## Seminar information archive

Seminar information archive ～08/07｜Today's seminar 08/08 | Future seminars 08/09～

#### Lie Groups and Representation Theory

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

Tight maps, a classification (ENGLISH)

**Oskar Hamlet**(Chalmers University)Tight maps, a classification (ENGLISH)

[ Abstract ]

Tight maps/homomorphisms were introduced during the study of rigidity properties of surface groups in Hermitian Lie groups. In this talk I'll discuss the properties of tight maps, their connection to rigidity theory and my work classifying them.

Tight maps/homomorphisms were introduced during the study of rigidity properties of surface groups in Hermitian Lie groups. In this talk I'll discuss the properties of tight maps, their connection to rigidity theory and my work classifying them.

### 2012/11/12

#### Algebraic Geometry Seminar

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

On Fano fourfolds with nef vector bundles $Λ^2T_X$ (JAPANESE)

**Kazunori Yasutake**(Kyushu University)On Fano fourfolds with nef vector bundles $Λ^2T_X$ (JAPANESE)

[ Abstract ]

By using results about extremal contractions on smooth fourfolds, we give a classification of fano fourfolds whose the second exterior power of tangent bundles are numerically effective.

By using results about extremal contractions on smooth fourfolds, we give a classification of fano fourfolds whose the second exterior power of tangent bundles are numerically effective.

#### Seminar on Geometric Complex Analysis

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

Residues of meromorphic differential forms (ENGLISH)

**A.G. Aleksandrov**(Institute of Control Sciences, Russian Acad. of Sci.)Residues of meromorphic differential forms (ENGLISH)

[ Abstract ]

The purpose of the talk is to discuss several interesting aspects

of the classical residue theory originated by H. Poincar\\'e, J. de Rham and J. Leray and their followers. Focus topics of our studies are some of the less known applications, developed by the author in the past few years in complex analysis, topology and geometry of singular varieties and in the theory of differential equations. Almost all considerations are based essentially on properties of a special class of meromorphic differential forms called logarithmic or multi-logarithmic forms.

The purpose of the talk is to discuss several interesting aspects

of the classical residue theory originated by H. Poincar\\'e, J. de Rham and J. Leray and their followers. Focus topics of our studies are some of the less known applications, developed by the author in the past few years in complex analysis, topology and geometry of singular varieties and in the theory of differential equations. Almost all considerations are based essentially on properties of a special class of meromorphic differential forms called logarithmic or multi-logarithmic forms.

### 2012/11/10

#### Harmonic Analysis Komaba Seminar

13:00-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Perturbed Besov spaces by short-range type potential

in exterior domains (JAPANESE)

Optimal constants and extremisers for some smoothing estimates (JAPANESE)

Spectral stability of the p-Laplacian (JAPANESE)

**Tokio Matsuyama**(Chuo University) 13:00-14:20Perturbed Besov spaces by short-range type potential

in exterior domains (JAPANESE)

[ Abstract ]

In this talk we will define perturbed Besov spaces by a short-range potential over exterior domains. These spaces will be available for obtaining the Strichartz estimates of wave equation with a potential in exterior domains.

We will pay attention to observe the equivalence relation between the perturbed Besov spaces and the free ones.

In this talk we will define perturbed Besov spaces by a short-range potential over exterior domains. These spaces will be available for obtaining the Strichartz estimates of wave equation with a potential in exterior domains.

We will pay attention to observe the equivalence relation between the perturbed Besov spaces and the free ones.

**Sugimoto Mitsuru**(Nagoya University) 14:40-16:00Optimal constants and extremisers for some smoothing estimates (JAPANESE)

[ Abstract ]

Our purpose is to study the optimal constant and extremising initial data for a broad class of smoothing estimates for solutions of linear dispersive equations.

Firstly, we discuss the existence/nonexistence of extremisers and then we provide an explicit formula and new observations for the optimal constant.

The talk is based on joint work with Neal Bez (University of Birmingham).

Our purpose is to study the optimal constant and extremising initial data for a broad class of smoothing estimates for solutions of linear dispersive equations.

Firstly, we discuss the existence/nonexistence of extremisers and then we provide an explicit formula and new observations for the optimal constant.

The talk is based on joint work with Neal Bez (University of Birmingham).

**Victor I. Burenkov**(Russia/United Kingdom) 16:30-17:50Spectral stability of the p-Laplacian (JAPANESE)

[ Abstract ]

Dependence of the eigenvalues of the p-Laplacian upon domain perturbation will be under discussion. Namely Lipschitz-type estimates for deviation of the eigenvalues following a domain perturbation will be presented. Such estimates are obtained for the class of open sets admitting open sets with arbitrarily strong degeneration and are expressed in terms of suitable measures of vicinity of two open sets, such as the \\lq\\lq atlas distance" between these sets or the \\lq\\lq lower Hausdor-Pompeiu

deviation" of their boundaries. In the case of open sets with Holder continuous boundaries, our results essentially improve a result known for the rst eigenvalue [2].

Joint work with P. D. Lamberti. The results were recently published in [1].

Supported by the grant of RFBR (project 08-01-00443).

References:

[1] V.I. Burenkov, P.D. Lamberti, Spectral stability of the p-Laplacian, Nonlinear Analysis, 71, 2009, 2227-2235.

[2] J. Fleckinger, E.M. Harrell and F. de Thelin, Boundary behaviour and estimates for solutions for equations containing the p-Laplacian, Electronic Journal of Dierential Equations, 38, 1999, 1-19.

Dependence of the eigenvalues of the p-Laplacian upon domain perturbation will be under discussion. Namely Lipschitz-type estimates for deviation of the eigenvalues following a domain perturbation will be presented. Such estimates are obtained for the class of open sets admitting open sets with arbitrarily strong degeneration and are expressed in terms of suitable measures of vicinity of two open sets, such as the \\lq\\lq atlas distance" between these sets or the \\lq\\lq lower Hausdor-Pompeiu

deviation" of their boundaries. In the case of open sets with Holder continuous boundaries, our results essentially improve a result known for the rst eigenvalue [2].

Joint work with P. D. Lamberti. The results were recently published in [1].

Supported by the grant of RFBR (project 08-01-00443).

References:

[1] V.I. Burenkov, P.D. Lamberti, Spectral stability of the p-Laplacian, Nonlinear Analysis, 71, 2009, 2227-2235.

[2] J. Fleckinger, E.M. Harrell and F. de Thelin, Boundary behaviour and estimates for solutions for equations containing the p-Laplacian, Electronic Journal of Dierential Equations, 38, 1999, 1-19.

### 2012/11/09

#### 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

### 2012/11/07

#### Classical Analysis

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

WKB analysis of the Painlev\\'e functions and parameteric Stokes phenomena (JAPANESE)

**Kohei IWAKI**(Kyoto University)WKB analysis of the Painlev\\'e functions and parameteric Stokes phenomena (JAPANESE)

### 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.

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