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

Last updated November 29, 2004
bW@
͖r
͐

1012 -- 056, 16:30 -- 18:00

Bob Penner (University of Southern California, Los Angeles CA)

Arc complexes

Complexes of arcs, or equivalently of suitable graphs, arise in several related contexts in the study of Riemann's moduli space. For instance, there is a cell decomposition of the product of moduli space with an open simplex, where cells are indexed by appropriate families of arcs in the underlying surface. Furthermore, the specification of an ideal triangulation determines global coordinates on the decorated'' Teichm\"uller space of the underlying surface, and these coordinates enjoy useful transformation and other properties. In this, the first of several lectures on these topics, we shall survey these and other aspects of the utility of arc or graph complexes in the study of moduli space.

g|W[ΗjZ~i[ɈāCȉ̓ŁCPennerɂC ڂeƂɊ֘AbɂĂ̍u\肳Ă܂B
@@POPRijPTFOO--PUFRO@@PQQ
@@POPTijPTFOO--PUFRO@@PQQ

1019 -- 056, 16:30 -- 18:00

c j (ww@ȊwȁCCOE)

The colored Jones polynomial of satellite links

Skein theory p Masbaum Vogel ɂ colored Jones polynomial ̌vZ@ЉCsatellite link ɑ΂vZ瓾 mutation Ɋւʂ^Cɑ̐ϗ\zƂ̊ւɂĐ܂B

21ICOEvOu

g|W[ΗjZ~i[ɂāA ww@Ȋw21ICOEvOuL̂悤ɊJÂ܂B

112 -- Ȋw 056, 16:30  18:00

Representing open 3-manifolds as 3-fold branched covering spaces and some examples

abstract

11811@Wul̂̃g|W[̖ցv

1116 -- 122, 17:00 -- 18:30

㓡 i (ww@w)

Unobstructed deformations of topological calibrations iʑIȃLu[V̔QIȕόɂ)

abstract: In this talk, I shall focus on geometric structures defined by closed differential forms and develop a systematic approach of the deformation problem of these structures. Introducing suitable cohomology groups, I establish a criterion of unobstructed deformations from the cohomological point of view.
Then as an application, I obtain a unified construction of moduli spaces of Calabi-Yau, HyperKaehler, G_2 and Spin(7) structures.
I also apply this approach to certain geometric structures on complex solvable manifolds, which are quotients of solvable Lie groups by discrete subgroups.
Further (if I have enough time), I discuss the problem as to when geometric structures with singularities can be deformed to be smooth ones. I show that the vanishing of certain cohomology classes with compact support is crucial to the smoothing problem.
The first part of this talk will be based on the paper Moduli spaces of topological calibrations, International J. Math. 15 (2004), 211-257

1130 -- 126, 16:30  18:00

Alexander Stoimenow (ww@ȊwȁC JSPS)

Properties of closed 3-braids

Abstract: Work of Bennequin, and later Birman-Menasco, on braided Seifert surfaces allowed to identify from 3-braid representations their closure links. However, many properties remain non-trivial to decide. Xu gave an algorithm to determine the genus. I will use her algorithm to describe the skein, Jones, and Alexander polynomial of 3-braid links and to determine which links are fibered. The study of Jones polynomials allows to prove that no non-trivial 3-braid links have trivial polynomial. Finally, depending on time, I will discuss some applications of the unitarization of the Burau representation, that concern also braids on more strings.

127 -- 056, 16:30  18:00

Gregor Masbaum (Univ. Paris VII)

Integral lattices in TQFT

Abstract: We will describe joint work with Pat Gilmer where we find explicit bases for naturally defined lattices in the vector spaces associated to surfaces by the SO(3) TQFT at an odd prime. These lattices form an "Integral TQFT" in an appropriate sense. Some applications relating quantum invariants to classical 3-manifold topology will be given.

1214 -- 056, 16:30  18:00

D. Kotschick (Ludwig-Maximillian-Univ. Munchen)

Asymptotic volumes, entropies, and some applications

Abstract: The first part of the lecture will survey the definitions and basic properties of certain topological invariants of manifolds that arise naturally from asymptotic considerations in geometry and dynamics. I will explain a chain of inequalities between such invariants which centers on the asymptotic volume, or minimal volume entropy. In the second part of the lecture we shall discuss a couple of very specific geometric problems in which the asymptotic volume and its upper and lower bounds have played a role recently.

1221 -- 126, 16:30  18:00

Jozef H. Przytycki (The George Washington University)

Khovanov Homology: categorification of the Kauffman bracket relation

Abstract: We define Khovanov homology, $H_{i,j,k}$ for links in products of surfaces and an interval and in twisted $I$-bundles over unorientable surfaces (excluding $RP^2$). We show how to stratify this homology so that (in the product case, $F\times I$) it categorifies the Kauffman bracket skein module (KBSM) of $F\times I$. That is, for any link $L$ in $F\times I$ we can recover coefficients of $L$ in the standard basis $B(F)$ of the KBSM of $F\times I$. In other words if $L = \sum_b a_b(A) b$ where the sum is taken over all basic elements, $b\in B(F)$, then each coefficient $a_b(A)$ can be recovered from polynomial Euler characteristics of the stratified Khovanov homology. In the case of unorientable F we are able to recover coefficients $a_b(A)$ only partially. We propose another basis of the KBSM for which categorification seems to be possible even for unorientable F (we use cores of M\"obius bands several times even if they intersect one another).

118 -- 056, 16:30  18:30

y Lk (ww@Ȋw)

Hyperplane arrangements and the determinant of a period Matrix

abstract: We study multivariable hypergeometric integrals of type $\int_{\Delta}Udx_1\cdots dx_n$, $U=\exp{(-\frac{1}{2} (x_1^2+\cdots+x_n^2))}f_1^{\lambda_1}\cdots f_m^{\lambda_m},$ where $\lambda_i(1\le i \le m)$ is a complex number and $f_i(1\le i \le m)$ is degree one polynomial defined in an $n$ dimensional vector space. The function $U$ define a local system $L$, and we give a twisted homology group for $L$. Aomoto conjectured on the base of the twisted cohomology for $L.$

q (ww@Ȋw)

Stratification of the complement of hyperplane arrangement and homology of local systems

abstract: Through some stratifications of $M(/mathcal A)$, the complementary space of hyperplane arrangement, two kinds of finite dimensional complexes are constructed. They can be used to compute $H_* ^{lf}(M(/mathcal A),/mathcal L )$, the locally finite homology groups of $M(/mathcal A)$ with coefficients in a rank 1 local system $/mathcal L$. In addition, some description of the canonical map from $H_* (M(/mathcal A),/mathcal L )$ to $H_* ^{lf}(M(/mathcal A),/mathcal L )$ are given. As an application of above results, we study the homological representations of the braid groups. The noncommutative polynomial rings are introduced to describe those representations.

ߋ̃vO

To Seminar Information