# Geometry and Topology Seminar 2012

Current contact: Dave Futer or Matthew Stover

The Seminar usually takes place on Wednesdays at 2:45 PM in Room 617 on the sixth floor of Wachman Hall.

• Tuesday January 17, 2012 at 15:30, Wachman 617
GeoTop Seminar

Henry Segerman, University of Melbourne, Some mathematical sculptures

I will talk about some $3D$ printed mathematical sculptures I have designed. I'll say a little about the mathematical ideas behind them, and how they were produced. In the second half, I'll talk about sculptures of space filling curves, how wobbly they are, and fractal graph structures designed to be more robust.

• Tuesday January 17, 2012 at 17:00, Wachman 617
GeoTop Seminar

Jean Sun, Yale University, Growth, projections and bounded generation of mapping class groups

We investigate the non-bounded generation of subgroups of mapping class groups through the hierarchy in curve complexes developed by Masur and Minsky (2000). We compare the subsurface projections to nearest point projections in curve complexes and extend Behrstock's inequality to include geodesics in curve complexes of subsurfaces in the Inequality on Triples in Bestivina-Bromberg-Fujiwara (2010). Based on this inequality, we can estimate translation lengths of words in the form $g_1^{n_1}\cdots g_k^{n_k}$ when $\sum |n_k|$ is sufficiently large for any given sequence $(g_i)_1^k$ in a mapping class group. With a growth argument, we further show that any subgroup of a mapping class group is boundedly generated if and only if it is virtually abelian.

• Tuesday January 24, 2012 at 17:00, Wachman 617
GeoTop Seminar

Tian Yang, Rutgers University, The skein algebra and the decorated Teichmuller space

The Kauffman bracket skein module $K(M)$ of a $3$-manifold $M$ is defined by Przytycki and Turaev as an invariant for framed links in $M$ satisfying the Kauffman skein relation. For a compact oriented surface $S$, it is shown by Bullock--Frohman--Kania-Bartoszynska and Przytycki-Sikora that $K(S\times [0,1])$ is a quantization of the $SL_2\mathbb{C}$-characters of the fundamental group of $S$ with respect to the Goldman--Weil--Petersson Poisson bracket.

In a joint work with J. Roger, we define a skein algebra of a punctured surface as an invariant for not only framed links but also framed arcs in $S\times [0,1]$ satisfying the skein relations of crossings both in the surface and at punctures. This algebra quantizes a Poisson algebra of loops and arcs on $S$ in the sense of deformation of Poisson structures. This construction provides a tool to quantize the decorated Teichmuller space; and the key ingredient in this construction is a collection of geodesic lengths identities in hyperbolic geometry which generalizes/is inspired by Penner's Ptolemy relation, the trace identity and Wolpert's cosine formula.

• Tuesday January 31, 2012 at 16:30, Wachman 617
GeoTop Seminar

Harold Sultan, Columbia University, Asymptotic geometry of Teichmuller space and divergence

I will talk about the asymptotic geometry of Teichmuller space equipped with the Weil-Petersson metric. In particular, I will give a criterion for determining when two points in the asymptotic cone of Teichmuller space can be separated by a point; motivated by a similar characterization in mapping class groups by Behrstock-Kleiner-Minsky-Mosher and in right angled Artin groups by Behrstock-Charney. As a corollary, I will explain a new way to uniquely characterize the Teichmuller space of the genus two once punctured surface amongst all Teichmuller space in that it has a divergence function which is superquadratic yet subexponential.

• Tuesday January 31, 2012 at 17:30, Wachman 617
GeoTop Seminar

Inanc Baykur, Max Planck Institute for Mathematics, Bonn, Surface bundles and Lefschetz fibrations

Surface bundles and Lefschetz fibrations over surfaces constitute a rich source of examples of smooth, symplectic, and complex manifolds. Their sections and multisections carry interesting information on the smooth structure of the underlying four-manifold. In this talk I will discuss several problems and results on (multi)sections of surface bundles and Lefschetz fibrations; joint with Mustafa Korkmaz and Naoyuki Monden. In the second part of the talk I will demonstrate the contrast(s) between symplectic and holomorphic fibrations. The talk will feature various construction techniques, where mapping class group factorizations will play a leading role.

• Tuesday February 14, 2012 at 16:00, Wachman 617
GeoTop Seminar

Thomas Church, Stanford University, Representation theory and homological stability

• Tuesday February 14, 2012 at 17:30, Wachman 617
GeoTop Seminar

Mark Sapir, Vanderbilt University, Asymptotic properties of mapping class groups

We study asymptotic cones of mapping class groups. The main result states that the asymptotic cones equivariantly embed into a direct product of finitely many $\mathbb{R}$--trees. Several known and new algebraic properties of the mapping class group follow. This is joint work with J. Behrstock and C. Drutu.

• Tuesday February 21, 2012 at 17:00, Wachman 617
GeoTop Seminar

Thomas Koberda, Harvard University, Mapping class groups and finite covers

I will give a survey of results concerning the actions of a mapping class on the homology of various finite covers to which it lifts. I will draw connections to 3-manifold theory, especially largeness, growth of torsion homology and Alexander polynomials.

• Tuesday February 28, 2012 at 16:00, Wachman 617
GeoTop Seminar

PATCH seminar, joint with Bryn Mawr, Haverford, and Penn

Morwen Thistlethwaite, University of Tennessee, Finding and deforming representations of 3-manifold groups.

Some assorted methods are described for finding exact specifications of representations of 3-manifold groups into classical matrix groups. These include (i) a method for finding hyperbolic structures on links that does not involve an ideal triangulation of the link complement, and (ii) deformations away from the hyperbolic structure of certain closed hyperbolic 3-manifolds.

• Tuesday February 28, 2012 at 17:30, Wachman 617
GeoTop Seminar

PATCH seminar, joint with Bryn Mawr, Haverford, and Penn:

Nancy Hingston, The College of New Jersey and IAS, Loop products and dynamics.

A metric on a compact manifold M gives rise to a length function on the free loop space LM whose critical points are the closed geodesics on M in the given metric. Morse theory gives a link between Hamiltonian dynamics and the topology of loop spaces, between iteration of closed geodesics and the algebraic structure given by the Chas-Sullivan product on the homology of LM. Geometry reveals the existence of a related product on the cohomology of LM.

A number of known results on the existence of closed geodesics are naturally expressed in terms of nilpotence of products. We use products to prove a resonance result for the loop homology of spheres.

I will not assume any prior knowledge of loop products.

Mark Goresky, Hans-Bert Rademacher, and (work in progress) Ralph Cohen and Nathalie Wahl are collaborators.

• Tuesday March 13, 2012 at 17:00, Wachman 617
GeoTop Seminar

An important line of research in 3-dimensional topology is the study of the behavior of the rank of the fundamental groups of the finite sheeted covers of an orientable hyperbolic 3-manifold. In this talk I will present some outstanding open problems and recent developments in the area. I will also construct what seems to be the first examples of such manifolds which have co-final towers of finite sheeted covers for which the rank of the fundamental groups grow linearly with the degree of the covers.

• Thursday March 15, 2012 at 16:30, PATCH seminar, at Penn, DRL room 4C8
GeoTop Seminar

Eli Grigsby, Boston College, A relationship between representation-theoretic and Floer-theoretic braid invariants

Given a braid, one can associate to it a collection of “categorified” braid invariants in two apparently different ways: “algebraically,” via the representation theory of Uq(sl2) (using ideas of Khovanov and Seidel) and “geometrically," via Floer theory (specifically, Ozsvath-Szabo´s Heegaard Floer homology package as extended by Lipshitz-Ozsvath-Thurston). Both collections of invariants are strong enough to detect the trivial braid. I will discuss what we know so far about the connection between these invariants, focusing on the relationship between the representation theory and the Floer theory. This is joint ongoing work with Denis Auroux and Stephan Wehrli.

• Thursday March 15, 2012 at 17:30, PATCH seminar, at Penn, DRL room 4C8
GeoTop Seminar

Gerard Misiolek, Notre Dame University and IAS, Right-invariant metrics on diffeomorphism groups

I will focus on metrics of Sobolev type. As pointed out by V. Arnold, motions of an ideal fluid in a compact manifold M correspond to geodesics of a right-invariant L^2 metric on the group of volume-preserving diffeomorphisms of M. I will discuss recent results on the structure of singularities of the associated exponential map. Time permitting I will also describe the geometry of an H^1 metric on the space of densities on M and its relation to geometric statistics.

• Tuesday March 20, 2012 at 17:00, WAchman 617
GeoTop Seminar

Genevieve Walsh, Tufts University, Right-angled Coxeter groups and acute triangulations.

A triangulation of $S^2$ yields a right-angled Coxeter group whose defining graph is the one-skeleton of that triangulation. In this case, the Coxeter group is the orbifold-fundamental group of a reflection orbifold which is finitely covered by a 3-manifold. We investigate the relationship between acute triangulations of $S^2$ and the geometry of the associated right-angled Coxeter group.

This is joint work in progress with Sam Kim.

• Tuesday March 27, 2012 at 17:00, Wachman 617
GeoTop Seminar

Igor Rivin, Temple University, How many ways can you fiber a manifold?

Can a manifold fiber in more than one way? Can a group be an extension in two ways? Can we restrict the fiber and base types? I will give a quick survey of some results on these questions.

• Tuesday April 3, 2012 at 17:00, Wachman 617
GeoTop Seminar

Scott Wolpert, University of Maryland, Weil-Petersson Riemannian and symplectic geometry

We discuss the correspondence between Weil-Petersson geometry on Teichmuller space $T$ and the hyperbolic geometry of surfaces, the unions of thrice punctured spheres. A theme is that the mapping class group is the symmetry group of geometries of $T$.

• Tuesday April 17, 2012 at 17:00, Wachman 617
GeoTop Seminar

Jeff Brock, Brown University, Fat, exhausted integer homology spheres

Since Perelman's groundbreaking proof of the geometrization conjecture for three-manifolds, the possibility of exploring tighter correspondences between geometric and algebraic invariants of three-manifolds has emerged. In this talk, we address the question of how homology interacts with hyperbolic geometry in 3-dimensions, providing examples of hyperbolic integer homology spheres that have large injectivity radius on most of their volume. (Indeed such examples can be produced that arise as $(1,n)$-Dehn filling on knots in the three-sphere). Such examples fit into a conjectural framework of Bergeron, Venkatesh and others providing a counterweight to phenomena arising in the setting of arithmetic Kleinian groups. This is joint work with Nathan Dunfield.

• Friday April 20, 2012 at 15:00, PATCH seminar, at Haverford College, KINSC H108

Liam Watson, UCLA, L-Spaces and Left-Orderability

• Friday April 20, 2012 at 16:30, PATCH seminar, at Haverford College, KINSC H108

David Gay, University of Georgia, Using Morse 2-Functions to Trisect 4-manifolds

• Tuesday April 24, 2012 at 15:30, Wachman 617
GeoTop Seminar

-Note different time- Special undergrad talk:

Joel David Hamkins, City University of New York, Fun and paradox with large numbers, logic and infinity

Are there some real numbers that in principle cannot be described? What is the largest natural number that can be written or described in ordinary type on a 3x5 index card? Which is bigger, a googol-bang-plex or a googol-plex-bang? Is every natural number interesting? Is every true statement provable? Does every mathematical problem ultimately reduce to a computational procedure? Is every sentence either true or false or neither true nor false? Can one complete a task involving infinitely many steps? We will explore these and many other puzzles and paradoxes involving large numbers, logic and infinity, and along the way, learn some interesting mathematics.

• Tuesday September 4, 2012 at 15:30, Wachman 617
GeoTop Seminar

-Note different time-

Brian Rushton, Temple University, An introduction to subdivision rules and Cannon's conjecture

Hyperbolic 3-space has a useful sphere at infinity, and any group acting geometrically on it has a sphere at infinity as well. It is not known if the converse is true; this is Cannon's conjecture about Gromov hyperbolic groups with a 2-sphere at infinity. Subdivision rules were developed in an attempt to solve this conjecture. We will discuss the background of Cannon's conjecture, subdivision rules, and what it means for a subdivision rule to be conformal.

• Tuesday September 11, 2012 at 16:00, Wachman 617
GeoTop Seminar

Brian Rushton, Temple University, An introduction to subdivision rules and Cannon's conjecture (Part 2)

Hyperbolic 3-space has a useful sphere at infinity, and any group acting geometrically on it has a sphere at infinity as well. It is not known if the converse is true; this is Cannon's conjecture about Gromov hyperbolic groups with a 2-sphere at infinity. Subdivision rules were developed in an attempt to solve this conjecture. We will discuss the background of Cannon's conjecture, subdivision rules, and what it means for a subdivision rule to be conformal.

• Friday September 21, 2012 at 14:00, Wachman 527
GeoTop Seminar

PATCH seminar, joint with Bryn Mawr, Haverford, and Penn -Note different location-

Thomas Koberda, Yale University, The complex of curves for a right-angled Artin group

I will discuss an analogue of the curve complex for right-angled Artin groups and describe some of its properties. I will then show how it guides parallel results between the theory of mapping class groups and the theory of right-angled Artin groups. Joint with Sang-hyun Kim.

• Friday September 21, 2012 at 15:30, Wachman 617
GeoTop Seminar

PATCH seminar, joint with Bryn Mawr, Haverford, and Penn

Eriko Hironaka, Florida State University, Small dilatation pseudo-Anosov mapping classes

A pseudo-Anosov mapping classes on a compact finite-type oriented surface S has the property that the growth rate of lengths of an essential simple closed curve under iterations of the mapping class is exponential, and the growth rate is independent of the choice of curve and the of the choice of metric. This growth rate is called that dilatation of the mapping class. In this talk, we discuss the problem of describing small dilatation pseudo-Anosov mapping classes, i.e., those such that the dilatation raised to the topological Euler characteristic of the surface is bounded. We describe small dilatation mapping classes in terms of deformations within fibered faces, and give some explicit examples. We finish the talk with a conjecture concerning the "shape" of small dilatation mapping classes.

• Friday September 21, 2012 at 17:00, Wachman 617
GeoTop Seminar

PATCH seminar, joint with Bryn Mawr, Haverford, and Penn

Andrew Putman, Rice University, Stability in the homology of congruence subgroups

I'll discuss some recent results which uncover new patterns in the homology groups of congruence subgroups of $SL_n(\mathbb{Z})$ and related groups.

• Tuesday October 2, 2012 at 16:00, Wachman 617
GeoTop Seminar

Bill Floyd, Virginia Tech, Finite subdivision rules and rational maps

A finite subdivision rule gives an essentially combinatorial method for recursively subdividing planar complexes. The theory was developed (as part of an approach to Cannon's conjecture) as a tool for studying the recursive structure at infinity of Gromov-hyperbolic groups, but it is becoming increasingly useful for studying postcritically finite rational maps. I'll give an overview (with lots of graphic images) of some of the connections between finite subdivision rules and rational maps.

• Tuesday October 9, 2012 at 16:00, Wachman 617
GeoTop Seminar

Patricia Cahn, University of Pennsylvania, Algebras counting intersections and self-intersections of curves

Goldman and Turaev discovered a Lie bialgebra structure on the vector space generated by free homotopy classes of loops on an oriented surface. Goldman's Lie bracket gives a lower bound on the minimum number of intersection points of two loops in two given free homotopy classes. Turaev's Lie cobracket gives a lower bound on the minimum number of self-intersection points of a loop in a given free homotopy class. Chas showed that these bounds are not equalities in general. We show that for other operations, namely, the Andersen-Mattes-Reshetikhin Poisson bracket and a new operation $\mu$, the corresponding bounds are always equalities. Some of this is joint work with Vladimir Chernov.

• Tuesday October 16, 2012 at 16:00, Wachman 617
GeoTop Seminar

Stefan Friedl, Universität zu Köln, The virtual fibering theorem for 3-manifolds

In 2007, Agol showed that any irreducible 3-manifold such that its fundamental groups is 'virtually RFRS' is virtually fibered. I will give a somewhat different proof using complexities of sutured manifolds. This is joint work with Takahiro Kitayama.

• Tuesday October 23, 2012 at 14:30, Wachman 527
GeoTop Seminar

-Note different location and time-

Christian Millichap, Temple University, How many hyperbolic 3-manifolds can have the same volume?

The work of Jorgensen and Thurston shows that there is a finite number $N(v)$ of orientable hyperbolic 3-manifolds with any given volume $v$. In this talk, we construct examples showing that the number of hyperbolic knot complements with a given volume v can grow at least factorially fast with $v$. A similar statement holds for closed hyperbolic 3-manifolds, obtained via Dehn surgery. Furthermore, we give explicit estimates for lower bounds of $N(v)$ in terms of $v$ for these examples. These results improve upon the work of Hodgson and Masai, which describes examples that grow exponentially fast with $v$. Our constructions rely on performing volume preserving mutations along Conway spheres and on the classification of Montesinos knots.

• Tuesday October 23, 2012 at 16:00, Wachman 617
GeoTop Seminar

Julien Roger, Rutgers University, Ptolemy groupoids, shear coordinates and the augmented Teichmuller space

Given a punctured surface $S$, its Ptolemy groupoid is a natural object associated to ideal triangulations on the surface. The action of the mapping class group on ideal triangulations extends to a homomorphism to this groupoid. Using hyperbolic geometry, in our context shear coordinates on Teichmuller space, this can be used to construct representations of the mapping class group in terms of rational functions. This was studied first by R. Penner using the closely related $\lambda$-length coordinates.

In this talk we will describe how this construction behaves when pinching simple closed curves on $S$. This has combinatorial implications, with the construction of ideal triangulations on pinched surfaces and the effect on the Ptolemy groupoid, and geometrical, with a natural extension of shear coordinates to the augmented Teichmuller space. In both cases we explain how this applies to the action of the mapping class group. If time permits we will describe some possible applications to the study of quantum Teichmuller theory.

• Friday October 26, 2012 at 14:30, PATCH seminar, at Bryn Mawr, Park Science Building room 336
GeoTop Seminar

Emmy Murphy, MIT, Loose Legendrian knots in high dimensional contact manifolds

The goal of this talk will be to define loose Legendrian knots in high dimensions, and state their classification. No prior knowledge of contact topology will be assumed; we will start by defining and drawing pictures of Legendrian knots in high dimensions. We will then define what it means for a Legendrian to be loose, and prove some of their basic existence properties, such as their $C^0$ density and their existence in any formal isotopy class. We will then state their classification up to Legendrian isotopy, and discuss various applications of their classification to high dimensional symplectic/contact topology. Time permitting, we will contrast with the 3-dimensional setting, and present some relevant open questions.

• Friday October 26, 2012 at 16:00, PATCH seminar, at Bryn Mawr, Park Science Building room 336
GeoTop Seminar

Dave Futer, Temple University, The virtual Haken conjecture

In 1968, Friedhelm Waldhausen posed the following conjecture: every closed, aspherical 3-manifold has a finite-sheeted cover containing an incompressible surface. After more than 40 years with essentially minimal progress, this conjecture fell in Spring 2012, due to the combined efforts of Ian Agol, Jeremy Kahn, Vladimir Markovic, and Daniel Wise, plus significant input from several others.

In addition to proving Waldhausen's conjecture, their solution established several other stunning and unexpected results about 3--manifolds, particularly hyperbolic 3-manifolds. The ingredients of the proof range from ergodic theory to group theory. I will survey some of the context of the conjecture and give a top-level outline of the proof.

• Tuesday November 13, 2012 at 16:00, Wachman 617
GeoTop Seminar

Viveka Erlandsson, CUNY Graduate Center, The Margulis region in hyperbolic 4-space

Given a discrete subgroup of the isometries of n-dimensional hyperbolic space there is always a region kept precisely invariant under the stabilizer of a parabolic fixed point, called the Margulis region. In dimensions 2 and 3 this region is always a horoball, In higher dimensions this is no longer true due to the existence of screw parabolic elements. There are examples of discrete groups acting on hyperbolic 4-space containing a screw parabolic element for which there is no precisely invariant horoball. Hence the Margulis region must have some other shape. In this talk we describe the asymptotic shape of this region. If time allows we show that for a certain class of screw parabolic elements, the region is quasi-isometric to a horoball. This is joint work with Saeed Zakeri.

• Tuesday November 20, 2012 at 16:00, Wachman 617
GeoTop Seminar

Joseph Maher, CUNY College of Staten Island, Statistics for Teichmuller geodesics

We describe two ways of picking a geodesic "at random" in a space, one coming from the standard Lebesgue measure on the visual sphere, and the other coming from random walks. The spaces we're interested in are hyperbolic space and Teichmuller space, together with some discrete group action on the space. We investigate the growth rate of word length as you move along the geodesic, and we show these growth rates are different depending on how you choose the geodesic. This is joint work with Vaibhav Gadre and Giulio Tiozzi.

• Thursday November 29, 2012 at 15:30, PATCH seminar, at Penn, DRL room 4C8
GeoTop Seminar

John Pardon, Stanford University, Totally disconnected groups (not) acting on three-manifolds

Hilberts Fifth Problem asks whether every topological group which is a manifold is in fact a (smooth!) Lie group; this was solved inthe affirmative by Gleason and Montgomery-Zippin. A stronger conjectureis that a locally-compact topological group which acts faithfully on a manifold must be a Lie group. This is the Hilbert-Smith Conjecture, which in full generality is still wide open. It is known, however (as acorollary to the work of Gleason and Montgomery-Zippin) that it suffices to rule out the case of the additive group of p-adic integers acting faithfully on a manifold. I will present a solution in dimension three. The proof uses tools from low-dimensional topology, for example incompressible surfaces, minimal surfaces, and a property of the mapping class group.

• Thursday November 29, 2012 at 17:00, PATCH seminar, at Penn, DRL room 4C8
GeoTop Seminar

Larry Guth, MIT, Contraction of areas and homotopy-type of mappings

I'm going to talk about connections between the geometry of a map and its homotopy type. Suppose that we have a map from the unit $m$-sphere to the unit $n$-sphere. We say that the $k$-dilation of the map is $< L$ if each $k$-dimensional surface with $k$-dim volume $V$ is mapped to an image with $k$-dim volume at most $LV$. Informally, if the $k$-dilation of a map is less than a small $\epsilon$, it means the map strongly shrinks each $k$-dimensional surface. Our main question is: can a map with very small $k$-dilation still be homotopically non-trivial?

Here are the main results. If $k > (m+1)/2$, then there are homotopically non-trivial maps from $S^m$ to $S^{m-1}$ with arbitrarily small $k$-dilation. But if $k \leq (m+1)/2$, then every homotopically non-trivial map from $S^m$ to $S^{m-1}$ has $k$-dilation at least $c(m) > 0$.

• Tuesday December 4, 2012 at 16:00, Wachman 617
GeoTop Seminar

Richard Kent, University of Wisconsin, Geometric subgroups of mapping class groups

Farb and Mosher introduced the notion of convex cocompactness from the theory of Kleinian groups to the study of mapping class groups of surfaces. This notion bears upon questions such as Gromov's weak hyperbolization conjecture for groups and the question of the existence of hyperbolic surface bundles over surfaces. I will discuss these questions and related attempts to find purely pseudo-Anosov surface subgroups of mapping class groups.