[English]
16:30 -- 18:00 Ȋwȓ(wLpX)
Tea: 16:00 -- 16:30 R[

Last updated January 17, 2008
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109 -- 056, 16:30 -- 18:00

K (sww@w)

Classification of codimension-one locally free actions of the affine group of the real line.

Abstract: By GA, we denote the group of affine and orientation-preserving transformations of the real line. In this talk, I will report on classification of locally free action of GA on closed three manifolds, which I obtained recently. In 1979, E.Ghys proved that if such an action preserves a volume, then it is smoothly conjugate to a homogeneous action. However, it was unknown whether non-homogeneous action exists. As a consequence of the classification, we will see that the unit tangent bundle of a closed surface of higher genus admits a finite-parameter family of non-homogeneous actions.

1016 -- 056, 17:00 -- 18:00

l (HƑww@Hw)

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1023 -- 002, 16:30 -- 18:00

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1030 -- 056, 17:00 -- 18:00

c [j (Éww@Ȋw)

$L_{\infty}$ action on Lagrangian filtered $A_{\infty}$ algebras.

Abstract: I will discuss $L_{\infty}$ actions on Lagrangian filtered $A_{\infty}$ algebras@by cycles of the ambient symplectic manifold. In the course of the construction, I like to remark that the stable map compactification is not sufficient in some case when we consider ones from genus zero bordered Riemann surface. Also, if I have time, I like to discuss some relation to (absolute) Gromov-Witten invariant and other applications.
(This talk is based on my joint work with K.Fukaya, Y-G Oh and K. Ono.)

116 -- 056, 16:30 -- 18:00

(ww@Ȋw)

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1120 -- 056, 16:30 -- 18:00

a (HƑww@Hw)

A certain slice of the character variety of a knot group and the knot contact homology

AbstractF For a knot $K$ in 3-sphere, we can consider representations of the knot group $G_K$ into $SL(2,\mathbb{C})$. Their characters construct an algebraic set. This is so-called the $SL(2,\mathbb{C})$-character variety of $G_K$ and denoted by $X(G_K)$.
In this talk, we introduce a slice (a subset) $S_0(K)$ of $X(G_K)$. In fact, this slice is closely related to the A-polynomial and the abelian knot contact homology. For example, the A-polynomial $A_K(m,l)$ of a knot $K$ is a two-variable polynomial knot invariant defined by using the character variety $X(G_K)$. Then we can show that for any {\it small knot} $K$, the number of irreducible components of $S_0(K)$ gives an upper bound of the maximal degree of the A-polynomial $A_K(m,l)$ in terms of the variable $l$. Moreover, for any 2-bridge knot $K$, we can show that the coordinate ring of $S_0(K)$ is exactly the degree 0 abelian knot contact homology $HC_0^{ab}(K)$.
We will mainly explain these facts.

1127 -- 056, 16:30 -- 18:00

Έ (sw͌)

A quandle cocycle invariant for handlebody-links

[joint work with Masahide Iwakiri (Osaka City University)]

Abstract: A handlebody-link is a disjoint union of circles and a finite trivalent graph embedded in a closed 3-manifold. We consider it up to isotopies and IH-moves. Then it represents an ambient isotopy class of handlebodies embedded in the closed 3-manifold. In this talk, I explain how a quandle cocycle invariant is defined for handlebody-links.

124 -- 056, 16:30 -- 18:00

G (ww@Ȋw)

Morse theory for abelian hyperkahler quotients

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1211 -- 056

16:30 -- 17:30
Xavier Gómez-Mont (CIMAT, Mexico)

A Singular Version of The Poincaré-Hopf Theorem

Abstract: The Poincaré-Hopf Theorem asserts that the Euler Characteristic of a compact manifold is the sum of the indices of any vector field on it with isolated singularities. A hypersurface in real or complex number space may be considered as the limit of the smooth hypersurfaces obtained from nearby regular values. The singularity contains ghiddenh topology, which is unfolded by a smooth regeneration. At the singularity one has an algebraic invariant, the Jacobi Algebra, which is obtained by considering analytic functions modulo the partial derivatives. It contains topological information of the singularity.

One may consider vector fields tangent to a hypersurface with isolated singularities, and define topologically an index, which coincides with the sum of the Poincaré-Hopf indices of a regeneration of it tangent to a nearby smooth hypersurface.

I will explain how to compute the index of a vector field X tangent to an isolated hypersurface singularity V using Homological Algebra, as the Euler Characteristic of the homology of the complex obtained by contracting differential forms on V with the vector field X. The formula contains several terms, but the higher order terms may be translated from the invariants of the singular point to invariants in the Jacobi Algebra, making this translation a local version of the Poincaré-Hopf Theorem.

I will also explain how some of these ideas can be extended to complete intersections.

17:40 -- 18:40
Miguel A. Xicotencatl (CINVESTAV, Mexico)

Chen Ruan cohomology of cotangent orbifolds and Chas-Sullivan string topology

(Joint with: A. Gonzalez, E. Lupercio, C. Segovia, and B. Uribe)
Abstract: At the end of 90's, two theories of topology were invented roughly at the same time and attracted considerable interest in the mathematical community. One is the Chas-Sullivan's loop product on the homology of loop space and the second one is Chen-Ruan's stringy cohomology of orbifold. It was an observation of Chen that inertia orbifold (which carries Chen-Ruan cohomology) is the space of constant loops of an orbifold. Therefore, two theories should interact. In this work we show that for an interesting family of orbifolds, the virtual orbifold cohomology, turns out to be a subalgebra of the homology of the loop orbifold, and is isomorphic, as algebras, to the Chen-Ruan orbifold cohomology of its cotangent orbifold.

1218 -- 056, 16:30 -- 18:00

R.C. Penner (USC and Aarhus University)

Groupoid lifts of representations of mapping classes

Abstract: The "Ptolemy groupoid" is the fundamental path groupoid of the dual to the ideal cell decomposition of the decorated Teichmueller space of a punctured or bordered surface, and the "Torelli groupoid" is thesimilar discretization of the fundamental path groupoid of the quotient by the Torelli subgroup of mapping classes that acts identically on the first integral homology of the surface. Mapping classes can be represented as appropriate elements of the Ptolemy groupoid and likewise for elements of the Torelli group in the Torelli groupoid.

A natural series of questions is to wonder which representations of mapping class groups, Torelli groups, and their subgroups can be lifted to the groupoid level. In a series of joint works with J. Andersen, A. Bene, N. Kawazumi, and S. Morita, we have given explicit lifts of a number of classical representations: The Johnson representations of the classical and higher Torelli groups and the symplectic representation of the mapping class group all lift to the Torelli groupoid. Furthermore, the Nielsen representation of the mapping class group as automorphisms of a free group lifts to the Ptolemy groupoid, and hence so too does any representation of the mapping class group that factors through its action on the fundamental group of the surface such as the Magnus representation. We shall survey these various groupoid lifts and discuss current and potential future applications.

115 -- 056, 16:30 -- 17:30

ѓc C (ww@Ȋw)

Adiabatic limits of eta-invariants and the Meyer functions

Abstract: The Meyer function is the function defined on the hyperelliptic mapping class group, which gives a signature formula for surface bundles over surfaces. In this talk, we introduce certain generalizations of the Meyer function by using eta-invariants and we discuss the uniqueness of this function and compute the values for Dehn twists.

129 -- 056, 16:30 -- 17:30

c \ (ww@Ȋw)

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17:30 -- 18:30

ؑ Nl (ww@Ȋw)

A Diagrammatic Construction of Third Homology Classes of Knot Quandles

Abstract:There exists a family of third (quandle / rack) homology classes, called the shadow (fundamental / diagram) classes, of the knot quandle, which are obtained from the shadow colourings of knot diagrams. We will show the construction of these homology classes, and also show their relation to the shadow quandle cocycle invariants of knots and that to other third homology classes.

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