Taylor, Temple University
The rank of a group is the minimal cardinality of a generating set. While simple to define, this quantity is notoriously difficult to calculate, even for well behaved groups. In this talk, I’ll introduce some algebraic and geometric properties of hyperbolic group extensions and discuss how their bundle structure can be used to understand rank in this setting.
David Futer, Temple University
Abstract: The study of hyperbolic manifolds often begins with the thick-thin decomposition. Given a number \(\epsilon > 0\), we decompose a manifold into the \(\epsilon\)-thin part (points on essential loops of length less than \(\epsilon\)), and the \(\epsilon\)-thick part (everything else). The Margulis lemma says that there is a universal number \(\epsilon_n\), depending only on the dimension, such that the thin part of every hyperbolic \(n\)-manifold has very simple topology.
In dimension 3, we still do not know the optimal Margulis constant \(\epsilon_3\). Part of the problem is that while the topology is simple, the geometry of \(\epsilon\)-thin tubes can be quite complicated. I will describe some results that control and estimate the geometry, which has applications to narrowing down the value of the Margulis constant. This is joint work with Jessica Purcell and Saul Schleimer.