Tuesday Seminar on Topology (April -- September, 2012)
[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 August 3, 2012
Information :@
Toshitake Kohno
Nariya Kawazumi
April 10 -- Room 056, 16:30 -- 18:00
Takuya Sakasai (The University of Tokyo)
On homology of symplectic derivation Lie algebras of
the free associative algebra and the free Lie algebra
Abstract:
We discuss homology of symplectic derivation Lie algebras of
the free associative algebra and the free Lie algebra
with particular stress on their abelianizations (degree 1 part).
Then, by using a theorem of Kontsevich,
we give some applications to rational cohomology of the moduli spaces of
Riemann surfaces and metric graphs.
This is a joint work with Shigeyuki Morita and Masaaki Suzuki.
April 17 -- Room 002, 16:30 -- 18:00
Eriko Hironaka (Florida State University)
Pseudo-Anosov mapping classes with small dilatation
Abstract:
A mapping class is a homeomorphism of an oriented surface
to itself modulo isotopy. It is pseudo-Anosov if the lengths of essential
simple closed curves under iterations of the map have exponential growth
rate. The growth rate, an algebraic integer of degree bounded with
respect to the topology of the surface, is called the dilatation of the
mapping class. In this talk we will discuss the minimization problem
for dilatations of pseudo-Anosov mapping classes, and give two general
constructions of pseudo-Anosov mapping classes with small dilatation.
April 24 -- Room 056, 16:30 -- 18:00
Dylan Thurston (Columbia University)
Combinatorial Heegaard Floer homology
Abstract:
Heegaard Floer homology is a powerful invariant of 3- and 4-manifolds.
In 4 dimensions, Heegaard Floer homology (together with the
Seiberg-Witten and Donaldson equations, which are conjecturally
equivalent), provides essentially the only technique for
distinguishing smooth 4-manifolds. In 3 dimensions, it provides much
geometric information, like the simplest representatives of a given
homology class.
In this talk we will focus on recent progress in making Heegaard Floer
homology more computable, including a complete algorithm for computing
it for knots.
May 1 -- Room 056, 16:30 -- 18:00
Hisashi Kasuya (The University of Tokyo)
Minimal models, formality and hard Lefschetz property of
solvmanifolds with local systems
Abstract:
For a simply connected solvable Lie group G with a
cocompact discrete subgroup {\Gamma}, we consider the space of
differential forms on the solvmanifold G/{\Gamma} with values in certain
flat bundle so that this space has a structure of a differential graded
algebra(DGA). We construct Sullivan's minimal model of this DGA. This
result is an extension of Nomizu's theorem for ordinary coefficients in
the nilpotent case. By using this result, we refine Hasegawa's result of
formality of nilmanifolds and Benson-Gordon's result of hard Lefschetz
properties of nilmanifolds.
May 8 -- Room 056, 16:30 -- 18:00
Tadashi Ishibe (The University of Tokyo, JSPS)
Infinite examples of non-Garside monoids having fundamental elements
Abstract:
The Garside group, as a generalization of Artin groups,
is defined as the group of fractions of a Garside monoid.
To understand the elliptic Artin groups, which are the fundamental
groups of the complement of discriminant divisors of the semi-versal
deformation of the simply elliptic singularities E_6~, E_7~ and E_8~,
we need to consider another generalization of Artin groups.
In this talk, we will study the presentations of fundamental groups
of the complement of complexified real affine line arrangements
and consider the associated monoids.
It turns out that, in some cases, they are not Garside monoids.
Nevertheless, we will show that they satisfy the cancellation condition
and carry certain particular elements similar to the fundamental elements
in Artin monoids.
As a result, we will show that the word problem can be solved
and the center of them are determined.
May 22 -- Room 056, 17:10 -- 18:10
Hiroshi Iritani (Kyoto University)
Gamma Integral Structure in Gromov-Witten theory
Abstract:
The quantum cohomology of a symplectic
manifold undelies a certain integral local system
defined by the Gamma characteristic class.
This local system originates from the natural integral
local sysmem on the B-side under mirror symmetry.
In this talk, I will explain its relationships to the problem
of analytic continuation of Gromov-Witten theoy (potentials),
including crepant resolution conjecture, LG/CY correspondence,
modularity in higher genus theory.
May 29 -- Room 056, 16:30 -- 18:00
Inasa Nakamura (Gakushuin University, JSPS)
Triple linking numbers and triple point numbers
of torus-covering $T^2$-links
Abstract:
The triple linking number of an oriented surface link was defined as an
analogical notion of the linking number of a classical link. A
torus-covering $T^2$-link $\mathcal{S}_m(a,b)$ is a surface link in the
form of an unbranched covering over the standard torus, determined from
two commutative $m$-braids $a$ and $b$.
In this talk, we consider $\mathcal{S}_m(a,b)$ when $a$, $b$ are pure
$m$-braids ($m \geq 3$), which is a surface link with $m$-components. We
present the triple linking number of $\mathcal{S}_m(a,b)$ by using the
linking numbers of the closures of $a$ and $b$. This gives a lower bound
of the triple point number. In some cases, we can determine the triple
point numbers, each of which is a multiple of four.
June 5 -- Room 056, 16:30 -- 18:00
Yusuke Kuno (Tsuda College)
A generalization of Dehn twists
Abstract:
We introduce a generalization
of Dehn twists for loops which are not
necessarily simple loops on an oriented surface.
Our generalization is an element of a certain
enlargement of the mapping class group of the surface.
A natural question is whether a generalized Dehn twist is
in the mapping class group. We show some results related to this question.
This talk is partially based on a joint work
with Nariya Kawazumi (Univ. Tokyo).
June 12 -- Room 056, 16:30 -- 18:00
Takefumi Nosaka (RIMS, Kyoto University, JSPS)
Topological interpretation of the quandle cocycle invariants of links
Abstract:
Carter et al. introduced many quandle cocycle invariants
combinatorially constructed from link-diagrams. For connected quandles of
finite order, we give a topological meaning of the invariants, without
some torsion parts. Precisely, this invariant equals a sum of "knot
colouring polynomial" and of a Z-equivariant part of the Dijkgraaf-Witten
invariant. Moreover, our approach involves applications to compute "good"
torsion subgroups of the 3-rd quandle homologies and the 2-nd homotopy
groups of rack spaces.
June 19 -- Room 056, 17:10 -- 18:10
Yukio Matsumoto (Gakushuin University)
On the universal degenerating family of Riemann surfaces
over the D-M compactification of moduli space
Abstract:
It is usually understood that over the Deligne-
Mumford compactification of moduli space of Riemann surfaces of
genus > 1, there is a family of stable curves. However, if one tries to
construct this family precisely, he/she must first take a disjoint union
of various types of smooth families of stable curves, and then divide
them by their automorphisms to paste them together. In this talk we will
show that once the smooth families are divided, the resulting quotient
family contains not only stable curves but virtually all types of
degeneration of Riemann surfaces, becoming a kind of universal
degenerating family of Riemann surfaces.
July 10 -- Room 056, 16:30 -- 18:00
Marcus Werner (Kavli IPMU)
Topology in Gravitational Lensing
Abstract:
General relativity implies that light is deflected by masses
due to the curvature of spacetime. The ensuing gravitational
lensing effect is an important tool in modern astronomy, and
topology plays a significant role in its properties. In this
talk, I will review topological aspects of gravitational lensing
theory: the connection of image numbers with Morse theory; the
interpretation of certain invariant sums of the signed image
magnification in terms of Lefschetz fixed point theory; and,
finally, a new partially topological perspective on gravitational
light deflection that emerges from the concept of optical geometry
and applications of the Gauss-Bonnet theorem.
July 17 -- Room 056, 16:30 -- 18:00
Mutsuo Oka (Tokyo University of Science)
Contact structure of mixed links
Abstract:
A strongly non-degenerate mixed function has a Milnor open book
structures on a sufficiently small sphere. We introduce the notion of
a holomorphic-like mixed function
and we will show that a link defined by such a mixed function has a
canonical contact structure.
Then we will show that this contact structure for a certain
holomorphic-like mixed function
is carried by the Milnor open book.
July 24 -- Room 056, 16:30 -- 18:00
Greg McShane (Institut Fourier, Grenoble)
Orthospectra and identities
Abstract:
The orthospectra of a hyperbolic manifold with geodesic
boundary consists of the lengths of all geodesics perpendicular to the
boundary.
We discuss the properties of the orthospectra, asymptotics, multiplicity
and identities due to Basmajian, Bridgeman and Calegari. We will give
a proof that the identities of Bridgeman and Calegari are the same.
September 4 -- Room 056, 17:00 -- 18:00
Piotr Nowak (the Institute of Mathematics, Polish Academy of Sciences)
Poincaré inequalities, rigid groups and applications
Abstract:
Kazhdanfs property (T) for a group G can be expressed as a
fixed point property for affine isometric actions of G on a Hilbert
space. This definition generalizes naturally to other normed spaces. In
this talk we will focus on the spectral (aka geometric) method for
proving property (T), based on the work of Garland and studied earlier
by Pansu, Zuk, Ballmann-Swiatkowski, Dymara-Januszkiewicz
(glambda_1>1/2h conditions) and we generalize it to to the setting of
all reflexive Banach spaces.
As applications we will show estimates of the conformal dimension of the
boundary of random hyperbolic groups in the Gromov density model and
present progress on Shalomfs conjecture on vanishing of 1-cohomology
with coefficients in uniformly bounded representations on Hilbert spaces.