School

**Rigidity School, Tokyo 2015**

Date: November 20 (Fri) -- November 23 (Mon), 2015 On 20th, School starts in the afternoon.

Place: Graduate School of Mathematical Sciences, University of Tokyo Room 123 (Room 056 from 3 pm of 20th)

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

Supported by Leading Graduate Course for Frontiers of Mathematical Sciences and Physics, the University of Tokyo.

Speakers:

In Kang KIM (Korean Institute for Advanced Study)

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

Motoko KATO (University of Tokyo)

Takefumi KONDO (Kagoshima University)

Tetsu TOYODA (Suzuka College of Technology)

Masato MIMURA (Tohoku University)

Jun-ichi MUKUNO (Nagoya University)

Organizers:

Hiroyasu IZEKI (Keio University) izeki@math.keio.ac.jp

Masahiko KANAI (University of Tokyo) mkanai@ms.u-tokyo.ac.jp

Shin NAYATANI (Nagoya University) nayatani@math.nagoya-u.ac.jp

Program :

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)

Title & Abstract:

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.