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, LabastidaMarinoOoguriVafa conjecture. LMOV also gives the application of LichorishMillet 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 4manifolds 
Saturday 2/13 Note location and day 
Southern California Topology ColloquiumPomona College, ClaremontSee http://pzacad.pitzer.edu/math/topologySeminar/SCTC2010.html 
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 UCLAChristopher Douglas (Berkeley): 2dimensional algebra and 3dimensional local field theoryWitten 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) ChernSimons theory. Reshetikhin and Turaev showed that this theory extends to a (1+1+1)dimensional topological field theorythat is, there is a ChernSimonstype invariant associated to 3manifolds, 3manifolds with boundary, and 3manifolds with codimension2 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 3manifolds with codimension3 corners. I will describe a notion of 2dimensional 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 WittenReshetikhinTuraev quantum invariants are known to give (sometimes sharp) lower bounds to the Heegaard genus of a 3manifold. In order to study the broader effectiveness of such an approach, we consider the distribution of the complexvalued 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): Leftorderability
and
3manifold groups A group G is called leftorderable if it admits a left invariant total order. When G is nontrivial and the fundamental group of an irreducible compact orientable 3manifold, 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 leftorderability of 3manifold groups pi_{1}(M) and consider the question of what the property of leftorderability of pi_{1}(M) implies about the topology of M. I will outline a proof revealing that leftorderability of pi_{1}(M) does not imply that M contains an Rcovered 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 wordhyperbolic 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 Ahat genus. On a string manifold, the Witten genus is a modular form and is the equivariant index of the S^1equivariant 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 HohnStolz 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 CalTechYanki Lekili (MSRI): Quilted Floer homology of 3manifoldsWe introduce quilted Floer homology (QFH), a new invariant of 3manifolds equipped with a circle valued Morse function. This is a natural extension of Perutz's 4manifold 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 3manifolds A fundamental question in 3manifold topology is whether the rank of the fundamental group a 3manifold is equal to its Heegaard genus. We use hyperbolic JSJ pieces to construct closed 3manifolds with rank smaller than genus. 
Monday 4/26 
Eamonn
Tweedy
(UCLA): The R grading
on the Heegaard Floer homology of branched doublecovers Seidel and Smith constructed a link invariant called the symplectic Khovanov homology Kh_{symp}(L), which is defined from a braid whose braid closure is a diagram for L. Given a link L in S^{3}, one can form a 3manifold D(L) by taking the double cover of S^{3} branched over L. Manolescu showed that the grading on Kh_{symp}(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 twobridge link. We'll end with some further discussion and speculation about the R invariant. 
Monday 5/3 
Joint USC/UCLA/CalTech topology seminar at USCVera 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 5_{2} 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 ThurstonBennequin 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(n^{2}/2) Legendrian representations with maximal ThurstonBennequin 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 KhovanovRozansky 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. 
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
WittenReshetikhinTuraev Topological Quantum Field Theory in
particular provides us with the socalled 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 noncommutative geometry Gtorsors or principal homogeneous spaces are familiar objects in geometry. I'll present an extension of such objects to "noncommutative geometry", i.e., to the world of quantum groups or of Hopf algebras. When G is a finite group, noncommutative Gtorsors 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 scalefree 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 (scalefree) 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 scalefree 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 UCLAAt UCLA in Room MS 6229.Scott Morrison (UC Berkeley): Blob homology Blob homology is a new gadget that takes an nmanifold and an ncategory with duals, and produces a graded vector space. It's a simultaneous generalization of two important constructions: the 0th 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 twobridge 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 USCKefeng Liu (UCLA): Recent results on moduli spacesPaul Melvin (Bryn Mawr College): Degree formulas for higher order linking The linking number of a pair of closed curves in 3space can be expressed as the degree of a map from the 2torus to the 2sphere, 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 "mubar invariants"). In this talk I will describe a formula for Milnor's triple linking number as the "degree" of a map from the 3torus to the 2sphere. This is joint work with DeTurck, Gluck, Komendarczyk, Shonkwiler and VelaVick. 
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 nontrivial graph manifold is virtually positive. As a consequence for each closed orientable prime 3manifold N, the set of mapping degrees D(M,N) is finite for any 3manifold M unless N is finitely covered by either a torus bundle, or a trivial circle bundle, or the 3sphere. This is joint work with Pierre Derbez. 
Monday 11/23 
Rena
Levitt
(Pomona
College): Biautomaticity and CAT(0) Groups A closed, compact ndimensional 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 3complexes. 
Monday 11/30 3:30  5:30 
Elena
Pavelescu
(Rice
University): The selflinking number in annulus
and pants open book decompositions We construct an immersed surface for a nullhomologous 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 selflinking number formula associated to the surface, which extends Bennequin's selflinking formula for a braid in R^{3} 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 Z_{2} 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 A_{n} and D_{n} algebras. The latter algebra is the semiclassical limit of the twisted qYangian 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 finitedimensional geometrical reductions of the (generally infinitedimensional) 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 CatTechYi 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 3manifold 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 3manifold 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 higherorder 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 2torsion 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 S^{3} It has been known for over 40 years that every closed connected orientable 3manifold is obtaind by surgery on a link in S^{3}. 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 S^{3}: Let K be a knot in S^{3}, and let r and r' be two distinct rational numbers of same sign, allowing r to be infinite; then there is no orientationpreserving 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. HorieMatsumotoKitanoSuzuki 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. 