Colloquium（2012）

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2013　　

Speaker:Jun KATO (Nagoya University)

Title: クライン・ゴルドン方程式に対するストリッカーツ型評価とその応用

Date: February 20 (Wed.) 2013, 16：30〜17：30

Place: Dept. of Mathematics, General Research. Bldg., 301

Abstract:

Speaker:David Kalaj (University of Montenegro)

Title: On Quasi-inversions with respect to the boundaries of the starlike domains ( a joint work with Gendi Wang and Matti Vuorinen )

Date: January 16 (Wed.) 2013, 16：30〜17：30

Place: Dept. of Mathematics, General Research. Bldg., 301

Abstract: Given a bounded domain $D \subset {\mathbb R}^n$ strictly starlike with respect to $0 \in D\,,$ we define a quasi-inversion w.r.t. the boundary $\partial D \,.$ We show that the quasi-inversion is bi-Lipschitz w.r.t. the absolute ratio metric, Ferrand's metric and the chordal metric with the constants depending on the geometric structure of $\partial D$. Moreover, all the bi-Lipschitz constants tend to 1, when $\partial D$ approaches to the unit sphere in a suitable way. In order to do so we use the concept of $\alpha$-tangent condition on a domain which has been introduced in the classical paper of F. W. Gehring and J. V\"ais\"al\"a (Acta Math. 1965). This condition is shown to be equivalent to bi-Lipschitz and quasiconformal extension of polar parametrization of $\partial D$. In addition we show that the polar parametrization which is a mapping of the unit sphere onto $\partial D$ is bi-Lipschitz if and only if $D$ satisfies $\alpha$-tangent condition. We also prove that the global bi-Lipschitz constant is equal to the supremum of local Lipschitz constants.

Speaker:Yohei KOMORI (Waseda University)

Title: トーラス上の種数２のリーマン面の退化族について

Date: November 14 (Wed.) 2012, 16：30〜17：30

Place: Dept. of Mathematics, General Research. Bldg., 301

Abstract:

Speaker:Changzheng Li (Kavli Institute for the Physics and Mathematics of the Universe, The University of Tokyo)

Title: Quantum Pieri rules for complex/symplectic Grassmannians

Date: October 24 (Wed.) 2012, 16：30〜17：30

Place: Dept. of Mathematics, Sci.Bldg., 301

Abstract: The (small) quantum cohomology of a flag variety is a deformation of the classical cohomology ring, by incorporating so-called genus zero, three-point Gromov-Witten invariants. We will take a brief review on this subject. In particular, we will introduce a Z^2-filtered algebraic structure on the quantum cohomology of a complete flag variety, as a kind of generalization of the Leray-Serre spectral sequence. As an application, we will study the quantum Pieri rule for the tautological subbundle over a complex/symplectic Grassmannian (i.e., a Grassmannian of type A/C). This is my joint work with Naichung Conan Leung.

Speaker:Hideo KOZONO (Waseda University)

Title: Stationary Navier-Stokes equations in multi-connected domains

Date: August. 1 (Wed.) 2012, 16：30〜17：30

Place: Dept. of Mathematics, Sci.Bldg., 3040

Abstract: In multi-connected domains, it is still an open question whether there does exist a solution of the stationary Navier-Stoeks equations with the inhomogeneous boundary data whose total flux is zero. The relation between the nonlinear structure of the equations and the topological invariance of the domain plays an important role for the solvability of this problem. We prove that if the harmonic part of solenoidal extensions of the given boundary data associated with the second Betti number of the domain is orthogonal to non-trivial solutions of the Euler equations, then there exists a solution for any viscosity constant. The relation between Leary's inequality and the topological type of the domain is also clarified. This talk is based on the joint work with Prof.Taku Yanagisawa at Nara Women University.

Speaker:Megumi HARADA (McMaster University)

Title: Integrable systems, toric degenerations and Okounkov bodies

Date: July. 11 (Wed.) 2012, 16：30〜17：30

Place: Dept. of Mathematics, Sci.Bldg., 3040

Abstract: Let X be a smooth projective variety of dimension n over C equipped with a very ample line bundle L. Using the theory of Okounkov bodies and an associated toric degeneration, we construct -- under a mild technical hypothesis on X -- an integrable system on X in the sense of symplectic geometry. More precisely, we construct a collection of real-valued functions {H_1, ..., H_n} on X which are continuous on all of X, smooth on an open dense subset U of X, and pairwise Poisson-commute on U. Here the symplectic structure on X is the pullback of the Fubini-Study form on P(H^0(X, L)^*) via the Kodaira embedding. The image of the `moment map' (H_1, ..., H_n): X to R^n is precisely the Okounkov body \Delta = \Delta(R, v) associated to the homogeneous coordinate ring R of X, and an appropriate choice of valuation v on R. Our main technical tools come from algebraic geometry, differential (Kaehler) geometry, and analysis. Specifically, we use: a toric degeneration of X to a (not necessarily normal) toric variety X_0, the gradient-Hamiltonian vector field, and a subtle generalization of the famous Lojasiewicz gradient inequality for real-valued analytic functions. Since our construction is valid for a large class of projective varieties X, these results provide a rich source of new examples of integrable systems. This is joint work with Kiumars Kaveh. In this talk I hope to briefly indicate the broader context of these results and to give a flavor of the techniques in the proof.

Speaker:Shintaro KUROKI (Osaka City University Advanced Mathematical Institute)

Title: Rigidity problems in toric topology

Date: June. 20 (Wed.) 2012, 16：30〜17：30

Place: Dept. of Mathematics, Sci.Bldg., 3040

Abstract: Toric topology is the study of algebraic, combinatorial, differential, geometric, and homotopy theoretic aspects of a particular class of commutative group actions, whose quotients are highly structured. For example, the set of (quasi)toric manifolds is one of such classes. As is well known that their equivariant (homeomorphism) types are determined by their (H^{*}(BT)-algebraic structures of) equivariant cohomologies. This fact leads us to the study of the cohomological rigidity problem of (quasi)toric manifolds asked by Masuda and Suh in 2006, i.e., for two (quasi)toric manifolds M and M', is it true that if H^{*}(M) and H^{*}(M') are isomorphic then M and M' are homeomorphic (or diffeomorphic)? Many partial affirmitive answers are known, however this original problem is still open. In this talk, I would like to introduce several rigidity problems (not only cohomological rigidity problem) of particular classes of commutative group actions (not only (quasi)toric manifolds).

Speaker:Yoshiyuki KIMURA (Osaka City University Advanced Mathematical Institute)

Title: 次数付き箙多様体と量子クラスター代数

Date: May. 30 (Wed.) 2012, 16：30〜17：30

Place: Dept. of Mathematics, Sci.Bldg., 3040

Abstract: