## Colloquium (2017)

 Speaker Hideo Kozono (Waseda University) Title Method of Besov spaces and the Navier-Stokes equations Date October 25 (Wed.) 2017, 16:30〜17:30 Place Dept. of Mathematics, Faculty of Science Bldg., E408 Abstract We first introduce several basic notions of the Basov spaces such as paraproduct formula and the chain rule. The bilinear estimates related to the nonlinear structure on the Navier-Stokes equations and the $L^p-L^q$-type estimates of the Stokes semigroup are established.　 Then the problem on existence, uniqueness and regularity of the Navier-Stokes equations is discussed in the scaling invariant homogeneous Besov space. This is based on the joint work with Prof. Senjo Shimizu at Kyoto University.
 Speaker Noriyuki Abe (Hokkaido University) Title Homological structures of representations Date October 18 (Wed.) 2017, 16:30〜17:30 Place Dept. of Mathematics, Faculty of Science Bldg., E408 Abstract A linear action of a group (or a Lie algebra) is called a representation. To study representations, we have two steps: first we classify irreducible representations which are building blocks of representations and next we consider how a given representation is built up. For answering the second question, sometimes "grading" structure is useful. I will explain this mechanism and give examples.
 Speaker Yasuhiro Nakagawa (Saga University) Title On the exsitence problems for Kähler-Ricci solitons on certain toric bundles Date July 12 (Wed.) 2017, 16:30〜17:30 Place Dept. of Mathematics, Faculty of Science Bldg., E408 Abstract On the certain toric bundles, fiber spaces over Einstein-Kähler Fano manifolds whose fibers are toric Fano manifolds satisfying some conditions, we consider the existence problems for Einstein-Kähler metrics and Kähler-Ricci solitons, and propose some conjectures. We shall explain some examples on which these conjectures holds.
 Speaker Hideo Takioka (OCAMI) Title Infinitely many knots with the trivial $(2,1)$-cable $\Gamma$-polynomial Date June 28 (Wed.) 2017, 16:30〜17:30 Place Dept. of Mathematics, Faculty of Science Bldg., E408 Abstract It is known that there exist many polynomial invariants for knots. For example, Alexander-Conway, Jones, $\Gamma$, $Q$, HOMFLYPT, Kauffman polynomials are well known. These polynomials of the trivial knot are one. The problem is whether there exists a non-trivial knot such that these polynomials are one. It is known that there exists such a knot for the Alexander-Conway, $\Gamma$, $Q$ polynomials. However, it is still an open problem for the other polynomial invariants. Moreover, we consider the $(p,1)$-cable versions of these polynomial invariants for an integer $p(\geq 2)$. These $(p,1)$-cable versions of the trivial knot are one. The problem is whether there exists a non-trivial knot such that these $(p,1)$-cable versions are one. It is known that there exists such a knot for the Alexander-Conway polynomial. However, it is still an open problem for the other polynomial invariants. In this talk, we show that there exist infinitely many knots such that the $(2,1)$-cable version of the $\Gamma$-polynomial for the knots is one.
 Speaker Junjiro Noguchi (U.T./T.I.T. Emeritus) Title Weak Coherence Theorem and Levi's Problem Date May 24 (Wed.) 2017, 16:30〜17:30 Place Dept. of Mathematics, Faculty of Science Bldg., E408 Abstract pdf file
 Speaker Ken Abe (Osaka City University) Title Analysis of the Navier-Stokes equations in a space of bounded functions Date April 19 (Wed.) 2017, 16:30〜17:30 Place Dept. of Mathematics, Faculty of Science Bldg., E408 Abstract The initial-boundary value problem of the Navier-Stokes equations has been studied in a large literature in spaces of integrable functions, while few results are known in a space of bounded functions on which singular integral operators may not act as a bounded operator. In this talk, we introduce some local existence theorem in a space of bounded functions for domains such as bounded or exterior domains. This in particular implies that a minimum rate of potential singularities is type I even in the presence of boundaries.