Colloquium (2017)

Speaker Megumi Harada(McMaster University,OCAMI)
Title The cohomology of abelian Hessenberg varieties and the Stanley-Stembridge conjecture
Date November 29 (Wed.) 2017, 15:15-16:15
Place Dept. of Mathematics, Faculty of Science Bldg., E408
Abstract The topic of this talk touches upon a variety of research areas including combinatorics, Lie theory, geometry, and representation theory, and I will attempt to make the talk accessible to a correspondingly wide audience, including graduate students.

The famous Stanley-Stembridge conjecture in combinatorics states that the chromatic symmetric function of the incomparability graph of a so-called (3+1)-free poset is e-positive. In this talk, we will briefly describe this conjecture, and then we will explain how recent work of Shareshian-Wachs, Brosnan-Chow, among others, makes a rather surprising connection between this conjecture and the geometry and topology of Hessenberg varieties, together with a certain symmetric-group representation on the cohomology of Hessenberg varieties. In particular, it turns out (a graded version of) the Stanley-Stembridge conjecture would follow if it can be proven that the cohomology of regular semisimple Hessenberg varieties (in Lie type A) are permutation representations of a certain form. I will then describe joint work with Martha Precup which proves this statement for the special case of abelian Hessenberg varieties, the definition of which is inspired by the theory of abelian ideals in a Lie algebra, as developed by Kostant and Peterson. Our proof relies on the incomparability graph of a Hessenberg function and previous combinatorial results of Stanley, Gasharov, and Shareshian-Wachs, as well as previous results on the geometry and combinatorics of Hessenberg varieties of Martha Precup.
Speaker Masashi Yasumoto (OCAMI)
Title Discrete zero mean curvature surfaces in Euclidean and Minkowski spaces
Date November 8 (Wed.) 2017, 16:30〜17:30
Place Dept. of Mathematics, Faculty of Science Bldg., E408
Abstract The research field "Discrete Differential Geometry" has been growing rapidly ever since previously hidden geometric aspects of integrable systems were discovered. In particular, discrete versions of differential geometric objects have been investigated from various viewpoints (differential geometry, integrable systems, discrete geometry, complex analysis and so on).

In this talk we will introduce three kinds of discrete zero mean curvature surfaces in Euclidean and Minkowski 3-spaces. The first type in the Euclidean 3-space was described by Bobenko and Pinkall, which are called discrete minimal surfaces. The other two types in Minkowski 3-space are introduced by the speaker Yasumoto, which we call discrete maximal surfaces and discrete timelike minimal surfaces. These classes initially look similar, but their global behaviors are quite different. Through their comparisons, we will briefly introduce our analysis on such discrete surfaces.
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.
Last Modified on 2017.11.13