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Lecture Series（2015） |
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Speaker |
Hajime Urakawa(emeritus profesor, Tohoku University) |
Title |
Geometry of Harmonic Maps and Bi-harmonic Maps |
Date |
March 28 (Mon)2016
15:30-17:00
March 29 (Tue)2016
10:00-11:30
13:30-16:00
March 30 (Wed)2016
10:00-11:30
13:30-16:00
March 31 (Thu)2016
10:00-11:30
13:30-15:30
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| Place |
F415, Faculty of Science, Osaka City University |
| Abstract |
(1)部分多様体論から、エネルギーの第一変分公式と第二変分公式、２−エネルギーの第一変分公式と第二変分公式
(2)球面や複素射影空間などの２−調和超曲面（井ノ口順一、一山稔之両氏との共同研究）。ユークリッド空間内の２−調和超曲面（小磯憲史氏との共同研究）
(3)自乗可積テンション場をもつ２−調和写像（中内伸光氏との共同研究）
(4)ケーラー多様体内の２−調和ラグランジュ部分多様体、佐々木多様体内の２−調和ルジャンドリアン部分多様体
(5)リーマン計量の共形変形と２−調和写像（内藤久資氏との共同研究）
(6)調和写像や２−調和写像のバブリング現象（中内伸光氏との共同研究）
(7)対称空間内の２−調和等質超曲面の分類（大野晋司、酒井高司両氏との共同研究） |
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Speaker |
Marcos Alexandrino（Institute of Mathematics and Statistics,
University of Sao Paulo, Brasil) |
Title |
Introduction to Polar Foliations |
Date |
March 24 (Thu)2016
9:30-10:30
11:00-12:00
March 25 (Fri)2016
9:30-10:30
13:00-14:00
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| Place |
F404, Building F, Faculty of Science, Osaka City University |
| Abstract |
（１）Lecture 1: Motivations and main definitions(where I should establish
the main definitions and provide an overviewusing the concrete example
of conjugation action)
（２）Lecture 2: Differential and geometrical aspects(Palais-Terng problem,
slice theorem, equifocality of manifolds)
（３）Lecture 3: Dynamical aspects(Weyl groups their relation with fundamental
groups, coxeter orbifold structure of the leaf spaces, and a few words
about foliations with no closed leaf)
（４）Lecture 4: Singular theory: level sets and foliations(transnormal and
isoparametric maps).
This intensive lecture is held as one of activities under the JSPS Program
for Advancing Strategic International Networks to Accelerate the Circulation
of Talented Researchers (Oct.2014-Mar.2017) "Mathematical Science
of Symmetry, Topology and Moduli,Evolution of International Research Network
based on OCAMI"(Osaka City University - Kobe University - Waseda University) |
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Speaker |
Pierre Baumann (CNRS, the Institut de Recherche Mathematique A
vancee, Strasbourg) |
Title |
Mirkovic-Vilonen polytopes in finite or affine type |
Date |
March 22(Tue.) 2016
10:30-12:00
14:00-15:30
16:00-17:30
March 23(Wed.) 2016
10:30-12:00
14:00-15:30
16:00-17:30
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| Place |
F205, Faculty of Science, Osaka City University |
| Abstract |
This series of talks will focus on joint work Joel Kamnitzer (Toronto)
and Peter Tingley (Chicago).
Let us begin with two natural constructions that give birth to convex polytopes,
some of the oldest mathematical objects. 1) Given a hamiltonian action
of a compact torus T on a compact connected symplectic manifold, the image
by the moment map of any closed T-stable subset is a convex polytope. 2)
Given an abelian finite length category A and an object X of A, the classes
in the Grothendieck group K_0(A) of the subobjects of X are finitely many;
the convex hull of these classes in K_0(A) \otimes R is thus a convex polytope,
called the Harder-Narasimhan (HN) polytope of X.
Now we consider the following two constructions, apparently unrelated.
Let g be a semisimple complex Lie algebra and let h be the Cartan subalgebra
of g. For a dominant integral weight \lambda, we denote by V(\lambda) the
irreducible g-module of highest weight \lambda. 1) The geometric Satake
correspondence identifies V(\lambda) with the intersection homology of
a certain Schubert variety X_\lambda, and certain cycles in X_\lambda,
called Mirkovic-Vilonen (MV) cycles, form a basis of this intersection
homology. Since there is a hamiltonian action in this situation, we obtain
polytopes by taking the image of the MV cycles by the moment map. 2) Let
\Lambda be the preprojective algebra built on the Dynkin diagram of g (supposed
here to be simply laced). To each finite-dimensional \Lambda-module M,
we associate its HN polytope.
It is remarkable that both constructions yield polytopes whose edges are
in root directions. In construction 2, this fact follows from Iyama and
Reiten's tilting theory for modules over the preprojective algebra. Further,
the polytopes from construction 1 can be obtained from construction 2 by
taking M general enough in the representation space of \Lambda. The first
lectures will be devoted to the explanation of this coincidence. The main
point is that these polytopes can be described by combinatorial data arising
from the Kashiwara crystals of the representations V(\lambda).
The polytopes obtained here are called MV polytopes, because they were
first discovered through construction 1. One may wish to extend the definition
of these polytopes to the case of an affine Kac-Moody algebra g. We will
use construction 2 to achieve this goal. One difficulty to be overcome
is that tilting theory has not enough reach to describe the edges that
are parallel to the imaginary roots: one must here use other tools like
Hall functors, which embed the category of finite-dimensional representations
of the preprojective algebra of the Kronecker quiver into 2-Calabi-Yau
categories. At the end of the day, one understands that HN polytopes can
be decorated by families of partitions. The last lectures will be devoted
to a presentation of this construction. |
|
Speaker |
Yuichi Nohara （Kagawa University） |
Title |
Gelfand-Cetlin Systems and Mirror Symmetry of Flag Manifolds |
Date |
February 22(Mon.) 2016
15:30-18:00
February 23(Tue.) 2016
10:00-11:30
16:30-18:00
February 24(Wed.) 2016
10:00-11:30
13:30-15:00
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| Place |
E408, Faculty of Science, Osaka City University |
| Abstract |
| Gelfand-Cetlin系とは、Guillemin-Sternbergにより導入された旗多様体上の完全可積分系(Lagrangeトーラスファイブレーション)である。この講義では、Gelfand-Cetlin系のシンプレクティック幾何と、そのミラー対称性への応用について解説する。 |
|
Speaker |
Kentaro Wada (Shinshu University) |
Title |
q-Schur 代数の表現論 |
Date |
December 25(Fri.)
10:30-12:00
14:00-15:30
16:00-17:30
December 26(Sat.)
10:30-12:00
14:00-15:30
16:00-17:30
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| Place |
E408, Faculty of Science, Osaka City University |
| Abstract |
q-Schur 代数は対称群に付随した Iwahori-Hecke 代数の permutation module達の直和の自己準同型環として定義される有限次元代数であり，準遺伝(quasi-hereditary)代数としての構造を持つ。q-Schur
代数の有限次元表現のなす圏から Hecke 代数の有限次元表現のなす圏へ Schur 関手と呼ばれる射影対象上fully faithful である完全関手が構成でき，この関手を通じて両者の間の密接な関係を調べることができる。一方で，q-Schur
代数は，一般線形リー代数に付随した量子群の自然表現のテンソル積表現を用いた，量子群と Hecke 代数の間の Schur-Weyl 双対（一般線形群と対称群の間の
Schur-Weyl 双対の q-類似)を通じて，量子群の商代数として実現できることが知られている。量子群サイドから見れば，q-Schur 代数は量子群の"多項式表現"のなす圏を与える有限次元代数となる。今回は，量子群サイドから，q-Schur
代数の構造や表現論を解説する予定です。主に大学院生や非専門家を対象として，予備知識を特に仮定せず，量子群の表現論の基本的なことからお話しする予定です。
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|
Speaker |
Hideya Hashimoto (Meijo Univ.) |
Title |
Clifford環と八元数の幾何学への応用 |
Date |
September 10 (Thu.) 2015,
14:00 -- 15:30
16:00 -- 17:30
September 11 (Fri.)2015,
11:00 -- 12:30
14:30 -- 16:00
September 12 (Sat.)2015,
11:00 -- 12:30
14:30 -- 16:00
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| Place |
F405, Faculty of Science, Osaka City University |
| Abstract |
Clifford環と八元数に関連する幾何構造に関する問題点についてできるだけ分り易く解説する。例外型単純Lie群G_2とSpin(7)の具体的表現、及び、これらの群に関連したfibre
bundleの構造、等質空間の族の幾何学的特性についても解説する。特に興味深いTwistor空間を具体化する。G_2とSpin(7)の幾何学の持つ独特な面白さについて解説する予定である。
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|
Speaker |
Patrick Dehornoy（University of Caen, France） |
Title |
Using selfdistributivity for investigating and ordering braids |
Date |
May 18 (Mon.) 2015,
10:00 -- 11:00
11:15 -- 12:15
14:30 -- 15:30
15:45 -- 16:45
May 19 (Tue.) 2015,
10:00 -- 11:00
11:15 -- 12:15
14:30 -- 15:30
15:45 -- 16:45
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| Place |
E408, Faculty of Science, Osaka City University |
| Abstract |
The connection between the selfdistributive law LD : x(yz) =
(xy)(xz) (or its symmetric counterpart) and Reidemeister move III is
well-known. So, given an LD-system (or shelf) S, that is, any structure
satisfying the LD law, it is natural to try to use S to find
topologically interesting results. However, requiring compatibility with
Reidemeister moves I and II leads to restricting to LD-systems of a
(very) particular type, namely racks and quandles. Otherwise, technical
obstructions appear, and seem to discard all topological applications.
The aim of the minicourse is to explain how, at least in one case, it is
possible to overcome the obstructions and obtain a sort of Reidemeister
II compatibility for shelves that are not racks, namely free shelves in
that specific case. This approach led to the orderability of Artin's
braid groups, and our hope is that the techniques that will be explained
can be used again and lead to applications for the (many) known shelves
that are neither quandles, nor even racks.
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|
Speaker |
Yosuke Saito（OCAMI） |
Title |
Elliptic Ding-Iohara-Miki algebra and related materials |
Date |
May 12 (Tue.) 13:30-15:30
June 2 (Tue.) 13:30-15:30
June 9 (Tue.) 13:30-15:30
June 16 (Tue.) 13:30-15:30
June 30 (Tue.) 13:30-15:30
|
| Place |
F405, Faculty of Science, Osaka City University |
| Abstract |
In 2009, Feigin-Hashizume-Hoshino-Shiraishi-Yanagida showed that a kind
of quantum group called the Ding-Iohara-Miki algebra arises from the free
field realization of the Macdonald operator. Recently, this algebra has
been studied intensively by several researchers, and it is known that the
Ding-Iohara-Miki algebra can be applied to 5 dimensional AGT conjecture,
topological string theory, for instance.
On the other hand, there exists an elliptic analog of the Macdonald operator,
called the elliptic Ruijsenaars operator. I have accomplished to construct
the free field realization of the elliptic Ruijsenaars operator using new
boson operators. As a result of the free field realization, it is shown
that an elliptic analog of the Ding-Iohara-Miki algebra (the elliptic Ding-Iohara-Miki
algebra) arises.
In this talk, we will start from an overview of some results due to Feigin-Hashizume-Hoshino-Shiraishi-Yanagida,
and we will explain how to construct the free field realization of the
elliptic Ruijsenaars operator, and how the elliptic Ding-Iohara-Miki algebra
arises.
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