SAKASAI, Takuya

Title
Associate Professor
Field
Topology
Research interests
Mapping class groups of surfaces, characteristic classes of bundles, 3-dimensional topology
Current research

The mapping class group of a surface is defined as the group of isotopy classes of self-diffeomorphisms of the surface. This group appears in various fields of mathematics such as topology, geometry, algebra, complex analysis and mathematical physics and plays an important role there. My central research interest is the mapping class group from a view point of topology. More specifically, I am recently studying the following topics:

  1. The cohomology rings of the mapping class group and its subgroups can be regarded as the set of characteristic classes of surface bundles. They also give topological information on the moduli space of Riemann surfaces. By using structures of these groups, I would like to describe those cohomology rings in terms of representation theory. Recently I am investigating closely related infinite dimensional Lie algebras by theoretical consideration and computer experiments.
  2. We might expect that the mapping class group and its enlargement called the homology cobordism group provide a method for classifying 3-dimensional manifolds in a systematic way. As for the latter group, the structure is not so well understood. I intend to construct new invariants of knots and 3-dimensional manifolds from structures of these groups and noncommutative algebras arising from fundamental groups of 3-dimensional manifolds.
Selected publications
  1. Lagrangian mapping class groups from a group homological point of view. Algebraic & Geometric Topology 12 (2012), 267-291.
  2. A survey of Magnus representations for mapping class groups and homology cobordisms of surfaces. Handbook of Teichmüller theory volume III (2012), 531-594.
  3. (With Shigeyuki Morita and Masaaki Suzuki) Abelianizations of derivation Lie algebras of the free associative algebra and the free Lie algebra. Duke Mathematical Journal 162, (2013), 965-1002.
  4. The Magnus representation and homology cobordism groups of homology cylinders, Journal of Mathematical Sciences, the University of Tokyo 22, (2015), 741-770.
  5. (With Shigeyuki Morita and Masaaki Suzuki) Structure of symplectic invariant Lie subalgebras of symplectic derivation Lie algebras. Advances in Mathematics 282, (2015), 291-334.
Memberships and activities

The Mathematical Society of Japan
Awards
 2007 Takebe prize (Encouragement Prize), The Mathematical Society of Japan