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

Last updated January 20, 2010
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929 -- 056, 16:30 -- 18:00

Sergei Duzhin (Steklov Mathematical Institute, Petersburg Division)

Symbol of the Conway polynomial and Drinfeld associator

Abstract: The Magnus expansion is a universal finite type invariant of pure braids with values in the space of horizontal chord diagrams. The Conway polynomial composed with the short circuit map from braids to knots gives rise to a series of finite type invariants of pure braids and thus factors through the Magnus map. We describe explicitly the resulting mapping from horizontal chord diagrams on 3 strands to univariante polynomials and evaluate it on the Drinfeld associator obtaining a beautiful generating function whose coefficients are integer combinations of multple zeta values.

1013 -- 056, 16:30 -- 18:00

Tj (ww@Ȋw)

YԂɑ΂CX^gFloerzW[

Abstract: Y ̂3ĺAPYSU(2)呩ƂB̂ƂŁAY ̕sϗʂłCX^g Floer zW[AChern-SimonsĊ֐MorsezW[ƂĒ邱ƂłBƂ́AP̕RڑSĊłƂƂłBȕRڑ݂Ƃ́Aiꕔ̏ꍇājĂȂB YԂ̊{Q͉łƂƂAYԏ̕Rڑ͑SĉłB̘bŁAYԂɑ΂ăCX^g Floer zW[𓱓B܂A̌vZЉB

1020 -- 056, 16:30 -- 18:00

gc F (ww@Hw)

Torus fibrations and localization of index

Abstract: I will describe a localization of index of a Dirac type operator. We make use of a structure of torus fibration, but the mechanism of the localization does not rely on any group action. In the case of Lagrangian fibration, we show that the index is described as a sum of the contributions from Bohr-Sommerfeld fibers and singular fibers. To show the localization we introduce a deformation of a Dirac type operator for a manifold equipped with a fiber bundle structure which satisfies a kind of acyclic condition. The deformation allows an interpretation as an adiabatic limit or an infinite dimensional analogue of Witten deformation.
Joint work with Hajime Fujita and Mikio Furuta.

1027 -- 002, 16:30 -- 18:00

Alex Bene (IPMU)

A new appearance of the Morita-Penner cocycle

Abstract: In this talk, I will recall the Morita-Penner cocycle on the dual fatgraph complex for a surface with one boundary component. This cocycle, when restricted to paths representing elements of the mapping class group, represents the extended first Johnson homomorphism \tau_1, thus can be viewed as a (in some specific sense canonical) "groupoid extension" of \tau_1. There are now several different contexts in which this cocycle can be constructed, and in this talk I will briefly review several of them, including one discovered in the context of finite type invariants of homology cylinders in joint work with J.E. Andersen, J-B. Meilhan, and R.C. Penner.

1110 -- 056, 16:30 -- 18:00

Alexander Getmanenko (IPMU)

Resurgent analysis of the Witten Laplacian in one dimension

Abstract: I will recall Witten's approach to the Morse theory through properties of a certain differential operator. Then I will introduce resurgent analysis -- an asymptotic method used, in particular, for studying quantum-mechanical tunneling. In conclusion I will discuss how the methods of resurgent analysis can help us "see" pseudoholomorphic discs in the eigenfunctions of the Witten Laplacian.

1117 -- 056, 16:30 -- 18:00

c qb (Vw)

On the SO(N) and Sp(N) free energy of a closed oriented 3-manifold

Abstract: We give an explicit formula of the SO(N) and Sp(N) free energy of a lens space and show that the genus g terms of it are analytic in a neighborhood at zero, where we can choose the neighborhood independently of g. Moreover, it is proved that for any closed oriented 3-manifold M and any g, the genus g terms of SO(N) and Sp(N) free energy of M coincide up to sign.

1124 -- 056, 16:30 -- 18:00

Adam Clay (University of British Columbia)

A topological approach to left orderable groups

Abstract: A group G is said to be left orderable if there is a strict total ordering of its elements such that g < h implies fg < fh for all f, g, h in G. Left orderable groups have been useful in solving many problems in topology, and now we find that topology is returning the favour: the set of all left orderings of a group is denoted by LO(G), and it admits a natural topology, under which LO(G) becomes a compact topological space. In general, the structure of the space LO(G) is not well understood, although there are surprising results in a few special cases. For example, the space of left orderings of the braid group B_n for n > 2 contains isolated points (yet it is uncountable), while the space of left orderings of the fundamental group of the Klein bottle is finite.
Twice in recent years, the space of left orderings has been used very successfully to solve difficult open problems from the field of left orderable groups, even though the connection between the topology of LO(G) and the algebraic properties of G was still unclear. I will explain the newest understanding of this connection, and highlight some potential applications of further advances.

121 -- 056, 16:30 -- 18:00

Andrei Pajitnov (Univ. de Nantes)

Non-Abelian Novikov homology

Abstract: Classical construction of S.P. Novikov associates to each circle-valued Morse map a chain complex defined over a ring of Laurent power series in one variable.
In this survey talk we shall explain several results related to the construction and properties of non-Abelian generalizations of the Novikov complex.

GCOEN[Y

128 -- 002, 17:30 -- 19:00

Giovanni Felder (ETH Zurich)

Gaudin subalgebras and stable rational curves.

Abstract: We show that Abelian subalgebras of maximal dimensions spanned by generators of the n-th Kohno-Drinfeld Lie algebra are classified by the Grothendieck-Knudsen moduli space of stable rational curves with n+1 marked points. I will explain the relation with Gaudin integrable systems of statistical mechanics and the representation theory of the symmetric group in the formulation of Vershik and Okounkov. The talk is based on joint work with Leonardo Aguirre and Alexander Veselov.

1215 -- 056, 17:00 -- 18:00 (LieQ_E\_Z~i[ƍ)

c (w)

Open Problems in Discrete Geometric Analysis

Abstract: Discrete geometric analysis is a hybrid field of several traditional disciplines: graph theory, geometry, theory of discrete groups, and probability. This field concerns solely analysis on graphs, a synonym of "1-dimensional cell complex". In this talk, I shall discuss several open problems related to the discrete Laplacian, a "protagonist" in discrete geometric analysis. Topics dealt with are 1. Ramanujan graphs, 2. Spectra of covering graphs, 3. Zeta functions of finitely generated groups.

1222 -- 056, 16:30 -- 18:00

[ FG (ww@Ȋw)

Relative DG-category, mixed elliptic motives and elliptic polylog

Abstract: We consider a full subcategory of mixed motives generated by an elliptic curve over a field, which is called the category of mixed elliptic motives. We introduce a DG Hopf algebra such that the categroy of mixed elliptic motives is equal to that of comodules over it. For the construction, we use the notion of relative DG-category with respect to GL(2). As an application, we construct an mixed elliptic motif associated to the elliptic polylog. It is a joint work with Kenichiro Kimura.

15 -- 056, 16:30 -- 17:30

L (ww@Ȋw)

$A_{\infty}$^P[[l̂̑̐ϑ

Abstract : $A_{\infty}$^P[[l̂Ƃ́AAnderson-Kronheimer-LeBrunyGotoɂč\ꂽzW[QƂȂRpNgȑl̏4P[[vʂłBł́Ǎvʂ̑̐ϑɒڂB[}l̂̑̐ϑƂ́A_𒆐SƂ锼a$r>0$̋̑̐ς̑QߋłB4̒P[[vʂł́A̐ϑ傪$r^4$ɂȂiALEԁjƁA$r^3$ɂȂiTaub-NUTԁj͂悭mĂB{uł́Ae$3 17:30 -- 18:30 MY (ww@Ȋw) On the Runge theorem for instantons Abstract: A classical theorem of Runge in complex analysis asserts that a meromorphic function on a domain in the Riemann sphere can be approximated, over compact subsets, by rational functions, that is, meromorphic functions on the Riemann sphere. This theorem can be paraphrased by saying that any solution of the Cauchy-Riemann equations on a domain in the Riemann sphere can be approximated, over compact subsets, by global solutions. In this talk we will present an analogous result in which the Cauchy-Riemann equations on Riemann surfaces are replaced by the Yang-Mills instanton equations on oriented 4-manifolds. We will also mention that the Runge theorem for instantons can be applied to develop Yang-Mills gauge theory on open 4-manifolds. 112 -- 056, 16:30 -- 17:30  (ww@Ȋw) Index problem for generically-wild homoclinic classes in dimension three Abstract : In the sphere of non-hyperbolic differentiable dynamical systems, one can construct an example of a homolinic class which does not admit any kind of dominated splittings (a weak form of hyperbolicity) in a robust way. In this talk, we discuss the index (dimension of the unstable manifold) of the periodic points inside such homoclinic classes from a$C^1$-generic viewpoint. 17:30 -- 18:30 G (ww@Ȋw) On a generalized suspension theorem for directed Fukaya categories Abstract: The directed Fukaya category$\mathrm{Fuk} W$of exact Lefschetz fibration$W : X \to \mathbb{C}$proposed by Kontsevich is a categorification of the Milnor lattice of$W$. This is defined as the directed$A_\infty$-category$\mathrm{Fuk} W = \mathrm{Fuk}^\to \mathbb{V}$generated by a distinguished basis$\mathbb{V}$of vanishing cycles. Recently Seidel has proved that this is stable under the suspension$W + u^2$as a consequence of his foundational work on the directed Fukaya category. We generalize his suspension theorem to the$W + u^d$case by considering partial tensor product$\mathrm{Fuk} W \otimes' \mathcal{A}_{d-1}$, where$\mathcal{A}_{d-1}$is the category corresponding to the$A_n\$-type quiver. This also generalizes a recent work by the author with Kazushi Ueda.

119 -- 056, 17:00 -- 18:00

(Éw)

QpɂǏƂ̑㐔IɏȖʂ̎ւ̉p

Abstract: u㐔IɏȖʂ̃KEXʑ̏Ol͍XQRHv ƂN̖ւ̐VAv[ɂĂb܂B ȋȖʂɑ΂ Cohn-Vossen s̗ގA{̈ ؂[̏Wcɑ΂čl̂łBWc Cohn-Vossen s̒莮Ƃ̏ؖ̃ACfBAqׂ܂Bؖ̍ ƂȂ̂\ɂ^ɂWۂłBWc Cohn-Vossen sueffectiveΐ̕v𓾂܂B㐔IɏȖʂ Pʉ~Տ̂鐳֐̂ӂ܂ɕ߂āA Ɂueffectiveΐ̕vKpƁAʔ] 𒊏oł܂B

126 -- 056, 17:00 -- 18:00

I F (MBw)

On the (co)chain type levels of spaces

Abstract: Avramov, Buchweitz, Iyengar and Miller have introduced the notion of the level for an object of a triangulated category. The invariant measures the number of steps to build the given object out of some fixed object with triangles. Using this notion in the derived category of modules over a (co)chain algebra, we define a new topological invariant, which is called the (co)chain type level of a space. In this talk, after explaining fundamental properties of the invariant, I describe the chain type level of the Borel construction of a homogeneous space as a computational example.
I will also relate the chain type level of a space to algebraic approximations of the L.-S. category due to Kahl and to the original L.-S. category of a map.

22 -- 056, 16:30 -- 18:00 (LieQ_E\_Z~i[ƍ)

Fanny Kassel (Univ. Paris-Sud, Orsay)

Deformation of compact quotients of homogeneous spaces

Abstract: Let G/H be a reductive homogeneous space. In all known examples, if G/H admits compact Clifford-Klein forms, then it admits "standard" ones, by uniform lattices of some reductive subgroup L of G acting properly on G/H. In order to obtain more generic Clifford-Klein forms, we prove that for L of real rank 1, if one slightly deforms in G a uniform lattice of L, then its action on G/H remains properly discontinuous. As an application, we obtain compact quotients of SO(2,2n)/U(1,n) by Zariski-dense discrete subgroups of SO(2,2n) acting properly discontinuously.
http://www.ms.u-tokyo.ac.jp/~toshi/seminar/ut-seminar2010.html#20100202kassel

216 -- 056, 17:30 -- 18:30

Dieter Kotschick (Univ. Munchen)

Characteristic numbers of algebraic varieties

Abstract: The Chern numbers of n-dimensional smooth projective varieties span a vector space whose dimension is the number of partitions of n. This vector space has many natural subspaces, some of which are quite small, for example the subspace spanned by Hirzebruch--Todd numbers, the subspace of topologically invariant combinations of Chern numbers, the subspace determined by the Hodge numbers, and the subspace of Chern numbers that can be bounded in terms of Betti numbers. I shall explain the relation between these subspaces, and characterize them in several ways. This leads in particular to the solution of a long- standing open problem originally formulated by Hirzebruch in the 1950s.

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224 () -- 370, 15:00 -- 16:30

Robert Penner (Aarhus University / University of Southern Californiaj

Protein Moduli Space

Abstract: Recent joint works with J. E. Andersen and others provide explicit discrete and continuous models of protein geometry. These models are inspired by corresponding constructions in the study of moduli spaces of flat G-connections on surfaces, in particular, for G=PSL(2,R) and G=SO(3). These models can be used for protein classification as well as for folding prediction, and computer experiments towards these ends will be discussed.

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