Catherine Pfaff

UCSB RTG Topology Seminar (Fall 2017-Spring 2018)

The topology seminar is co-organized by myself (Catherine Pfaff) and Stephen Bigelow. It is funded by the Research Training Groups (RTG) in Topology and Geometry funded by the Division of Mathematical Sciences (DMS) of the National Science Foundation (NSF) and University of California at Santa Barbara.

Directions and maps for UCSB and the Mathematics Department.

KITP's guide to local accommodation

Time: Tu 3:30-4:30 (except as noted)

Place: South Hall 6635 (except as noted)

FALL

10/3/2017, No Seminar

10/10/2017, Stephen Bigelow (University of California, Santa Barbara)

Title: Quantum computing with tangled trivalent graphs

Abstract: The Jones polynomial is a knot invariant, but started life as a representation of the braid groups. For a special value of the parameter, it can be extended to a representation of the category of tangled trivalent graphs. Freedman, Larsen and Wang showed that this could be used to do quantum computation. I will describe this strange kind of machine code in the language of tangled trivalent graphs, and the tricks that go into programming a quantum gate. This is joint work with Claire Levaillant.

10/17/2017, Catherine Pfaff (University of California, Santa Barbara)

Title: Pretalk To ``From measured Foliations to Teichmüller Space: using the right trousers''

Abstract: In ``From measured Foliations to Teichmüller Space'' we will introduce a new method for assigning to a marked hyperbolic metric on a closed surface S of genus g ≥ 2 a measured foliation on S. These foliations will provide a new naturality to the compactification of the Teichmüller space by projective measured foliations and allow one to decompose this Teichmüller space into copies of the Teichmüller space of a once-punctured torus. (Joint work in progress with Daryl Cooper.) This talk with be a ``pretalk'' where we familiarize the audience with the more classic picture.

10/24/2017, Francis Bonahon (University of Southern California)

Title: Coordinates for real projective structures on surfaces

Abstract: I will discuss certain coordinate systems for the space of convex projective structures on a closed surface, which are well-behaved with respect to a geodesic lamination on the surface. This is joint work with Guillaume Dreyer, and draws its origins in earlier breakthroughs of Fock and Goncharov for the case of punctured surfaces.

10/31/2017, Eleni Panagiotou (University of California, Santa Barbara)

Title: Topological Methods for Polymeric Materials: Characterizing the Relationship Between Polymer Entanglement and Viscoelasticity.

Abstract:
joint work with K C Millett and P J Atzberger
We draw on mathematical results from topology to develop quantitative methods for polymeric materials to characterize the relationship between polymer chain entanglement and bulk viscoelastic responses. We generalize the mathematical notion of the Linking Number and Writhe to be applicable to open (linear) chains. We show how our results can be used in practice by performing fully three-dimensional computational simulations of polymeric chains entangled in weaves of a few distinct topologies and with varying levels of chain densities. We investigate relationships between our topological characteristics for chain entanglement and viscoelastic responses by performing Lees-Edwards simulations of the rheology over a broad range of frequencies. Our topological measures of entanglement indicate the global topology is the dominant factor in characterizing mechanical properties. We find an almost linear relation between the mean absolute Writhe and the loss tangent and an almost inverse linear relation between the mean absolute Periodic Linking Number and the loss tangent. These results indicate the potential of our topological methods in providing a characterization of the level of chain entanglement useful in understanding the origins of mechanical responses in polymeric materials.

11/7/2017, Joshua Pankau (University of California Santa Barbara)

Title: Salem number stretch factors of pseudo-Anosov maps

Abstract: Associated to every pseudo-Anosov (pA) homeomorphism of a closed orientable surface is a real number, greater than 1, called the stretch factor. Thurston showed that the stretch factor of any pA map is an algebraic unit, but it is still an open question which algebraic units appear as stretch factors. In this talk we will look at a construction of pA maps due to Thurston, and focus on a certain type of algebraic unit known as a Salem number. I will discuss my recent result where I showed every Salem number has a power that is the stretch factor of a pA map coming from Thurston's construction.

11/14/2017 Nick Castro (University of California, Davis)

Title: Interactions Between 3- and 4-manifolds via Trisections

Abstract: A trisection of a smooth 4-manifold X is the 4-dimensional analog of a Heegaard splitting of a 3-manifold, decomposing X into three diffeomorphic, codimension 0 submanifolds whose intersections encode the complexity of X. In the case that X has non-empty boundary, a trisection induces a fiber bundle over the circle on the bounding 3-manifold(s) known as an open book decomposition. In this talk, I will give the basic definition of trisections of 4-manifolds with and without boundary, as well as define trisection diagrams. I will give an algorithm which explicitly determines the open book decomposition induced by a trisection (diagram). This is joint work with David Gay and Juanita Pinzón-Caicedo. If time permits, I will discuss the gluing theorem which highlights the importance of open book decompositions, and the algorithm, in the theory of relative trisections.

11/21/2017 No Seminar (Thanksgiving Week)

11/28/2017, Kathleen Hake (University of California Santa Barbara)

Title: Geometry of Polygonal Knot Space

Abstract: A $n$-sided polygon in R³ can be described as a point in R³ⁿ by listing in order the coordinates of it vertices. In this way, the space of n-sided polygons embedded in R³ is a manifold in which points correspond to piecewise linear knots and paths correspond to isotopies which preserve the geometric structure of these knots. Restricting to polygons of unit edge length gives a submanifold consisting of equilateral knots. We will discuss aspects of the topology of the space of equilateral hexagons as well as its symplectic structure. Additionally we will prove that at least 11/12 of the space of equilateral hexagons consists of unknots.

12/5/2017, Dmitry Gekhtman (Caltech)

Title: Holomorphic retractions onto Teichmüller disks

Abstract: There is a finite-dimensional space of ways to equip a closed surface with the structure of a complex manifold. This space of complex structures is called the Teichmüller space of the surface. It turns out that the Teichmüller space is itself a complex manifold, whose analytic structure reflects the surface's topology and geometry. Certain polygonal decompositions of the surface give rise to holomorphically embedded copies of the unit disk inside of Teichmüller space. These embedded disks are called Teichmüller disks. In this talk, we will describe recent partial answers to the following question: Which Teichmüller disks are holomorphic retracts of Teichmüller space?

WINTER

1/9/2018, 2-3pm, Carolyn Abbott (University of California, Berkeley)

Title: Ordering actions on hyperbolic metric spaces

Abstract: Every group admits at least one action by isometries on a hyperbolic metric space, and certain classes of groups admit many different actions on different hyperbolic metric spaces (in fact, often uncountably many). One such class of groups is the class of so-called acylindrically hyperbolic groups, which contains many interesting groups, such as mapping class groups, Out(F_n), and right-angled Artin and Coxeter groups, among many others. In this talk, I will describe how to put a partial order on the set of actions of a given group on hyperbolic spaces which, in some sense, measures how much information about the group the action provides. This partial order defines a "poset of actions" of the given group. I will then define the class of acylindrically hyperbolic groups and give some structural properties of the resulting poset of actions for such groups. In particular, I will discuss for which (classes of) groups the poset contains a largest element.

1/16/2018, Nicholas Vlamis (University of Michigan)

Title: Homomorphisms of pure mapping class groups to the integers

Abstract: 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 is joint work with Javier Aramayona and Priyam Patel.

1/23/2018, Daryl Cooper (University of California, Santa Barbara)

Title: Realizing Thurston's Compactification of Teichmüller Space with Quadratic Differentials

Abstract: Teichmüller space is the parameter space of isotopy classes of hyperbolic structures on a closed surface S of genus g. It if homeomorphic to R^6g-6. Thurston compactified it using equivalence classes of measured foliations. This yields a closed disc of dimension 6g-6. We will describe a new way to do this, by removing the equivalence relations and realising both hyperbolic metrics and measured foliations as explicit (no isotopy) (semi-)metrics on S. It is based on new coordinates for these spaces. Joint with Catherine Pfaff.

1/30/2018, Giuseppe Martone (University of Southern California)

Title: Hitchin representations and configurations of apartments

Abstract: Hitchin singled out a preferred component in the character variety of representations from the fundamental group of a surface to PSL(n,R). When n=2, this Hitchin component coincides with the Teichmüller space consisting of all hyperbolic metrics on the surface. Later Labourie showed that Hitchin representations share many important differential geometric and dynamical properties. Morgan and Shalen provided an algebro-geometric interpretation of Thurston's compactification of the Teichmüller space in terms of valuations on character varieties. Parreau extended this construction to a compactification of the Hitchin component whose boundary points are described by actions of the fundamental group of the surface on a building. This generalizes the actions on trees occurring for the Morgan-Shalen compactification. In this talk, we offer a new presentation for the Parreau compactification, which is based on certain positivity properties discovered by Fock and Goncharov. More precisely, we use the Fock-Goncharov construction to describe the intersection patterns of apartments in invariant subsets of the building that arise in the boundary of the Hitchin component.

2/6/2018, Catherine Pfaff (University of California, Santa Barbara)

Title: Counting conjugacy classes of fully irreducibles

Abstract: It is well known (due to Margulis, for example) that in a classical hyperbolic dynamical system, the number of closed curves of a bounded length increases exponentially with respect to the bound on the length of the curves. In 2011 Eskin and Mirzakhani proved that the same holds for the moduli space of a closed surface. In joint work with Ilya Kapovich, we prove that the outer automorphism group of the free group (and consequently the moduli space of graphs) exhibits a new behavior where this growth rate is in fact double exponential. To our knowledge, this is the only known example of such behavior.

2/13/2018, Beatrice Pozzetti (Universität Heidelberg)

Title: Ultralimits of maximal representations

Abstract: Maximal representations are an instance of the so-called higher (rank) Teichmüller theories: they form families of discrete subgroups of the Lie group Sp(2n,R), that are isomorphic to the fundamental group of a surface. Interestingly these subgroups share many geometric properties with holonomies of hyperbolizations. I will discuss joint work with Marc Burger, Alessandra Iozzi and Anne Parreau, in which we study actions on affine buildings arising as ultralimits of maximal representations and observe a clear distinction between phenomena already present in the boundary of the Teichmüller space and new flat features.

2/20/2018, Respite for the Talk Weary

2/27/2018, Respite for the Talk Weary

3/6/2018, Michael Kapovich (University of California, Davis)

Title: Discrete isometry groups of higher rank symmetric spaces

Abstract: I will review basic geometry of higher rank symmetric spaces, and then discuss discrete isometry groups of such spaces, which are higher rank generalizations of Kleinian groups. We will see how some of the classical notions and results translate in the higher rank setting and how do they connect to the broader geometric group theory. For instance: What is the higher rank analogue of the convergence property for Moebius transformations? How to define limit sets and domains of discontinuity in higher rank? How to define geometrically finite discrete groups in higher rank? What could quasiconvexity mean? Why would a person interested in RAAGs care about any of this? The talk is based on my work with Bernhard Leeb and Joan Porti.

3/13/2018, Ayaka Shimizu (National Institute of Technology, Gunma College)

Title: Some strategies for the reductivity problem on knot projections

Abstract: An inverse-half-twisted splice operation, HS^-1, is a local transformation on knot projections on the sphere. By a finite number of HS^-1s, we can obtain a reducible knot projection from any nontrivial knot projection, where a knot projection is reducible if we can put a circle on S² so that the circle intersects the knot projection transversely at just a crossing point. The reductivity of a knot projection is the minimal number of HS^-1 required to obtain a reducible knot projection from the knot projection. It has been shown by the speaker that the reductivity is four or less for any nontrivial knot projection. However, it is unknown if there exists a knot projection whose reductivity is four. In this talk, we see some strategies for this reductivity problem.

SPRING

4/3/2018, Talk Cancelled

4/10/2018, Jennifer Schultens (University of California, Davis)

Title: The Kakimizu complex of Seifert fibered spaces

Abstract: Seifert fibered spaces are 3-manifolds characterized by their decompositions into circular fibers. We will consider Seifert fibered spaces and surfaces that lie in Seifert fibered spaces. Moreover, we will define the Kakimizu complex and see how it helps us to understand surfaces in Seifert fibered spaces.

4/17/2018, Respite for the Talk Weary

4/24/2018, Matthew Cordes (Technion – Israel Institute of Technology)

Title: Geometric approximate group theory

Abstract: In this talk I will introduce the notion of an approximate group and discuss work in progress with Tobias Hartnick & Vera Tonic about developing geometric group theory style tools to analyze approximate groups.

5/1/2018, Heejoung Kim (University of Illinois, Urbana-Champaign)

Title: Stable subgroups and Morse subgroups of mapping class groups

Abstract: The notion of a ``quasiconvex" subgroup of a word-hyperbolic group G plays an important role in the theory of hyperbolic groups. This notion has several equivalent characterizations in that context, in terms of being ``undistorted", in terms of the action on the boundary, in terms of being ``rational" with respect to automatic structures on G, in terms of the contracting properties of the projection maps, etc. For an arbitrary finitely generated group G, there are two recent generalizations of the notion of a quasiconvex subgroup: a ``stable" subgroup and a ``Morse" subgroup. In this talk, we will discuss these two notions and their different properties. We prove that the properties of being Morse and being stable coincide for a subgroup of infinite index in the mapping class group of an oriented, connected, finite type surface with negative Euler characteristic.

5/8/2018, Thomas Koberda (University of Virginia)

Title: Diffeomorphism groups of intermediate regularity

Abstract: Let M be the interval or the circle. For each real number α \in [1,∞), write α=k+τ, where k is the floor function of α. I will discuss a construction of a finitely generated group of diffeomorphisms of M which are C^k and whose k^th derivatives are τ-Hölder continuous, but which are admit no algebraic smoothing to any higher Hölder continuity exponent. In particular, such a group cannot be realized as a group of C^k+1 diffeomorphisms of M. I will discuss the construction of countable simple groups with the same property, and give some applications to continuous groups of diffeomorphisms. This is joint work with Sang-hyun Kim.

5/15/2018, Abigail Thompson (University of California Davis)

Title: Trisections and surgery on links

Abstract: Gay and Kirby have shown that a closed 4-manifodl admits a ``trisection”, which is a decomposition of the manifold into three 4-dimensional handlebodies. Using this set-up, many interesting questions about 4-manifolds can be translated into questions about certain types of Dehn surgeries on links in some number of copies of ¹ X ². I’ll describe this translation, and examples of surgery questions that arise, as well as some progress on the simplest of these. This is joint work with Rob Kirby.

5/22/2018, Catherine Pfaff (University of California, Santa Barbara)

Title: Random automorphisms of free groups and what happens when you iterate them.

Abstract: Two of the most natural and interesting questions one can ask about an automorphism group is what a random (in a random walk sense) element of the automorphism group looks like and what happens as one repeatedly applies the automorphism to an element of the group (asymptotic conjugacy class invariants). In the mapping class group circumstance, these questions (and their intersection) have been thoroughly studied with results dating back to Nielsen and Thurston, and then more recently with Dahmani, Horbez, Maher, Rivin, Sisto, Tiozzo, etc. While some is known in the case of the outer automorphism group of the free group, little to nothing has been known about the most basic questions in the intersection of the main classes of questions, i.e. understanding the asymptotic conjugacy class invariants of random (outer) automorphisms of free groups. Together with Ilya Kapovich, Joseph Maher, and Samuel Taylor, we give a fairly detailed answer to this question.

5/29/2018, Ryan Blair (California State University, Long Beach)

Title: A Computable Definition of Bridge Number and Applications to the Meridional Rank Conjecture.

Abstract: The Meridional Rank Conjecture states that the bridge number of a knot and the meridional rank of a knot are always equal. We define the Wirtinger of a knot which is a diagrammatically defined invariant naturally bounded below by the meridional rank and above by the bridge number. We show that the Wirtinger number is equal to the bridge number and use this new definition to calculate the bridge number of 450,000+ tabulated knots.

6/5/2018, Emily Stark (Technion – Israel Institute of Technology)

Title: Cannon-Thurston maps in non-positive curvature

Abstract: 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 boundary of the group, and (2) to study the structure of its finitely generated subgroups. Cannon--Thurston boundary maps allow one to combine these approaches. Mitra (Mj) generalized work of Cannon and Thurston to prove the existence of Cannon--Thurston maps for any normal hyperbolic subgroup 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.

6/12/2018, Kathleen Hake (University of California Santa Barbara)

Title: Geometric Properties of Polygonal Knot Space

Abstract: For a positive integer n greater than 3, the collection of n-sided polygons embedded in 3-space defines the space of geometric knots. In this talk, we will consider the subspace of equilateral knots, consisting of embedded n-sided polygons with unit length edges. Paths in this space determine isotopies of polygons, so path-components correspond to equilateral knot types. When n is less than 6, the space of equilateral knots is connected. Therefore, we examine the space of equilateral hexagons. Using techniques from symplectic geometry, we can parametrize the space of equilateral hexagons with a set of measure preserving action-angle coordinates. With this coordinate system, we provide new bounds on the knotting probability of equilateral hexagons. Additionally, we study physical properties of equilateral knots, especially thickness.

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