Tuesday Seminar on Topology (September, 2009 -- February, 2010)
[Japanese]
[Past Programs]
16:30 -- 18:00 Graduate School of Mathematical Sciences,
The University of Tokyo
Tea: 16:00 -- 16:30 Common Room
Last updated January 20, 2010
Information :@
Toshitake Kohno
Nariya Kawazumi
September 29 -- Room 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.
October 13 -- Room 056, 16:30 -- 18:00
Hirofumi Sasahira (The University of Tokyo)
Instanton Floer homology for lens spaces
Abstract:
Let Y be an oriented closed 3-manifold and P be an SU(2)-bundle on Y. Under a certain condition, instanton Floer homology for Y can be defined as the Morse homology of the Chern-Simons functional. The condition is that all flat connections on P are irreducible. When there is a reducible flat connection on P, instanton Floer homology is not defined in general.
Since the fundamental group of a lens sapce is commutative, all flat connections on the lens space are reducible. In this talk I will introduce instanton Floer homology for lens spaces. I also show calculations for some lens spaces.
October 20 -- Room 056, 16:30 -- 18:00
Takahiko Yoshida (Meiji University)
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.
October 27 -- Room 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.
November 10 -- Room 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.
November 17 -- Room 056, 16:30 -- 18:00
Toshie Takata (Niigata University)
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.
November 24 -- Room 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.
December 1 -- Room 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.
GCOE lecture series
December 8 -- Room 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.
December 15 -- Room 056, 17:00 -- 18:00 (to be held jointly with
seminar on Lie groups and representation theory)
Toshikazu Sunada (Meiji University)
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.
December 22 -- Room 056, 16:30 -- 18:00
Tomohide Terasoma (The University of Tokyo)
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.
January 5 -- Room, 16:30 -- 17:30
Kota Hattori (The University of Tokyo)
The volume growth of hyperkaehler manifolds of type $A_{\infty}$
Abstract : Hyperkaehler manifolds of type $A_{\infty}$ were constructed due to Anderson-Kronheimer-LeBrun and Goto. These manifolds are 4-demensional, noncompact and their homology groups are infinitely generated. We focus on the volume growth of these hyperkaehler metrics. Here, the volume growth is asymptotic behavior of the volume of a ball of radius $r0$ with the center fixed. There are known examples of hyperkaehler manifolds whose volume growth is $r^4$ (ALE space) or $r^3$ (Taub-NUT space). In this talk we show that there exists a hyperkaehler manifold of type $A_{\infty}$ whose volume growth is $r^c$ for a given $3
17:30 -- 18:30
Shinichiroh Matsuo (The University of Tokyo)
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.
January 12 -- Room 056, 16:30 -- 17:30
Katsutoshi Shinohara (The University of Tokyo)
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
Masahiro Futaki (The University of Tokyo)
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.
January 19 -- R00m 056, 17:00 -- 18:00
Ryoichi Kobayashi (Nagoya University)
Localization via group action and its application to
the period condition of algebraic minimal surfaces
Abstract:
The optimal estimate for the number of exceptional
values of the Gauss map of algebraic minimal surfaces is a long
standing problem. In this lecture, I will introduce new ideas
toward the solution of this problem. The ``collective Cohn-Vossen
inequality" is the key idea. From this we have effective
Nevanlinna's lemma on logarithmic derivative for a certain class
of meromorphic functions on the disk. On the other hand, we can
construct a family holomorphic functions on the disk from the
Weierstrass data of the algebraic minimal surface under
consideration, which encodes the period condition.
Applying effective Lemma on logarithmic derivative to these
functions, we can extract an intriguing inequality.
January 26 -- Room 056, 17:00 -- 18:00
Katsuhiko Kuribayashi (Shinshu University)
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.
February 2 -- Room 056, 16:30 -- 18:00 (to be held jointly with
seminar on Lie groups and representation theory)
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
February 16 -- Room 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.
Special Lecture
February 24 (Wednesday) -- Room 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.