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| 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 |
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 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 |
| 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): |
| Monday 3/1 Note location |
Joint USC/UCLA/CalTech topology seminar at UCLAAt UCLA in Room MS XXXX.Christopher Douglas (Berkeley): |
| Monday 3/8 |
Joel
Louwsma
(CalTech): |
| Monday 3/15 |
Spring break |
| Monday 3/22 |
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| Monday 3/29 |
Rachel
Roberts
(Washington
University): |
| Monday 4/5 |
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| Monday 4/12 |
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| Monday 4/19 |
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| Monday 4/26 |
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| Monday 5/3 |
| 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.
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| 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 UCLAAt 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 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 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 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 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. |