**スクール**

**Rigidity School, Tokyo 2015**

**日時：２０１５年１１月２０日（金）〜１１月２３日（月） ２０日（金）は午後開始の予定です。**

**場所：東京大学数理科学研究科 (駒場キャンパス) 数理科学科棟１２３号室（２０日の午後３時以降は０５６号室）**

http://www.ms.u-tokyo.ac.jp/access/index.html

**宿泊施設は各自で手配して下さるようお願いいたします。**

**この研究集会は、東京大学数物フロンティア・リーディング大学院の援助のもとに開催されます。**

**同じ週に東大数理におきまして日仏共同研究（タイヒミュラー空間と写像類群）による小研究集会が開催されます。詳しくは以下のサイトをご覧下さい。**

http://www.ms.u-tokyo.ac.jp/~kawazumi/1511franco-japonais.html

**講演者：**

**In Kang KIM (Korean Institute for Advanced Study)**

**Piotr NOWAK (Polish Academy of Sciences / University of Warsaw)**

**加藤本子 (東京大学)**

**近藤剛史 (鹿児島大学)**

**豊田哲 (鈴鹿高専)**

**見村万佐人 (東北大学)**

**椋野純一 (名古屋大学)**

**問合せ先・世話人：**

**井関裕靖 (慶応大学) izeki@math.keio.ac.jp**

**金井雅彦 (東京大学) mkanai@ms.u-tokyo.ac.jp**

**納谷信 (名古屋大学) nayatani@math.nagoya-u.ac.jp**

**プログラム:**

**November 20 (Fri)**

**13:30--14:20 Nowak (1)**

**14:40--15:30 Kato**

**16:10--17:00 Mukuno**

**November 21 (Sat)**

**9:30--10:20 Kim(1)**

**10:40--11:30 Nowak(2)**

**13:30--14:20 Kondo**

**14:40--15:30 Kim(2)**

**16:10--17:00 Nowak(3)**

**18:30-- Party**

**November 22 (Sun)**

**9:30--10:20 Nowak(4)**

**10:40--11:30 Kim(3)**

**13:30--14:20 Toyoda**

**14:40--15:30 Nowak(5)**

**16:10--17:00 Kim(4)**

**November 23 (Mon)**

**9:30--10:20 Mimura**

**10:40--11:30 Kim(5)**

タイトル、アブストラクト:

In Kang Kim

Title: Rigidity, simplicial volume and bounded cohomology

Abstract: Recently there has been some advancement in the study of (local) rigidity and flexibility of lattices in semi-simple Lie groups. Rigidity can be studied using cohomology theory as developed by Weil and many others, and also can be investigated by other tools such as harmonic forms, bounded cohomology, barycenter methods. Gromov simplicial volume can also be calculated using these techniques for some cases. We will try to presents these methods as elementary as possible and give some detailed proofs if time permits.

Piotr Nowak

Title: Rigidity of groups and higher index theory

Abstract: The coarse Baum-Connes conjecture is a statement in index theory that relates the coarse homology of a metric space with the K-theory of its Roe algebra via a conjectured isomorphism. Whenever it is true for a finitely generated group G, it implies the Novikov conjecture for G. The coarse Baum-Connes conjecture may fail however in some cases, in the presence of property (T)-type phenomena: certain expanders are counterexamples to the conjecture. In the lectures we will describe the appropriate context in which property (T) appears in the setting of K-theory, discuss the mentioned above failure of the conjecture for expanders as well as describe new candidates for such counterexamples.

Motoko Kato

Title: Serre's property FA of Higher dimensional Thompson groups

Abstract: In 2004, Brin defined a family of infinite simple groups nV of which 1V is Thompson's group V. nV is a group which consists of partially affine, partially orientation preserving bijections between n-dimensional cubes. Hennig and Matucci obtained a presentation for nV and showed that they are finitely presented groups. In this talk we prove that each nV has Serre's property FA by using the presentation. This is a generalization of the corresponding result of Farley, who studied Thompson's group V.

Takefumi Kondo

Title: Nonlinear spectral gaps with respect to CAT(0) spaces

Abstract: As a nonlinear analog of the spectral gap of the combinatorial Laplacian of a finite graph, we can define the nonlinear spectral gap of a finite graph with respect to a metric space and this quantity plays important roles in rigidity problem and metric embedding problem. In this talk we will describe how we can determine the exact values of the nonlinear spectral gaps with respect to CAT(0) spaces for some class of finite weighted graphs.

Tetsu Toyoda

Title: On a question of Gromov about the Wirtinger inequalities

Abstract: In the study of measure concentration inequalities on CAT(0) spaces, M. Gromov introduced the conditions called the Wirtinger condition and the Cycl(0) condition for finite points on a metric space. It is known that if every four points on a geodesic space satisfies the Cycl(0) condition then any finitely many (at least four) number of points satisfies the Wirtinger condition. Gromov asked if such implication holds true without assuming a space is geodesic. We answer this question affirmatively. This is a joint work with Takefumi Kondo (Kagoshima University) and Takato Uehara (Saga University).

Masato Mimura

Title: Strong algebraization of fixed point property

Abstract: We obtain a purely algebraic criterion for fixed point properties on Banach spaces under relative fixed point properties. This is a strengthening of Shalom's algebraization in ICM 2006, in the sense of that we completely remove any form of "bounded generation" assumption. Several application may, in addition, be discussed.

Jun-ichi Mukuno

Title: On the automorphism groups of unbounded homogeneous domains with the boundaries of light cone type

Abstract: We consider the holomorphic automorphism group of an unbounded homogeneous complex domain with the boundary of light cone type. In this talk, we determine the automorphism group of the domain, and as a corollary we show that there does not exist no compact quotient of the domain. Furthermore we present the characterization of the domain by the automorphism group. This is a joint work with Yoshikazu Nagata.