## Seminar information archive

Seminar information archive ～03/04｜Today's seminar 03/05 | Future seminars 03/06～

#### Tuesday Seminar on Topology

17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Growth rate of the number of periodic points for generic dynamical systems (JAPANESE)

**Masayuki Asaoka**(Kyoto University)Growth rate of the number of periodic points for generic dynamical systems (JAPANESE)

[ Abstract ]

For any hyperbolic dynamical system, the number of periodic

points grows at most exponentially and the growth rate

reflects statistic property of the system. For dynamics far

from hyperbolicity, the situation is different. In 1999,

Kaloshin proved genericity of super-exponential growth in the

region where dense set of dynamical systems exhibits homoclinic

tangency (so called the Newhouse region).

How does the number of periodic points grow for generic

partially hyperbolic dynamical systems? Such systems are known

to be far from homoclinic tangency. Is the growth at most

exponential like hyperbolic system, or super-exponential by

a mechanism different from homoclinic tangency?

The speaker, Katsutoshi Shinohara, and Dimitry Turaev proved

super-exponential growth of the number of periodic points for

generic one-dimensional iterated function systems under some

reasonable conditions. Such systems are models of dynamics

of partially hyperbolic systems in neutral direction. So, we

expect genericity of super-exponential growth in a region of

partially hyperbolic systems.

In this talk, we start with a brief history of the problem on

growth rate of the number of periodic point and discuss two

mechanisms which lead to genericity of super-exponential growth,

Kaloshin's one and ours.

For any hyperbolic dynamical system, the number of periodic

points grows at most exponentially and the growth rate

reflects statistic property of the system. For dynamics far

from hyperbolicity, the situation is different. In 1999,

Kaloshin proved genericity of super-exponential growth in the

region where dense set of dynamical systems exhibits homoclinic

tangency (so called the Newhouse region).

How does the number of periodic points grow for generic

partially hyperbolic dynamical systems? Such systems are known

to be far from homoclinic tangency. Is the growth at most

exponential like hyperbolic system, or super-exponential by

a mechanism different from homoclinic tangency?

The speaker, Katsutoshi Shinohara, and Dimitry Turaev proved

super-exponential growth of the number of periodic points for

generic one-dimensional iterated function systems under some

reasonable conditions. Such systems are models of dynamics

of partially hyperbolic systems in neutral direction. So, we

expect genericity of super-exponential growth in a region of

partially hyperbolic systems.

In this talk, we start with a brief history of the problem on

growth rate of the number of periodic point and discuss two

mechanisms which lead to genericity of super-exponential growth,

Kaloshin's one and ours.

### 2015/05/11

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Integral Kahler Invariants and the Bergman kernel asymptotics for line bundles

**Kengo Hirachi**(The Univ. of Tokyo)Integral Kahler Invariants and the Bergman kernel asymptotics for line bundles

[ Abstract ]

On a compact Kahler manifold, one can define global invariants by integrating local invariants of the metric. Assume that a global invariant thus obtained depends only on the Kahler class. Then we show that the integrand can be decomposed into a Chern polynomial (the integrand of a Chern number) and divergences of one forms, which do not contribute to the integral. We apply this decomposition formula to describe the asymptotic expansion of the Bergman kernel for positive line bundles and to show that the CR Q-curvature on a Sasakian manifold is a divergence. This is a joint work with Spyros Alexakis (U Toronto).

On a compact Kahler manifold, one can define global invariants by integrating local invariants of the metric. Assume that a global invariant thus obtained depends only on the Kahler class. Then we show that the integrand can be decomposed into a Chern polynomial (the integrand of a Chern number) and divergences of one forms, which do not contribute to the integral. We apply this decomposition formula to describe the asymptotic expansion of the Bergman kernel for positive line bundles and to show that the CR Q-curvature on a Sasakian manifold is a divergence. This is a joint work with Spyros Alexakis (U Toronto).

#### Tokyo Probability Seminar

16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)

Phase transitions for controlled Markov chains on infinite graphs (JAPANESE)

**Naoyuki Ichihara**(College of Science and Engineering, Aoyama Gakuin University)Phase transitions for controlled Markov chains on infinite graphs (JAPANESE)

#### Algebraic Geometry Seminar

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Deformations of weak Fano varieties (日本語 or English)

https://sites.google.com/site/tarosano222/

**Taro Sano**(Kyoto University)Deformations of weak Fano varieties (日本語 or English)

[ Abstract ]

A smooth projective variety often has obstructed deformations.

Nevertheless, important varieties such as Fano varieties and

Calabi-Yau varieties have unobstructed deformations.

In this talk, I explain about unobstructedness of deformations of weak

Fano varieties, in particular a weak Q-Fano 3-fold.

I also present several examples to show delicateness of this unobstructedness.

[ Reference URL ]A smooth projective variety often has obstructed deformations.

Nevertheless, important varieties such as Fano varieties and

Calabi-Yau varieties have unobstructed deformations.

In this talk, I explain about unobstructedness of deformations of weak

Fano varieties, in particular a weak Q-Fano 3-fold.

I also present several examples to show delicateness of this unobstructedness.

https://sites.google.com/site/tarosano222/

### 2015/05/08

#### Geometry Colloquium

10:00-11:30 Room #126 (Graduate School of Math. Sci. Bldg.)

On Perelman type functionals for the Ricci Yang-Mills flow (Japanese)

**Masashi Ishida**(Osaka University)On Perelman type functionals for the Ricci Yang-Mills flow (Japanese)

[ Abstract ]

In his works on the Ricci flow, Perelman introduced two functionals with monotonicity

formulas under the Ricci flow. The monotonicity formulas have many remarkable geometric applications. On the other hand, around 2007, Jeffrey Streets and Andrea Young independently and simultaneously introduced a new geometric flow which is called the Ricci Yang-Mills flow. The new flow can be regarded as the Ricci flow coupled with the Yang-Mills

heat flow. In this talk, we will introduce new functionals with monotonicity formulas under the Ricci Yang-Mills flow and discuss its applications.

In his works on the Ricci flow, Perelman introduced two functionals with monotonicity

formulas under the Ricci flow. The monotonicity formulas have many remarkable geometric applications. On the other hand, around 2007, Jeffrey Streets and Andrea Young independently and simultaneously introduced a new geometric flow which is called the Ricci Yang-Mills flow. The new flow can be regarded as the Ricci flow coupled with the Yang-Mills

heat flow. In this talk, we will introduce new functionals with monotonicity formulas under the Ricci Yang-Mills flow and discuss its applications.

### 2015/05/07

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

The group of parenthesized braids (ENGLISH)

**Patrick Dehornoy**(Univ. de Caen)The group of parenthesized braids (ENGLISH)

[ Abstract ]

We describe a group B obtained by gluing in a natural way two well-known

groups, namely Artin's braid group B_infty and Thompson's group F. The

elements of B correspond to braid diagrams in which the distances

between the strands are non uniform and some rescaling operators may

change these distances. The group B shares many properties with B_infty:

as the latter, it can be realized as a subgroup of a mapping class

group, namely that of a sphere with a Cantor set removed, and as a group

of automorphisms of a free group. Technically, the key point is the

existence of a self-distributive operation on B.

We describe a group B obtained by gluing in a natural way two well-known

groups, namely Artin's braid group B_infty and Thompson's group F. The

elements of B correspond to braid diagrams in which the distances

between the strands are non uniform and some rescaling operators may

change these distances. The group B shares many properties with B_infty:

as the latter, it can be realized as a subgroup of a mapping class

group, namely that of a sphere with a Cantor set removed, and as a group

of automorphisms of a free group. Technically, the key point is the

existence of a self-distributive operation on B.

### 2015/05/02

#### Harmonic Analysis Komaba Seminar

13:30-17:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Two-weight Morrey norm inequality and the sequential testing

(日本語)

The topology of the dual space of ${\mathcal S}_0$

(日本語)

**Hitoshi Tanaka**(Univ Tokyo) 13:30-15:00Two-weight Morrey norm inequality and the sequential testing

(日本語)

[ Abstract ]

In this talk we extend Sawyer's two-weight theory to Morrey spaces and give a characterization of two-weight Morrey norm inequalities for the (general) Hardy-Littlewood maximal operators in terms of the sequential testing due to H\"{a}nninen, Hyt\"{o}nen and Li.

We also introduce the description of the K\"othe dual of Morrey type spaces generated by a basis of measurable functions.

The second topic is based on a joint work with Professors Sawano (Tokyo Metropolitan University) and Masty{\l}o (Adam Mickiewicz University and Institute of Mathematics).

In this talk we extend Sawyer's two-weight theory to Morrey spaces and give a characterization of two-weight Morrey norm inequalities for the (general) Hardy-Littlewood maximal operators in terms of the sequential testing due to H\"{a}nninen, Hyt\"{o}nen and Li.

We also introduce the description of the K\"othe dual of Morrey type spaces generated by a basis of measurable functions.

The second topic is based on a joint work with Professors Sawano (Tokyo Metropolitan University) and Masty{\l}o (Adam Mickiewicz University and Institute of Mathematics).

**Yoshihiro Sawano**(Tokyo Metropolitan University.) 15:30-17:00The topology of the dual space of ${\mathcal S}_0$

(日本語)

[ Abstract ]

Based on the notation of my Japanese book, I will consider the topology of ${\mathcal S}_0'$, the dual of ${\mathcal S}_0$.

In view of the linear isomorphism ${\mathcal S}_0' \sim {\mathcal S}/{\mathcal P}$, we can consider two different topologies;

1) the weak-* topology

and

2) the quotient topology in ${\mathcal S}/{\mathcal P}$.

We aim to show that these two topologies are the same. This will be an errortum of my Japanese book.

This work is done jointly with Takahiro Noi and Shohei Nakamura in Tokyo Metropolitan University.

Based on the notation of my Japanese book, I will consider the topology of ${\mathcal S}_0'$, the dual of ${\mathcal S}_0$.

In view of the linear isomorphism ${\mathcal S}_0' \sim {\mathcal S}/{\mathcal P}$, we can consider two different topologies;

1) the weak-* topology

and

2) the quotient topology in ${\mathcal S}/{\mathcal P}$.

We aim to show that these two topologies are the same. This will be an errortum of my Japanese book.

This work is done jointly with Takahiro Noi and Shohei Nakamura in Tokyo Metropolitan University.

### 2015/04/28

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Verify hyperbolicity of 3-manifolds by computer and its applications. (JAPANESE)

**Hidetoshi Masai**(The University of Tokyo, JSPS)Verify hyperbolicity of 3-manifolds by computer and its applications. (JAPANESE)

[ Abstract ]

In this talk I will talk about the program called HIKMOT which

rigorously proves hyperbolicity of a given triangulated 3-manifold. To

prove hyperbolicity of a given triangulated 3-manifold, it suffices to

get a solution of Thurston's gluing equation. We use the notion called

interval arithmetic to overcome two types errors; round-off errors,

and truncated errors. I will also talk about its application to

exceptional surgeries along alternating knots. This talk is based on

joint work with N. Hoffman, K. Ichihara, M. Kashiwagi, S. Oishi, and

A. Takayasu.

In this talk I will talk about the program called HIKMOT which

rigorously proves hyperbolicity of a given triangulated 3-manifold. To

prove hyperbolicity of a given triangulated 3-manifold, it suffices to

get a solution of Thurston's gluing equation. We use the notion called

interval arithmetic to overcome two types errors; round-off errors,

and truncated errors. I will also talk about its application to

exceptional surgeries along alternating knots. This talk is based on

joint work with N. Hoffman, K. Ichihara, M. Kashiwagi, S. Oishi, and

A. Takayasu.

#### Lie Groups and Representation Theory

17:00-18:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Restricting automorphic forms to geodesic cycles (English)

**Bent Orsted**(Aarhus University and the University of Tokyo)Restricting automorphic forms to geodesic cycles (English)

[ Abstract ]

We find estimates for the restriction of automorphic forms on hyperbolic manifolds to compact geodesic cycles in terms of their expansion into eigenfunctions of the Laplacian. Our method resembles earlier work on products of automorphic forms by Bernstein and Reznikov, and it uses Kobayashi's new symmetry-breaking kernels. This is joint work with Jan M\"o{}llers.

We find estimates for the restriction of automorphic forms on hyperbolic manifolds to compact geodesic cycles in terms of their expansion into eigenfunctions of the Laplacian. Our method resembles earlier work on products of automorphic forms by Bernstein and Reznikov, and it uses Kobayashi's new symmetry-breaking kernels. This is joint work with Jan M\"o{}llers.

### 2015/04/27

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Variational formulas for canonical differentials and application (Japanese)

**Sachiko Hamano**(Fukushima Univ.)Variational formulas for canonical differentials and application (Japanese)

[ Abstract ]

We prove the variational formulas of the second order for $L_1$- and $L_0$-canonical differentials, which with the remarkable contrast are our first example in the case of the deforming non-planar open Riemann surface. As a direct application, we show the rigidity of the Euclidean radius of the moduli disk on open torus under pseudoconvexity. The main part of this talk is a joint work with Masakazu Shiba and Hiroshi Yamaguchi.

We prove the variational formulas of the second order for $L_1$- and $L_0$-canonical differentials, which with the remarkable contrast are our first example in the case of the deforming non-planar open Riemann surface. As a direct application, we show the rigidity of the Euclidean radius of the moduli disk on open torus under pseudoconvexity. The main part of this talk is a joint work with Masakazu Shiba and Hiroshi Yamaguchi.

#### Algebraic Geometry Seminar

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Lagrangian embeddings of cubic fourfolds containing a plane (日本語)

**Genki Ouchi**(University of Tokyo/IPMU)Lagrangian embeddings of cubic fourfolds containing a plane (日本語)

#### Numerical Analysis Seminar

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

A method of verified computations for solutions to semilinear parabolic equations using an analytic semigroup (日本語)

**Akitoshi Takayasu**(Waseda University)A method of verified computations for solutions to semilinear parabolic equations using an analytic semigroup (日本語)

### 2015/04/24

#### Geometry Colloquium

10:00-11:30 Room #126 (Graduate School of Math. Sci. Bldg.)

On K-stability and the volume functions of Q-Fano varieties (JAPANESE)

**Kento Fujita**(Kyoto University)On K-stability and the volume functions of Q-Fano varieties (JAPANESE)

[ Abstract ]

For Fano manifolds X, it is known that X admits K\"ahler-Einstein metrics if and only if the polarized pair

(X, -K_X) is K-polystable. In this talk, I will introduce a new effective stability named "divisorial stability" for Fano manifolds, which is weaker than K-stability and stronger than slope stability along divisors. I will show that:

1. We can easily test divisorial stability via the volume functions.

2. There is a relationship between divisorial stability and the structure property of Okounkov bodies of anti-canonical divisors.

3. For toric Fano manifolds, the existence of K\"ahler-Einstein metrics is equivalent to divisorial semistability.

For Fano manifolds X, it is known that X admits K\"ahler-Einstein metrics if and only if the polarized pair

(X, -K_X) is K-polystable. In this talk, I will introduce a new effective stability named "divisorial stability" for Fano manifolds, which is weaker than K-stability and stronger than slope stability along divisors. I will show that:

1. We can easily test divisorial stability via the volume functions.

2. There is a relationship between divisorial stability and the structure property of Okounkov bodies of anti-canonical divisors.

3. For toric Fano manifolds, the existence of K\"ahler-Einstein metrics is equivalent to divisorial semistability.

#### Colloquium

16:50-17:50 Room #123 (Graduate School of Math. Sci. Bldg.)

Rigidity of conformal functionals on spheres (ENGLISH)

**Bent Oersted**(Aarhus University and University of Tokyo)Rigidity of conformal functionals on spheres (ENGLISH)

[ Abstract ]

On a compact smooth manifold one may construct a Riemannian metric in many different ways. Each metric gives rise to natural elliptic operators such as the Laplace-Beltrami operator and corresponding spectral invariants, e.g. the eigenvalues, the trace of the heat semigroup, and the zeta function. In

this lecture we shall consider such functionals on the space of metrics on the sphere, combining conformal differential geometry and representation theory of semisimple Lie groups to obtain results about local extremal properties of special functionals. This is based on joint work with Niels Martin Moeller.

On a compact smooth manifold one may construct a Riemannian metric in many different ways. Each metric gives rise to natural elliptic operators such as the Laplace-Beltrami operator and corresponding spectral invariants, e.g. the eigenvalues, the trace of the heat semigroup, and the zeta function. In

this lecture we shall consider such functionals on the space of metrics on the sphere, combining conformal differential geometry and representation theory of semisimple Lie groups to obtain results about local extremal properties of special functionals. This is based on joint work with Niels Martin Moeller.

### 2015/04/23

#### Applied Analysis

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

The importance of being just late (ENGLISH)

**Bernold Fiedler**(Free University of Berlin)The importance of being just late (ENGLISH)

[ Abstract ]

Delays are a ubiquitous nuisance in control. Delays increase finite-dimensional phase spaces to become infinite-dimensional. But, are delays all that bad?

Following an idea of Pyragas, we attempt noninvasive and model-independent stabilization of unstable p-periodic phenomena $u(t)$ by a friendly delay $r$ . Our feedback only evaluates differences $u(t-r)-u(t)$. When the time delay $r$ is chosen to be an integer multiple $np$ of the minimal period $p$, the difference and the feedback vanish alike: the control strategy becomes noninvasive on the target periodic orbit.

We survey promise and limitations of this idea, including applications and an example of delay control of delay equations.

The results are joint work with P. Hoevel, W. Just, I. Schneider, E. Schoell, H.-J. Wuensche, S. Yanchuk, and others. See also

http://dynamics.mi.fu-berlin.de/

Delays are a ubiquitous nuisance in control. Delays increase finite-dimensional phase spaces to become infinite-dimensional. But, are delays all that bad?

Following an idea of Pyragas, we attempt noninvasive and model-independent stabilization of unstable p-periodic phenomena $u(t)$ by a friendly delay $r$ . Our feedback only evaluates differences $u(t-r)-u(t)$. When the time delay $r$ is chosen to be an integer multiple $np$ of the minimal period $p$, the difference and the feedback vanish alike: the control strategy becomes noninvasive on the target periodic orbit.

We survey promise and limitations of this idea, including applications and an example of delay control of delay equations.

The results are joint work with P. Hoevel, W. Just, I. Schneider, E. Schoell, H.-J. Wuensche, S. Yanchuk, and others. See also

http://dynamics.mi.fu-berlin.de/

#### Infinite Analysis Seminar Tokyo

17:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Parabolic analogue of periodic Kazhdan-Lusztig polynomials (JAPANESE)

**Hideya Watanabe**(Department of Mathematics, Tokyo Institute of Technology, Graduate school of science and Engineering)Parabolic analogue of periodic Kazhdan-Lusztig polynomials (JAPANESE)

[ Abstract ]

We construct a parabolic analogue of so-called periodic modules, which are modules over the Hecke algebra

associated with an affine Weyl group.

These modules have a basis similar to Kazhdan-Lusztig basis.

Our construction enables us to see the relation between (ordinary)periodic KL-polynomials and parabolic ones.

We construct a parabolic analogue of so-called periodic modules, which are modules over the Hecke algebra

associated with an affine Weyl group.

These modules have a basis similar to Kazhdan-Lusztig basis.

Our construction enables us to see the relation between (ordinary)periodic KL-polynomials and parabolic ones.

### 2015/04/22

#### Operator Algebra Seminars

16:45-18:15 Room #122 (Graduate School of Math. Sci. Bldg.)

Construction of minimal skew products of amenable minimal dynamical systems

**Yuhei Suzuki**(Unvi. Tokyo)Construction of minimal skew products of amenable minimal dynamical systems

### 2015/04/21

#### Tuesday Seminar of Analysis

16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Residue current techniques with application to a general theory of

linear delay-differential equations with constant coefficients (Japanese)

**Saiei Matsubara**(Graduate School of Mathematical Sciences, the University of Tokyo)Residue current techniques with application to a general theory of

linear delay-differential equations with constant coefficients (Japanese)

[ Abstract ]

We introduce the ring of differential operators with constant coefficients and commensurate time lags (we use the terminology D$\Delta$ operators from now) initially defined by H. Gl\"using-L\"ur\ss en for ordinary $D\Delta$ operators and observe that various function modules enjoy good cohomological properties over this ring. %After revising the notion of the residue current in the spirit of M. Andersson and E. Wulcan, we introduce the multidimensional version of the ring D$\Delta$ operators.

Combining this ring theoretic observation with the integral representation technique developed by M. Andersson, we solve a certain type of division with bounds. In the last chapter, we prove the injectivity property of various function modules over this ring as well as spectral synthesis type theorems for $D\Delta$ equations.

We introduce the ring of differential operators with constant coefficients and commensurate time lags (we use the terminology D$\Delta$ operators from now) initially defined by H. Gl\"using-L\"ur\ss en for ordinary $D\Delta$ operators and observe that various function modules enjoy good cohomological properties over this ring. %After revising the notion of the residue current in the spirit of M. Andersson and E. Wulcan, we introduce the multidimensional version of the ring D$\Delta$ operators.

Combining this ring theoretic observation with the integral representation technique developed by M. Andersson, we solve a certain type of division with bounds. In the last chapter, we prove the injectivity property of various function modules over this ring as well as spectral synthesis type theorems for $D\Delta$ equations.

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Orbit equivalence relations arising from Baumslag-Solitar groups (JAPANESE)

**Yoshikata Kida**(The University of Tokyo)Orbit equivalence relations arising from Baumslag-Solitar groups (JAPANESE)

[ Abstract ]

This talk is about measure-preserving actions of countable groups on probability

measure spaces and their orbit structure. Two such actions are called orbit equivalent

if there exists an isomorphism between the spaces preserving orbits. In this talk, I focus

on actions of Baumslag-Solitar groups that have two generators, a and t, with the relation

ta^p=a^qt, where p and q are given integers. This group is well studied in combinatorial

and geometric group theory. Whether Baumslag-Solitar groups with different p and q can

have orbit-equivalent actions is still a big open problem. I will discuss invariants under

orbit equivalence, motivating background and some results toward this problem.

This talk is about measure-preserving actions of countable groups on probability

measure spaces and their orbit structure. Two such actions are called orbit equivalent

if there exists an isomorphism between the spaces preserving orbits. In this talk, I focus

on actions of Baumslag-Solitar groups that have two generators, a and t, with the relation

ta^p=a^qt, where p and q are given integers. This group is well studied in combinatorial

and geometric group theory. Whether Baumslag-Solitar groups with different p and q can

have orbit-equivalent actions is still a big open problem. I will discuss invariants under

orbit equivalence, motivating background and some results toward this problem.

#### Lie Groups and Representation Theory

17:00-18:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Norm computation and analytic continuation of vector valued holomorphic discrete series representations

(English)

**Ryosuke Nakahama**(the University of Tokyo, Department of Mathematical Sciences)Norm computation and analytic continuation of vector valued holomorphic discrete series representations

(English)

[ Abstract ]

The holomorphic discrete series representations is realized on the space of vector-valued holomorphic functions on the complex bounded symmetric domains. When the parameter is sufficiently large, then its norm is given by the converging integral, but when the parameter becomes small, then the integral does not converge. However, if once we compute the norm explicitly, then we can consider its analytic continuation, and can discuss its properties, such as unitarizability. In this talk we treat the results on explicit norm computation.

The holomorphic discrete series representations is realized on the space of vector-valued holomorphic functions on the complex bounded symmetric domains. When the parameter is sufficiently large, then its norm is given by the converging integral, but when the parameter becomes small, then the integral does not converge. However, if once we compute the norm explicitly, then we can consider its analytic continuation, and can discuss its properties, such as unitarizability. In this talk we treat the results on explicit norm computation.

### 2015/04/20

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Weighted Laplacians on real and complex complete metric measure spaces (Japanese)

**Akito Futaki**(The Univ. of Tokyo)Weighted Laplacians on real and complex complete metric measure spaces (Japanese)

[ Abstract ]

We compare the weighted Laplacians on real and complex (K¥"ahler) metric measure spaces. In the compact case K¥"ahler metric measure spaces are considered on Fano manifolds for the study of K¥"ahler Ricci solitons while real metric measure spaces are considered with Bakry-¥'Emery Ricci tensor. There are twisted Laplacians which are useful in both cases but look alike each other. We see that if we consider noncompact complete manifolds significant differences appear.

We compare the weighted Laplacians on real and complex (K¥"ahler) metric measure spaces. In the compact case K¥"ahler metric measure spaces are considered on Fano manifolds for the study of K¥"ahler Ricci solitons while real metric measure spaces are considered with Bakry-¥'Emery Ricci tensor. There are twisted Laplacians which are useful in both cases but look alike each other. We see that if we consider noncompact complete manifolds significant differences appear.

#### Tokyo Probability Seminar

16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)

TBA (JAPANESE)

**Tetsuya Hattori**(Faculty of Economics, Keio University)TBA (JAPANESE)

#### Algebraic Geometry Seminar

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Fano 5-folds with nef tangent bundles (日本語)

**Akihiro Kanemitsu**(University of Tokyo)Fano 5-folds with nef tangent bundles (日本語)

### 2015/04/17

#### Geometry Colloquium

10:00-11:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Mean dimension of the dynamical system of Brody curves (日本語)

**Masaki TSUKAMOTO**(Kyoto University)Mean dimension of the dynamical system of Brody curves (日本語)

[ Abstract ]

Mean dimension is a topological invariant of dynamical systems with infinite dimension and infinite entropy. Brody curves are Lipschitz entire holomorphic curves, and they form an infinite dimensional dynamical system. Gromov started the problem of estimating its mean dimension in 1999. We solve this problem by proving the exact mean dimension formula. Our formula expresses the mean dimension by the energy density of Brody curves. A key novel ingredient is an information theoretic approach to mean dimension introduced by Lindenstrauss and Weiss.

Mean dimension is a topological invariant of dynamical systems with infinite dimension and infinite entropy. Brody curves are Lipschitz entire holomorphic curves, and they form an infinite dimensional dynamical system. Gromov started the problem of estimating its mean dimension in 1999. We solve this problem by proving the exact mean dimension formula. Our formula expresses the mean dimension by the energy density of Brody curves. A key novel ingredient is an information theoretic approach to mean dimension introduced by Lindenstrauss and Weiss.

### 2015/04/15

#### Operator Algebra Seminars

16:45-18:15 Room #122 (Graduate School of Math. Sci. Bldg.)

On the tricategory of coordinate free conformal nets

(English)

**Juan Orendain**(UNAM/Univ. Tokyo)On the tricategory of coordinate free conformal nets

(English)

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