## Seminar information archive

Seminar information archive ～02/19｜Today's seminar 02/20 | Future seminars 02/21～

### 2012/11/06

#### Tuesday Seminar on Topology

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

Galois action on knots (JAPANESE)

**Furusho Hidekazu**(Nagoya University)Galois action on knots (JAPANESE)

[ Abstract ]

I will explain a motivic structure on knots.

Then I will explain that the absolute Galois group of

the rational number field acts non-trivially

on 'the space of knots' in a non-trivial way.

I will explain a motivic structure on knots.

Then I will explain that the absolute Galois group of

the rational number field acts non-trivially

on 'the space of knots' in a non-trivial way.

#### Tuesday Seminar of Analysis

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Resonance free domains for homoclinic orbits (ENGLISH)

**Thierry Ramond**(Univ. Paris, Orsay)Resonance free domains for homoclinic orbits (ENGLISH)

#### Lie Groups and Representation Theory

16:30-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

An explicit construction of spherical designs on S^3 (JAPANESE)

**Takayuki Okuda**(the University of Tokyo)An explicit construction of spherical designs on S^3 (JAPANESE)

[ Abstract ]

The existence of spherical t-designs on S^d for any t and d are proved by Seymour--Zaslavsky in 1984.

However, explicit constructions of spherical designs were not known for d > 2 and large t.

In this talk, for a given spherical t-design Y on S^2, we give an

algorithm to make a spherical 2t-design X on S^3 which maps Y by a Hopf map. In particular, by combining with the results of Kuperberg in 2005, we have an explicit construction of spherical t-designs on S^3 for any t.

The existence of spherical t-designs on S^d for any t and d are proved by Seymour--Zaslavsky in 1984.

However, explicit constructions of spherical designs were not known for d > 2 and large t.

In this talk, for a given spherical t-design Y on S^2, we give an

algorithm to make a spherical 2t-design X on S^3 which maps Y by a Hopf map. In particular, by combining with the results of Kuperberg in 2005, we have an explicit construction of spherical t-designs on S^3 for any t.

### 2012/11/05

#### Algebraic Geometry Seminar

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

The rationality of the moduli spaces of trigonal curves (JAPANESE)

**Shouhei Ma**(Nagoya University)The rationality of the moduli spaces of trigonal curves (JAPANESE)

### 2012/10/31

#### Geometry Colloquium

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Hofer-Zehnder capacity and a Hamiltonian circle action with noncontractible orbits (JAPANESE)

**Kei Irie**(Kyoto University)Hofer-Zehnder capacity and a Hamiltonian circle action with noncontractible orbits (JAPANESE)

[ Abstract ]

Hofer-Zehnder (HZ) capacity is an invariant of symplectic manifolds, which is important in symplectic topology and Hamiltonian dynamics. The energy-capacity inequality (due to Hofer and many others) claims that HZ capacity of a domain is bounded from above by its dispalcement energy.

In this talk, we prove a variant of this inequality, which is applicable to nondisplaceable domains. We also give some applications, including case of disc cotangent bundles.

Hofer-Zehnder (HZ) capacity is an invariant of symplectic manifolds, which is important in symplectic topology and Hamiltonian dynamics. The energy-capacity inequality (due to Hofer and many others) claims that HZ capacity of a domain is bounded from above by its dispalcement energy.

In this talk, we prove a variant of this inequality, which is applicable to nondisplaceable domains. We also give some applications, including case of disc cotangent bundles.

### 2012/10/30

#### Numerical Analysis Seminar

16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Error analysis of Galerkin's method for semilinear partial differential equations (JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

**Tadashi Kawanago**(Tokyo Institute of Technology)Error analysis of Galerkin's method for semilinear partial differential equations (JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

#### Tuesday Seminar on Topology

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

Applications of knot theory to molecular biology (JAPANESE)

**Koya Shimokawa**(Saitama University)Applications of knot theory to molecular biology (JAPANESE)

[ Abstract ]

In this talk we discuss applications of knot theory to studies of DNA

and proteins.

Especially we will consider (1)topological characterization of

mechanisms of site-specific recombination systems,

(2)modeling knotted DNA and proteins in confined regions using lattice

knots, and

(3)mechanism of topoisomerases and signed crossing changes.

In this talk we discuss applications of knot theory to studies of DNA

and proteins.

Especially we will consider (1)topological characterization of

mechanisms of site-specific recombination systems,

(2)modeling knotted DNA and proteins in confined regions using lattice

knots, and

(3)mechanism of topoisomerases and signed crossing changes.

#### Tuesday Seminar of Analysis

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

About the method of characteristics (ENGLISH)

**Francis Nier**(Univ. Rennes 1)About the method of characteristics (ENGLISH)

[ Abstract ]

While studying the mean field dynamics of a systems of bosons, one is led to solve a transport equation for a probability measure in an infinite dimensional phase-space. Since those probability measures are characterized after testing with cylindrical or polynomial observables, which make classes which are not invariant after composing with a nonlinear flow. Thus the standard method of characteristics for transport equations cannot be extended at once to the infinite dimensional case. A solution comes from techniques developed for optimal transport and a probabilistic interpretation of trajectories.

While studying the mean field dynamics of a systems of bosons, one is led to solve a transport equation for a probability measure in an infinite dimensional phase-space. Since those probability measures are characterized after testing with cylindrical or polynomial observables, which make classes which are not invariant after composing with a nonlinear flow. Thus the standard method of characteristics for transport equations cannot be extended at once to the infinite dimensional case. A solution comes from techniques developed for optimal transport and a probabilistic interpretation of trajectories.

#### Lectures

17:00-18:00 Room #470 (Graduate School of Math. Sci. Bldg.)

Discrete Topologgy of Cellular Microstructures

and Complicatedness Measurements for Cell Complexes (JAPANESE)

**Frank Lutz**(Technische Universität Berlin)Discrete Topologgy of Cellular Microstructures

and Complicatedness Measurements for Cell Complexes (JAPANESE)

[ Abstract ]

Our first aim is to use methods from discrete and geometric topology

to recover structural information from the composition of

monocrystalline materials that have a periodic foam structure

(such as gas hydrates and transition metal alloys) and also of

polycrystalline materials (such as metals and certain ceramics).

For more general complexes, even with a billion of faces, homological

information can be obtained with computational homology packages

such as CHomP or RedHom. These packages extensively use discrete Morse

theory as a preprocessing step. Although it is NP-hard to find optimal

discrete Morse functions, most data appears to be easy and it is

in fact hard to construct ``complicated'' examples. As we will see,

random discrete Morse theory will allow us to measure the

``complicatedness'' of complexes.

Our first aim is to use methods from discrete and geometric topology

to recover structural information from the composition of

monocrystalline materials that have a periodic foam structure

(such as gas hydrates and transition metal alloys) and also of

polycrystalline materials (such as metals and certain ceramics).

For more general complexes, even with a billion of faces, homological

information can be obtained with computational homology packages

such as CHomP or RedHom. These packages extensively use discrete Morse

theory as a preprocessing step. Although it is NP-hard to find optimal

discrete Morse functions, most data appears to be easy and it is

in fact hard to construct ``complicated'' examples. As we will see,

random discrete Morse theory will allow us to measure the

``complicatedness'' of complexes.

### 2012/10/29

#### Algebraic Geometry Seminar

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

The Mukai conjecture for log Fano manifolds (JAPANESE)

**Kento Fujita**(RIMS)The Mukai conjecture for log Fano manifolds (JAPANESE)

[ Abstract ]

The concept of log Fano manifolds is one of the most natural generalization of the concept of Fano manifolds. We will give some structure theorems of log Fano manifolds. For example, we will show that the Mukai conjecture for Fano manifolds implies the `log Mukai conjecture' for log Fano manifolds.

The concept of log Fano manifolds is one of the most natural generalization of the concept of Fano manifolds. We will give some structure theorems of log Fano manifolds. For example, we will show that the Mukai conjecture for Fano manifolds implies the `log Mukai conjecture' for log Fano manifolds.

#### Seminar on Geometric Complex Analysis

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

Value distribution of meromorphic functions on foliated manifolds,II (JAPANESE)

**Atsushi Atsuji**(Keio University)Value distribution of meromorphic functions on foliated manifolds,II (JAPANESE)

### 2012/10/26

#### Seminar on Probability and Statistics

14:50-16:00 Room #006 (Graduate School of Math. Sci. Bldg.)

Tuning parameter selection in sparse regression modeling (JAPANESE)

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2012/10.html

**HIROSE, Kei**(Graduate School of Engineering Science, Osaka University)Tuning parameter selection in sparse regression modeling (JAPANESE)

[ Abstract ]

In sparse regression modeling via regularization such as the lasso, it is important to select appropriate values of tuning parameters including regularization parameters. The choice of tuning parameters can be viewed as a model selection and evaluation problem. Mallows' Cp type criteria may be used as a tuning parameter selection tool in lasso type regularization methods, for which the concept of degrees of freedom plays a key role. In this talk, we propose an efficient algorithm that computes the degrees of freedom by extending the generalized path seeking algorithm. Our procedure allows us to construct model selection criteria for evaluating models estimated by regularization with a wide variety of convex and nonconvex penalties. The proposed methodology is investigated through the analysis of real data and Monte Carlo simulations. Numerical results show that Cp criterion based on our algorithm performs well in various situations.

[ Reference URL ]In sparse regression modeling via regularization such as the lasso, it is important to select appropriate values of tuning parameters including regularization parameters. The choice of tuning parameters can be viewed as a model selection and evaluation problem. Mallows' Cp type criteria may be used as a tuning parameter selection tool in lasso type regularization methods, for which the concept of degrees of freedom plays a key role. In this talk, we propose an efficient algorithm that computes the degrees of freedom by extending the generalized path seeking algorithm. Our procedure allows us to construct model selection criteria for evaluating models estimated by regularization with a wide variety of convex and nonconvex penalties. The proposed methodology is investigated through the analysis of real data and Monte Carlo simulations. Numerical results show that Cp criterion based on our algorithm performs well in various situations.

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2012/10.html

### 2012/10/23

#### Tuesday Seminar of Analysis

16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Topics in quantum entropy and entanglement (ENGLISH)

**Elliott Lieb**(Princeton Univ.)Topics in quantum entropy and entanglement (ENGLISH)

[ Abstract ]

Several recent results on quantum entropy and the uncertainty

principle will be discussed. This is partly joint work with Eric Carlen

on lower bounds for entanglement, which has no classical analog, in terms

of the negative of the conditional entropy, S1 - S12, whose negativity,

when it occurs, also has no classical analog. (see arXiv:1203.4719)

It is also partly joint work with Rupert Frank on the uncertaintly

principle for quantum entropy which compares the quantum von Neumann

entropy with the classical entropies with respect to two different

bases. We prove an extension to the product of two and three spaces, which

has applications in quantum information theory. (see arxiv:1204.0825)

Several recent results on quantum entropy and the uncertainty

principle will be discussed. This is partly joint work with Eric Carlen

on lower bounds for entanglement, which has no classical analog, in terms

of the negative of the conditional entropy, S1 - S12, whose negativity,

when it occurs, also has no classical analog. (see arXiv:1203.4719)

It is also partly joint work with Rupert Frank on the uncertaintly

principle for quantum entropy which compares the quantum von Neumann

entropy with the classical entropies with respect to two different

bases. We prove an extension to the product of two and three spaces, which

has applications in quantum information theory. (see arxiv:1204.0825)

#### Tuesday Seminar on Topology

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

A geometric approach to the Johnson homomorphisms (JAPANESE)

**Nariya Kawazumi**(The University of Tokyo)A geometric approach to the Johnson homomorphisms (JAPANESE)

[ Abstract ]

We re-construct the Johnson homomorphisms as an embeddig of the Torelli

group

into the completed Goldman-Turaev Lie bialgebra. Then the image is

included in the

kernel of the Turaev cobracket. In the case where the boundary is

connected,

the Turaev cobracket clarifies a geometric meaning of the Morita traces.

Time permitting, we also discuss the case of holed discs.

This talk is based on a joint work with Yusuke Kuno (Tsuda College).

We re-construct the Johnson homomorphisms as an embeddig of the Torelli

group

into the completed Goldman-Turaev Lie bialgebra. Then the image is

included in the

kernel of the Turaev cobracket. In the case where the boundary is

connected,

the Turaev cobracket clarifies a geometric meaning of the Morita traces.

Time permitting, we also discuss the case of holed discs.

This talk is based on a joint work with Yusuke Kuno (Tsuda College).

### 2012/10/22

#### Seminar on Geometric Complex Analysis

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

The second main therorem for entire curves into Hilbert modular surfaces (JAPANESE)

**Yusaku Tiba**(Grad. School of Math. Sci., Univ. of Tokyo)The second main therorem for entire curves into Hilbert modular surfaces (JAPANESE)

### 2012/10/19

#### Seminar on Probability and Statistics

14:50-16:00 Room #006 (Graduate School of Math. Sci. Bldg.)

Asymptotic expansion of ruin probability under Lévy insurance risks (JAPANESE)

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2012/09.html

**SHIMIZU, Yasutaka**(Graduate School of Engineering Science, Osaka University)Asymptotic expansion of ruin probability under Lévy insurance risks (JAPANESE)

[ Abstract ]

An asymptotic expansion formula of the ultimate ruin probability under L\\'evy insurance risks

is given as the loading factor tends to zero. The formula is obtained via the Edgeworth type expansion of

the compound geometric random sum. We give higher-order expansions of the ruin probability with a certain validity.

This allows us to evaluate quantile of the ruin function, which is nicely applied to estimate the VaR-type risk measure due to ruin.

[ Reference URL ]An asymptotic expansion formula of the ultimate ruin probability under L\\'evy insurance risks

is given as the loading factor tends to zero. The formula is obtained via the Edgeworth type expansion of

the compound geometric random sum. We give higher-order expansions of the ruin probability with a certain validity.

This allows us to evaluate quantile of the ruin function, which is nicely applied to estimate the VaR-type risk measure due to ruin.

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2012/09.html

### 2012/10/18

#### Seminar on Probability and Statistics

15:15-16:25 Room #006 (Graduate School of Math. Sci. Bldg.)

Quasi-Bayesian analysis of nonparametric instrumental variables models (JAPANESE)

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2012/08.html

**KATO, Kengo**(Department of Mathematics, Graduate School of Science, Hiroshima University)Quasi-Bayesian analysis of nonparametric instrumental variables models (JAPANESE)

[ Abstract ]

This paper aims at developing a quasi-Bayesian analysis

of the nonparametric instrumental variables model, with a focus on the

asymptotic properties of quasi-posterior distributions. In this paper,

instead of assuming a distributional assumption on the data generating

process, we consider a quasi-likelihood induced from the conditional

moment restriction, and put priors on the function-valued parameter.

We call the resulting posterior quasi-posterior, which corresponds to

``Gibbs posterior'' in the literature. Here we shall focus on sieve

priors, which are priors that concentrate on finite dimensional sieve

spaces. The dimension of the sieve space should increase as the sample

size. We derive rates of contraction and a non-parametric Bernstein-von

Mises type result for the quasi-posterior distribution, and rates of

convergence for the quasi-Bayes estimator defined by the posterior

expectation. We show that, with priors suitably chosen, the

quasi-posterior distribution (the quasi-Bayes estimator) attains the

minimax optimal rate of contraction (convergence, respectively). These

results greatly sharpen the previous related work.

[ Reference URL ]This paper aims at developing a quasi-Bayesian analysis

of the nonparametric instrumental variables model, with a focus on the

asymptotic properties of quasi-posterior distributions. In this paper,

instead of assuming a distributional assumption on the data generating

process, we consider a quasi-likelihood induced from the conditional

moment restriction, and put priors on the function-valued parameter.

We call the resulting posterior quasi-posterior, which corresponds to

``Gibbs posterior'' in the literature. Here we shall focus on sieve

priors, which are priors that concentrate on finite dimensional sieve

spaces. The dimension of the sieve space should increase as the sample

size. We derive rates of contraction and a non-parametric Bernstein-von

Mises type result for the quasi-posterior distribution, and rates of

convergence for the quasi-Bayes estimator defined by the posterior

expectation. We show that, with priors suitably chosen, the

quasi-posterior distribution (the quasi-Bayes estimator) attains the

minimax optimal rate of contraction (convergence, respectively). These

results greatly sharpen the previous related work.

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2012/08.html

#### Lectures

16:00-17:00 Room #470 (Graduate School of Math. Sci. Bldg.)

Multidimensional ill-posed problems (ENGLISH)

**Anatoly Yagola**(Lomonosov Moscow State University)Multidimensional ill-posed problems (ENGLISH)

[ Abstract ]

It is very important now to develop methods of solving multidimensional ill-posed problems using regularization procedures and parallel computers. The main purpose of the talk is to show how 2D and 3D Fredholm integral equations of the 1st kind can be effectively solved.

We will consider ill-posed problems on compact sets of convex functions [1] and functions convex along lines parallel to coordinate axes [2].

Recovery of magnetic target parameters from magnetic sensor measurements has attracted wide interests and found many practical applications. However, difficulties present in identifying the permanent magnetization due to the complications of magnetization distributions over the ship body, and errors and noises of measurement data degrade the accuracy and quality of the parameter identification. In this paper, we use a two step sequential solutions to solve the inversion problem. In the first step, a numerical model is built and used to determine the induced magnetization of the ship. In the second step, we solve a type of continuous magnetization inversion problem by solving 2D and 3D Fredholm integral

equations of the 1st kind. We use parallel computing which allows solve the inverse problem with high accuracy. Tikhonov regularization has been applied in solving the inversion problems. The proposed methods have been validated using simulation data with added noises [4, 6].

2D and 3D inverse problems also could be found in tomography [3] and electron microscopy [5]. We will demonstrate examples of applied problems and discuss methods of numerical solving.

This paper was supported by the Visby program and RFBR grants 11-01-00040–а and 12-01-91153-NSFC-a.

It is very important now to develop methods of solving multidimensional ill-posed problems using regularization procedures and parallel computers. The main purpose of the talk is to show how 2D and 3D Fredholm integral equations of the 1st kind can be effectively solved.

We will consider ill-posed problems on compact sets of convex functions [1] and functions convex along lines parallel to coordinate axes [2].

Recovery of magnetic target parameters from magnetic sensor measurements has attracted wide interests and found many practical applications. However, difficulties present in identifying the permanent magnetization due to the complications of magnetization distributions over the ship body, and errors and noises of measurement data degrade the accuracy and quality of the parameter identification. In this paper, we use a two step sequential solutions to solve the inversion problem. In the first step, a numerical model is built and used to determine the induced magnetization of the ship. In the second step, we solve a type of continuous magnetization inversion problem by solving 2D and 3D Fredholm integral

equations of the 1st kind. We use parallel computing which allows solve the inverse problem with high accuracy. Tikhonov regularization has been applied in solving the inversion problems. The proposed methods have been validated using simulation data with added noises [4, 6].

2D and 3D inverse problems also could be found in tomography [3] and electron microscopy [5]. We will demonstrate examples of applied problems and discuss methods of numerical solving.

This paper was supported by the Visby program and RFBR grants 11-01-00040–а and 12-01-91153-NSFC-a.

### 2012/10/17

#### Geometry Colloquium

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

p-Kazhdan constants and non-expanders (JAPANESE)

**Masato Mimura**(Tohoku University)p-Kazhdan constants and non-expanders (JAPANESE)

[ Abstract ]

In study of graphs and finitely generated groups (as Cayley graphs) as metric spaces with the path metrics, one basic idea is to "linearize" them, more precisely, to embed them into certain Banach spaces in some nice way. Special attention has been paid to embeddings of graphs into Hilbert spaces or l^p spaces. It is a well-known result that a "family of expanders", namely, a family of finite graphs (of unifromly bounded degree) with uniform lower bound of spectral gaps (equivalently, of Cheeger constants), does not coarsely embed into Hilbert spaces, or l^p spaces.

In this talk, we investigate a "family of NON-expanders" coming from Cayley graphs of a family of finitely generated groups. In this setting we define l^p-version of the Kazhdan constant and of the property tau constant for groups, and study the decay rate of p-spectral gap of non-expanders in terms of them. This gives some metric geometrical information on the family. Our main example will be the family of (Cayley graphs of SL_n(Z/k_nZ)), indexed by n>2, for (k_n)_n a sequence of natural numbers>2 and with respect to standard 4-element generating sets. We will start from basic definitions, such as ones of Cayley graphs, expander families, and Kazhdan constants.

In study of graphs and finitely generated groups (as Cayley graphs) as metric spaces with the path metrics, one basic idea is to "linearize" them, more precisely, to embed them into certain Banach spaces in some nice way. Special attention has been paid to embeddings of graphs into Hilbert spaces or l^p spaces. It is a well-known result that a "family of expanders", namely, a family of finite graphs (of unifromly bounded degree) with uniform lower bound of spectral gaps (equivalently, of Cheeger constants), does not coarsely embed into Hilbert spaces, or l^p spaces.

In this talk, we investigate a "family of NON-expanders" coming from Cayley graphs of a family of finitely generated groups. In this setting we define l^p-version of the Kazhdan constant and of the property tau constant for groups, and study the decay rate of p-spectral gap of non-expanders in terms of them. This gives some metric geometrical information on the family. Our main example will be the family of (Cayley graphs of SL_n(Z/k_nZ)), indexed by n>2, for (k_n)_n a sequence of natural numbers>2 and with respect to standard 4-element generating sets. We will start from basic definitions, such as ones of Cayley graphs, expander families, and Kazhdan constants.

#### Lectures

15:00-16:00 Room #370 (Graduate School of Math. Sci. Bldg.)

Inverse problems for differential operators on spatial networks (ENGLISH)

**Vjacheslav Yurko**(Saratov University)Inverse problems for differential operators on spatial networks (ENGLISH)

### 2012/10/16

#### Tuesday Seminar on Topology

17:10-18:10 Room #056 (Graduate School of Math. Sci. Bldg.)

Analytic torsion of log-Enriques surfaces (JAPANESE)

**Ken-Ichi Yoshikawa**(Kyoto University)Analytic torsion of log-Enriques surfaces (JAPANESE)

[ Abstract ]

Log-Enriques surfaces are rational surfaces with nowhere vanishing

pluri-canonical forms. We report the recent progress on the computation

of analytic torsion of log-Enriques surfaces.

Log-Enriques surfaces are rational surfaces with nowhere vanishing

pluri-canonical forms. We report the recent progress on the computation

of analytic torsion of log-Enriques surfaces.

### 2012/10/15

#### Seminar on Geometric Complex Analysis

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

An L^2 estimate on domains and application to Levi-flat surfaces (JAPANESE)

**Takeo Ohsawa**(Nagoya University)An L^2 estimate on domains and application to Levi-flat surfaces (JAPANESE)

#### Algebraic Geometry Seminar

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

On the moduli b-divisors of lc-trivial fibrations (JAPANESE)

**Yoshinori Gongyo**(University of Tokyo)On the moduli b-divisors of lc-trivial fibrations (JAPANESE)

[ Abstract ]

Roughly speaking, by using the semi-stable minimal model program, we prove that the moduli part of an lc-trivial fibration coincides with that of a klt-trivial fibration induced by adjunction after taking a suitable generically finite cover. As an application, we obtain that the moduli part of an lc-trivial fibration is b-nef and abundant by Ambro's result on klt-trivial fibrations. Moreover I may explain some applications of canonical bundle formulas. These are joint works with Osamu Fujino.

Roughly speaking, by using the semi-stable minimal model program, we prove that the moduli part of an lc-trivial fibration coincides with that of a klt-trivial fibration induced by adjunction after taking a suitable generically finite cover. As an application, we obtain that the moduli part of an lc-trivial fibration is b-nef and abundant by Ambro's result on klt-trivial fibrations. Moreover I may explain some applications of canonical bundle formulas. These are joint works with Osamu Fujino.

### 2012/10/12

#### Colloquium

16:30-17:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Homogenization on arbitrary manifolds (ENGLISH)

**Antonio Siconolfi**(La Sapienza - University of Rome)Homogenization on arbitrary manifolds (ENGLISH)

[ Abstract ]

We show that results on periodic homogenization for Hamilton-Jacobi equations can be generalized replacing the torus by an arbitrary compact manifold. This allows to reach a deeper understanding of the matter and unveils phenomena somehow hidden in the periodic case, for instance the fact that the ambient spaces of oscillating equations and that of the limit problem are different, and possess even different dimensions. Repetition structure for the base manifold, changes of scale in it and asymptotic analysis, which are the basic ingredients of homogenization, need substantial modification to work in the new frame, and this task is partially accomplished using tools from algebraic topology. An adapted notion of convergence allowing approximating entities and limit to lie in different spaces is also provided.

We show that results on periodic homogenization for Hamilton-Jacobi equations can be generalized replacing the torus by an arbitrary compact manifold. This allows to reach a deeper understanding of the matter and unveils phenomena somehow hidden in the periodic case, for instance the fact that the ambient spaces of oscillating equations and that of the limit problem are different, and possess even different dimensions. Repetition structure for the base manifold, changes of scale in it and asymptotic analysis, which are the basic ingredients of homogenization, need substantial modification to work in the new frame, and this task is partially accomplished using tools from algebraic topology. An adapted notion of convergence allowing approximating entities and limit to lie in different spaces is also provided.

### 2012/10/09

#### Tuesday Seminar on Topology

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

The growth series of pure Artin groups of dihedral type (JAPANESE)

**Michihiko Fujii**(Kyoto University)The growth series of pure Artin groups of dihedral type (JAPANESE)

[ Abstract ]

In this talk, I consider the kernel of the natural projection from

the Artin group of dihedral type to the corresponding Coxeter group,

that we call a pure Artin group of dihedral type,

and present rational function expressions for both the spherical and

geodesic growth series

of the pure Artin group of dihedral type with respect to a natural

generating set.

Also, I show that their growth rates are Pisot numbers.

This talk is partially based on a joint work with Takao Satoh.

In this talk, I consider the kernel of the natural projection from

the Artin group of dihedral type to the corresponding Coxeter group,

that we call a pure Artin group of dihedral type,

and present rational function expressions for both the spherical and

geodesic growth series

of the pure Artin group of dihedral type with respect to a natural

generating set.

Also, I show that their growth rates are Pisot numbers.

This talk is partially based on a joint work with Takao Satoh.

< Previous 123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143 Next >