# Geometry and Topology Seminar 2009-10

## Spring 2010

This semester, the seminar usually meets on Mondays, from 4:30 to 5:30 in KAP 245.

Monday 1/18
University holiday
Monday 1/25
Qingtao Chen (USC): Quantum invariants of links
The colored HOMFLY polynomial is a quantum invariant of oriented links in S associated with a collection of irreducible representations of each quantum group U_q(sl_N) for each component of the link. We will discuss in detail how to construct these polynomials and their general structure. Then we will discuss the new progress, Labastida-Marino-Ooguri-Vafa conjecture. LMOV also gives the application of Lichorish-Millet type formula for links. The corresponding theory of colored Kauffman polynomial and orthogonal LMOV conjecture could also be developed in a same fashion by using more complicated algebra structures.
Monday 2/1
Cancelled
Monday 2/8
Jian He (USC): Symplectic Invariants of  Subcritical Stein Manifolds
There are a wealth of symplectic invariants arising from holomorphic curve theory, however they are quite hard to compute. In the case of subcritical Stein manifolds, some of these invariants reduces to Morse theory. I will talk about how to define such invariants on Stein manifolds and their computation in the subcritical case (some of them still conjectural).
Friday 2/12
Note day

### Colloquium and Special Geometry/Topology Seminar

Rob Kirby (UC Berkeley): Broken fibrations on 4-manifolds
Theorem:  Every map from an n-manifold is homotopic to a broken fibration, and any two such are related by a simple set of moves.   The talk will explain the terms in the theorem,  give reasons why the theorem may be useful, describe its antecedents and relations to symplectic 4-manifolds, and maybe more.
Saturday 2/13
Note location and day

### Southern California Topology Colloquium

Pomona College, Claremont
Monday 2/15
University holiday
Monday 2/22
Julie Bergner (UC Riverside): Homotopy theories in topology
Homotopy theory originated in topology, with the study of topological spaces up to (weak) homotopy equivalence.  A more general notion of "homotopy theory" has been useful in a broad range of areas within algebra and algebraic geometry.  However, it is still providing new insights in topology, for example in Lurie's recent work on the Cobordism Hypothesis.  We'll look at the notion of abstract homotopy theory and some examples both classical and recent.
Monday 3/1
4:00 - 4:50
UCLA MS 6627
Note location

### Joint USC/UCLA/CalTech topology seminar at UCLA

Christopher Douglas (Berkeley): 2-dimensional algebra and 3-dimensional local field theory
Witten showed that the Jones polynomial invariants of knots can be computed in terms of partition functions of a (2+1)-dimensional topological field theory, namely the SU(2) Chern-Simons theory.  Reshetikhin and Turaev showed that this theory extends to a (1+1+1)-dimensional topological field theory---that is, there is a Chern-Simons-type invariant associated to 3-manifolds, 3-manifolds with boundary, and 3-manifolds with codimension-2 corners.
I will explain the notion of a local or (0+1+1+1)-dimensional topological field theory,  which has, in addition to the structure of a (1+1+1)-dimensional theory, invariants associated to 3-manifolds with codimension-3 corners.  I will describe a notion of 2-dimensional algebra that allows us to construct and investigate such local field theories. Along the way I will discuss the geometric classification of local field theories, and explain the dichotomy between categorification and algebraification.

Monday 3/8
Joel Louwsma (CalTech): Immersed surfaces in the modular orbifold
A hyperbolic conjugacy class in the modular group PSL(2,Z) corresponds to a closed geodesic in the modular orbifold. Some of these geodesics virtually bound immersed surfaces, and some do not; the distinction is related to the polyhedral structure in the unit ball of the stable commutator length norm. We prove the following stability theorem: for every hyperbolic element of the modular group, the product of this element with a sufficiently large power of a parabolic element is represented by a geodesic that virtually bounds an immersed surface. This is joint work with Danny Calegari.
Monday 3/15
Spring break
Monday 3/22
Helen Wong (Carleton College): Quantum invariants of random Heegaard splittings
The Witten-Reshetikhin-Turaev quantum invariants are known to give (sometimes sharp) lower bounds to the Heegaard genus of a 3-manifold. In order to study the broader effectiveness of such an approach, we consider the distribution of the complex-valued quantum invariants for a random Heegaard splitting, i.e. one associated to the result of a random walk in the mapping class group.  This is joint work with Nathan Dunfield.
Monday 3/29
Rachel Roberts (Washington University): Left-orderability and 3-manifold groups
A group G is called left-orderable if it admits a left invariant total order. When G is nontrivial and the fundamental group of an irreducible compact orientable 3-manifold, this is equivalent to the existence of a nontrivial representation of G as a group of orientation preserving homeomorphisms of R. In this talk I will review what is known about the left-orderability of 3-manifold groups pi1(M) and consider the question of what the property of left-orderability of pi1(M) implies about the topology of M. I will outline a proof revealing that left-orderability of pi1(M) does not imply that M contains an R-covered foliation. This is work joint with Sergio Fenley.
Monday 4/5
Henry Wilton (CalTech): A topological approach to conjugacy separability
A subset of a group is called separable if it is closed in the profinite topology.  Subgroup separability has a well known topological interpretation, in terms of lifting immersions to embeddings.  The study of separable conjugacy classes has, until recently, retained a more algebraic flavour.  I will describe some connections between various open questions about the separability properties of word-hyperbolic groups, and give a topological proof of the fact that conjugacy classes in surface groups are separable.  If time permits, I will explain the proof that surface groups are omnipotent (also proved by Bajpai).
Monday 4/12
Guillaume Dreyer (USC): Hitchin representations and length functions
Let S be a closed surface of negative Euler characteristic. We consider the space Rep_n(S) of homomorphisms from the fundamental group of S to SL_n(R), with n > 2. Hitchin gives a complete description of the connected components of the space Rep_n(S). A particularly interesting component is the one containing the Fuchsian representations, which is called the Hitchin component. Given a curve c on S and a representation r in the Hitchin component of Rep_n(S), we can consider the eigenvalues of r(c). We show how to extend these eigenvalue functions to length functions on the space of measured laminations on S, or more generally to the space of Holder geodesic currents. This is based on Labourie's dynamical characterization of those representations which are in the Hitchin component.
Monday 4/19
Fei Han (National University of Singapore): Generalized Witten Genus and Complete Intersections
The Witten genus is the loop space analogue of the Hirzebruch A-hat genus.  On a string manifold, the Witten genus is a modular form and is the equivariant index  of the S^1-equivariant Dirac operator on the free loop space. The homotopy refinement of the Witten genus leads to the theory of topological modular forms. Hohn and Stolz also conjectured that existence of a positive Ricci curvature metric on a string manifold implies the vanishing of the Witten genus. In this talk, I will discuss two extensions of the classical Witten genus. One concerns a generalization of string manifolds, which we call string^c manifolds and the other one is the mod 2 extension. We will present vanishing results for these generalized Witten genera on complete intersections and describe a possible mod 2 extension of the Hohn-Stolz conjecture. The talk is based on the joint work with Q.Chen and W. Zhang.
Friday
4/23
Note location and day

### Joint USC/UCLA/CalTech topology seminar at CalTech

Yanki Lekili (MSRI): Quilted Floer homology of 3-manifolds
We introduce quilted Floer homology (QFH), a new invariant of 3-manifolds equipped with a circle valued Morse function. This is a natural extension of Perutz's 4-manifold invariants, making it a (3+1) theory. We relate Perutz's theory to Heegaard Floer theory by showing an isomorphism between QFH and HF^+ for extremal spin^c structures with respect to the fibre of the Morse function. As applications, we give new calculations of Heegaard Floer theory and a characterization of sutured Floer homology.

Tao Li (Boston College):
Rank and genus of amalgamated 3-manifolds
A fundamental question in 3-manifold topology is whether the rank of the fundamental group a 3-manifold is equal to its Heegaard genus. We use hyperbolic JSJ pieces to construct closed 3-manifolds with rank smaller than genus.
Monday 4/26
Eamonn Tweedy (UCLA): The R grading on the Heegaard Floer homology of branched double-covers
Seidel and Smith constructed a link invariant called the symplectic Khovanov homology Khsymp(L), which is defined from a braid whose braid closure is a diagram for L.  Given a link L in S3, one can form a 3-manifold D(L) by taking the double cover of S3 branched over L.   Manolescu showed that the grading on Khsymp(L) induces a grading R on the Heegaard Floer hat homology of D(L), via a correspondence between Seidel and Smith's chain group generators and a set of Heegaard Floer chain group generators.  Seidel and Smith have recently proven that R is a link invariant (by proving that Manolescu's correspondence on chains actually respects differentials).  We mention here a concrete (independent) proof of the same fact, but from Manolescu's Heegaard Floer perspective.  Moreover, we show that this grading takes a simple form when L is a two-bridge link.  We'll end with some further discussion and speculation about the R invariant.
Monday 5/3

### Joint USC/UCLA/CalTech topology seminar at USC

Vera Vertesi (MSRI): Legendrian and transverse classification of twist knots.
In 1997 Chekanov gave the first example of a knot type whose Legendrian representations are not distinguishable using only the classical
invariants: the 52 knot. Epstein, Fuchs and Meyer extended his result by showing that there are at least n different Legendrian representations of the (2n+1)2 knot with maximal Thurston-Bennequin number. The aim of this talk to give a complete classification of Legendrian representations of twist knots. In particular the (2n+1)2 knot has exactly ceil(n2/2) Legendrian representations with  maximal Thurston-Bennequin number. This is a joint work (in progress) with John Etnyre and Lenhard Ng.

Daniel Krasner (Columbia and MSRI):
Graphical calculus of Soergel bimodules in Khovanov-Rozansky link homology
I will outline a graphical calculus of Soergel bimodules, developed by B. Elias and M. Khovanov, and describe how it can be used to construct an integral version of sl(n) and HOMLFYPT link homology, as well as prove functoriality of the latter.

## Fall 2009

This semester, the seminar usually meets on Mondays, from 4:30 to 5:30 in KAP 245.

Monday 8/31
No seminar this week, but see Dave Gabai's colloquium on Wednesday 9/2
Monday 9/7
University holiday
Monday 9/14
Francis Bonahon (USC): Quantum traces for quantum spaces of representations

The space of representations of surface groups into the matrix group SL(2, C) occurs in many different contexts and can be seen in several ways. The algebraic geometry point of view sees this space as an algebraic variety, whose coordinate ring is generated by trace functions. The topologists and hyperbolic geometers tend to prefer explicit coordinates, such as shear coordinates or cusp length coordinates.
Quantizations of this representation space have been introduced in the past 15 years, using the Kauffman skein algebra for the algebraic geometry framework, and quantum Teichmüller theory for the coordinate- based approach. Each point of view has its own advantages, and its own deficiencies. It was conjectured that these two quantizations were essentially equivalent, but a proof had remained elusive.
We establish a bridge between the two points of view. This is joint work with Helen Wong.

Monday 9/21
4:30 - 5:30 in KAP 414
Note room change
Jørgen Andersen (University of Aarhus): TQFT and the quantum geometry of moduli spaces

The Witten-Reshetikhin-Turaev Topological Quantum Field Theory in particular provides us with the so-called quantum representations of mapping class groups. The geometric construction of these involves geometric  quantization of moduli spaces, which produces in particular a holomorphic vector bundle over  Teichmüller space. This bundle supports a projectively ﬂat connection constructed using algebraic geometric techniques by Hitchin. We will present a differential geometric construction of this connection in a generalized setting. Further, we will discuss the relation between this construction and Toeplitz operators. We will also discuss applications of this including the asymptotic faithfulness of these quantum representations and their application in our proof that the mapping class groups do not have Kazhdan’s property T.
Monday 9/28
3:30 - 4:30
in KAP 245
Joint with algebra seminar
Christian Kassel (University of Strasbourg): Torsors in non-commutative geometry
G-torsors or principal homogeneous spaces are familiar objects in geometry. I'll present an extension of such objects to "non-commutative geometry", i.e., to the world of quantum groups or of Hopf algebras. When G is a finite group, non-commutative G-torsors are governed by a group that has both an arithmetic component and a geometric one. The arithmetic part is given by a classical Galois cohomology group; the geometric input is encoded in a (not necessarily abelian) group that takes into account all normal abelian subgroups of G of central type. Various examples will be exhibited.

Monday 10/5,
3:30 - 4:30 in KAP 414
Note room and time change
Joint seminar with CAMS Colloquium
Dmitri Krioukov (CAIDA, UC San Diego): Hyperbolic geometry of complex networks
We establish a connection between observed scale-free topology and hidden hyperbolic geometry of complex networks. The topologies of many complex networks in nature and society (biological networks, the Internet, social networks, etc.) share two common properties: (1) strong clustering, i.e., high concentration of triangular subgraphs, and (2) heterogenous (scale-free) node degree distributions, which often closely follow power laws. We show that these two common topological properties of complex networks can be explained by the existence of hidden spaces, which are: (1) metric, and (2) hyperbolic. Strong clustering in a network appears as a reflection of the triangle inequality in its hidden space, while the negative curvature of this hidden space affects the heterogeneity of the degree distribution. We also discuss implications for transport phenomena on networks, such as routing. Embedding a real scale-free network into an appropriate hyperbolic space allows for efficient geometric routing without global knowledge of the network topology, which holds a significant promise to find a variety of practical applications, such as improving the performance of Internet routing.

Friday 10/9
4:00 - 6:00
at UCLA in MS 6229
Note day and location

### Joint USC/UCLA/CalTech topology seminar at UCLA

At UCLA in Room MS 6229.

Scott Morrison (UC Berkeley): Blob homology
Blob homology is a new gadget that takes an n-manifold and an n-category with duals, and produces a graded vector space. It's a simultaneous generalization of two important constructions: the 0-th graded piece recovers the usual "skein module" invariant, and in the special case of n=1, where the manifold is the circle, blob homology reduces to Hochschild homology. I'll begin by reviewing these ideas, then give the definition of blob homology. Finally, I'll describe some of its nice formal properties, including an action of chains of diffeomorphisms generalizing the action of diffeomorphisms on a skein module, and a nice gluing formula in terms of A_\infty modules.

Yi Ni (CalTech):     Some applications of Heegaard Floer homology to Dehn surgery
In recent years, Heegaard Floer homology has become a very powerful tool for studying Dehn surgery. In this talk, we will discuss two kinds of such applications. One is to exploit the relationship between the sutured structure of the knot complement and longitudinal surgery, the other consists of some results about cosmetic surgeries.

Monday 10/12
Masaaki Suzuki (Akita University and USC): Epimorphisms between knot groups
Let G(K) denote the knot group of a knot K.  We say that K dominates K' if there exists an epimorphism from G(K) to G(K'). We determine this partial ordering of the set of prime knots with up to 11 crossings. In the latter half of the talk, we especially focus on epimorphisms between two-bridge knot groups.

Monday 10/19
No seminar this week.
Monday 10/26
Liam Watson (UCLA): Dehn surgery and Khovanov homology

Khovanov homology is a link invariant that has the Jones polynomial as graded Euler characteristic. One simple, new invariant furnished by this theory is homological width: this measures the number of diagonals supporting the Khovanov homology of a given link. This talk will discuss how to interpret homological width as an obstruction to certain exceptional surgeries on strongly invertible knots.
Monday 11/2
4:00 - 6:00
KAP 145
Note time and room

### Joint USC/UCLA/CalTech topology seminar at USC

Kefeng Liu (UCLA): Recent results on moduli spaces

Paul Melvin (Bryn Mawr College):
Degree formulas for higher order linking
The linking number of a pair of closed curves in 3-space can be expressed as the degree of a map from the 2-torus to the 2-sphere, by means of the linking integral Gauss wrote down in 1833.  In the early 1950's, John Milnor introduced a family of higher order linking numbers (the "mu-bar invariants").  In this talk I will describe a formula for Milnor's triple linking number as the "degree" of a map from the 3-torus to the 2-sphere.  This is joint work with DeTurck, Gluck, Komendarczyk, Shonkwiler and Vela-Vick.

Monday 11/9
No seminar this week.
Monday 11/16
Shicheng Wang (Peking University and CalTech): Graph manifolds have virtually positive Seifert volume
The Seifert volume of each closed non-trivial graph manifold is virtually positive. As a consequence for each closed orientable prime 3-manifold  N, the set of mapping degrees D(M,N) is finite for any 3-manifold M unless N is finitely covered by either a torus bundle, or a trivial circle bundle, or the 3-sphere. This is joint work with Pierre Derbez.
Monday 11/23
Rena Levitt (Pomona College): Biautomaticity and CAT(0) Groups
A closed, compact n-dimensional Riemannian manifold with strictly negative sectional curvatures has a contractible universal cover with unique geodesics, and a fundamental group whose word problem can be solved in linear time. Both of these properties have been generalized. The geometric properties lead to the theory of CAT(0) and nonpositively curved spaces, while the computational properties inspired the theory of automatic and biautomatic groups. This leads to the following question: are groups acting geometrically on CAT(0) spaces biautomatic? In this talk I will discuss these generalizations, and examples CAT(0) groups that are biautomatic, focusing on groups acting on CAT(0) simplicial 3-complexes.
Monday 11/30
3:30 - 5:30
Elena Pavelescu (Rice University): The self-linking number in annulus and pants open book decompositions
We construct an immersed surface for a null-homologous braid in an annulus open book decomposition. It is an extension of the so called Bennequin surface for a braid in $\R^3$. By resolving the singularities of the immersed surface, we obtain an embedded Seifert surface for the braid. We find a self-linking number formula associated to the surface, which extends Bennequin's self-linking formula for a braid in R3 with the standard structure. This is joint work with Keiko Kawamuro.

Leonid Chekhov (Russian Academy of Sciences): Twisted Yangians arising in Teichmüller theory
We generalize the combinatorial description of Riemann surfaces with holes in the Poincaré uniformization to the case where orbifold points of order 2 and 3 are present. In the case of n Z2 orbifold points on the disc (sphere with one hole) and annulus (sphere with two holes) we can close the obtained algebras of geodesic functions obtaining the An and Dn algebras. The latter algebra is the semiclassical limit of the twisted q-Yangian algebra for the orthogonal Lie algebra. We represent the mapping class group action in the both cases as an adjoint matrix action of the braid group, consider finite-dimensional geometrical reductions of the (generally infinite-dimensional) Yangian algebra, and construct central elements for these reduced cases. This is based on joint work with M. Mazzocco.
Friday 12/4
Note day and location

### Joint USC/UCLA/CalTech topology seminar at CatTech

Yi Liu (UC Berkeley): Tiny groups and the simplicial volume
A group is called tiny if it cannot map onto the fundamental group of any aspherical 3-manifold of negative Euler characteristic. For example, knot groups are tiny. In this talk we show that if a finitely presented tiny group G maps onto the fundamental group of a compact aspherical 3-manifold N, then the simplicial volume of N is bounded above in terms of G. This is joint work with Ian Agol.

Shelly Harvey (Rice University): Filtrations of the Knot Concordance Group
For each sequence of polynomials P={p1(t), ... }, we define a characteristic series of groups, called the derived series localized at P. Given a knot K, such a sequence of polynomials arises naturally as the sequence of orders of the higher-order Alexander modules of K. These new series yield filtrations of the smooth knot concordance group that refine the (n)-solvable filtration. We show that each of the successive quotients of this refined filtration contains 2-torsion and elements of infinite order.  These results generalize the p(t)-primary decomposition of the algebraic knot concordance group due to Milnor, Kervaire and Levine. This is joint work with Tim Cochran and Constance Leidy.
Monday 12/7
Zhongtao Wu (Princeton University): Cosmetic Surgery Conjecture on S3
It has been known for over 40 years that every closed connected orientable 3-manifold is obtaind by surgery on a link in S3.  However, a complete classification has remained elusive due to the lack of uniqueness of this surgery description.  In this talk, we discuss the following uniqueness theorem for Dehn surgery on a nontrivial knot in S3: Let K be a knot in S3, and let r and r' be two distinct rational numbers of same sign, allowing r to be infinite; then there is no orientation-preserving homeomorphism between the manifolds obtained by performing Dehn surgery of type r and r, respectively. In particular, this result implies the famous Knot Complement Theorem of Gordon and Luecke.
Monday
12/21
1:00 - 3:00
KAP 414
Note change in time and room
Teruaki Kitano (Soka University): Epimorphisms between knot groups and degree zero maps between knot exteriors
A partial order on the set of prime knots can be defined if there exists an epimorphism between knot groups. Horie-Matsumoto-Kitano-Suzuki determined this partial ordering of the set of prime knots with up to 11 crossings. There are some interesting examples in this list. In this talk, we take up examples of epimorphisms induced by degree zero maps.

Takayuki Morifuji (Tokyo University of Agriculture and Technology): A formula for the twisted Alexander polynomial of twist knots
In recent years, the twisted Alexander polynomial has been widely studied, and it has many applications to knot theory. In this talk, we will describe an explicit formula for the twisted Alexander polynomial of twist knots for nonabelian SL(2,C)-representations. Some properties of the polynomial will be discussed from the viewpoint of the representation space of the knot group.