Tuesday Seminar on Topology

17:00 -- 18:30 Graduate School of Mathematical Sciences, The University of Tokyo
Tea: 16:30 -- 17:00 Common Room

Last updated September 9, 2016
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Toshitake Kohno
Nariya Kawazumi
Takahiro Kitayama
Takuya Sakasai

April 5 -- Room 056, 17:00 -- 18:30

Takahiro Kitayama (The University of Tokyo)

Torsion invariants and representation varieties for non-positively curved cube complexes

Abstract: Applications of torsion invariants and representation varieties have been extensively studied for 3-manifolds. Twisted Alexander polynomials are known to detect the Thurston norm and fiberedness of a 3-manifold. Ideal points of character varieties are known to detect essential surfaces in a 3-manifold in a certain extension of Culler-Shalen theory. In view of cubulation of 3-manifolds one can expect that these results naturally extend to a wider framework and, in particular, the case of virtually special cube complexes. We formulate and discuss such analogous questions for non-positively curved cube complexes.

April 12 -- Room 056, 17:00 -- 18:30

Aniceto Murillo (Universidad de Malaga)

Homotopy theory of differential graded Lie algebras

Abstract: Having as motivation the Deligne's principle by which every deformation functor is governed by a differential graded Lie algebra, we build a homotopy theory for these algebras which extend the classical Quillen approach and let us model any (non necessarily 1-connected nor path connected) complex. This is joint work with Urtzi Buijs, Yves Félix and Daniel Tanré.

April 19 -- Room 056, 17:00 -- 18:30

Błażej Szepietowski (Gdansk University)

Topological rigidity of finite cyclic group actions on compact surfaces

Abstract: Two actions of a group on a surface are called topologically equivalent if they are conjugate by a homeomorphism of the surface. I will describe a method of enumeration (and classification) of topological equivalence classes of actions of a finite group on a compact surface, based on the combinatorial theory of noneuclidean crystallographic groups (NEC groups in short) and a relationship between the outer automorphism group of an NEC group and certain mapping class group. By this method we study topological equivalence of actions of a finite cyclic group on a compact surface, in the situation where the order of the group is large relative to the genus of the surface.

April 26 -- Room 056, 17:00 -- 18:30

Jun Ueki (The University of Tokyo)

Arithmetic topology on branched covers of 3-manifolds

Abstract: The analogy between 3-dimensional topology and number theory was first pointed out by Mazur in the 1960s, and it has been studied systematically by Kapranov, Reznikov, Morishita, and others. In their analogies, for example, knots and 3-manifolds correspond to primes and number rings respectively. The study of these analogies is called arithmetic topology now.
In my talk, based on their dictionary of analogies, we study analogues of idelic class field theory, Iwasawa theory, and Galois deformation theory in the context of 3-dimensional topology, and establish various foundational analogies in arithmetic topology.

May 10 -- Room 056, 17:00 -- 18:30

Yuka Kotorii (The University of Tokyo)

On Milnor's link-homotopy invariants for handlebody-links

Abstract: A handlebody-link is a disjoint union of handlebodies embedded in $S^3$ and HL-homotopy is an equivalence relation on handlebody-links generated by self-crossing changes. A. Mizusawa and R. Nikkuni classified the set of HL-homotopy classes of 2-component handlebody-links completely using the linking numbers for handlebody-links. In this talk, by using Milnor's link-homotopy invariants, we construct an invariant for handlebody-links and give a bijection between the set of HL-homotopy classes of n-component handlebody-links with some assumption and a quotient of the action of the general linear group on a tensor product of modules. This is joint work with Atsuhiko Mizusawa at Waseda University.

May 17 -- Room 056, 17:00 -- 18:30

Hidetoshi Masai (The University of Tokyo)

Some dynamics of random walks on the mapping class groups

Abstract: The dynamics of random walks on the mapping class groups on closed surfaces of genus >1 will be discussed. We define the topological entropy of random walks. Then we prove that the drift with respect to Thurston or Teichmüller metrics and the Lyapunov exponent all coincide with the topological entropy. This is a "random version" of pseudo-Anosov dynamics observed by Thurston and I will begin this talk by recalling the work of Thurston.

May 24 -- Room 056, 17:00 -- 18:30

Kokoro Tanaka (Tokyo Gakugei University)

Independence of Roseman moves for surface-knot diagrams

Abstract: Roseman moves are seven types of local modifications for surface-knot diagrams in 3-space which generate ambient isotopies of surface-knots in 4-space. In this talk, I will discuss independence among the seven Roseman moves. In particular, I will focus on Roseman moves involving triple points and on those involving branch points. The former is joint work with Kanako Oshiro (Sophia University) and Kengo Kawamura (Osaka City University), and the latter is joint work with Masamichi Takase (Seikei University).

May 31 -- Room 056, 17:00 -- 18:30

Benoît Guerville-Ballé (Tokyo Gakugei University)

A linking invariant for algebraic curves

Abstract: We construct a topological invariant of algebraic plane curves, which is in some sense an adaptation of the linking number of knot theory. As an application, we show that this invariant distinguishes a new Zariski pair of curves (ie a pair of curves having same combinatorics, yet different topology).

June 7 -- Room 056, 17:00 -- 18:30

Kenta Hayano (Keio University)

Topology of holomorphic Lefschetz pencils on the four-torus

Abstract: In this talk, we will show that two holomorphic Lefschetz pencils on the four-torus are (smoothly) isomorphic if they have the same genus and divisibility. The proof relies on the theory of moduli spaces of polarized abelian surfaces. We will also give vanishing cycles of some holomorphic pencils of the four-torus explicitly. This is joint work with Noriyuki Hamada (The University of Tokyo).

June 14 -- Room 056, 17:00 -- 18:30

Naohiko Kasuya (Aoyama Gakuin University)

Non-Kähler complex structures on R^4

Abstract: We consider the following problem. "Is there any non-Kähler complex structure on R^{2n}?" If n=1, the answer is clearly negative. On the other hand, Calabi and Eckmann constructed non-Kähler complex structures on R^{2n} for n ≥ 3. In this talk, I will construct uncountably many non-Kähler complex structures on R^4, and give the affirmative answer to the case where n=2. For the construction, it is important to understand the genus-one achiral Lefschetz fibration S^4 → S^2 found by Yukio Matsumoto and Kenji Fukaya. This is a joint work with Antonio Jose Di Scala and Daniele Zuddas.

June 21 -- Room 056, 17:00 -- 18:30

Noboru Ito (The University of Tokyo)

Spaces of chord diagrams of spherical curves

Abstract: In this talk, the speaker introduces a framework to obtain (possibly infinitely many) new topological invariants of spherical curves under local homotopy moves (several types of Reidemeister moves). They are defined by chord diagrams, each of which is a configurations of even paired points on a circle. We see that these invariants have useful properties.

June 28 -- Room 056, 17:00 -- 18:30

Masato Mimura (Tohoku University)

Strong algebraization of fixed point properties

Abstract: Concerning proofs of fixed point properties, we succeed in removing all forms of "bounded generation" assumptions from previous celebrated strategy of "algebraization" by Y. Shalom ([Publ. IHES, 1999] and [ICM proceedings, 2006]). Our condition is stated as the existence of winning strategies for a certain "Game."

July 12 -- Room 056, 17:00 -- 18:30

John Parker (Durham University)

Non-arithmetic lattices

Abstract: In this talk I will discuss arithmetic and non-arithmetic lattices and I will give a history of the problem of finding non-arithmetic lattices. I will also briefly describe the construction of new non-arithmetic lattices in SU(2,1) found in my joint workwith Martin Deraux and Julien Paupert.

July 19 -- Room 056, 17:00 -- 18:30

Yohsuke Watanabe (University of Hawaii)

The geometry of the curve graphs and beyond

Abstract: The curve graphs are locally infinite. However, by using Masur-Minsky's tight geodesics, one could view them as locally finite graphs. Bell-Fujiwara used a special property of tight geodesics and showed that the asymptotic dimension of the curve graphs is finite. In this talk, I will introduce a new class of geodesics which also has the property. If time permits, I will explain how such geodesics can be adapted in Out(F_n) setting.