## Seminar information archive

Seminar information archive ～08/21｜Today's seminar 08/22 | Future seminars 08/23～

#### Lectures

15:30-16:15 Room #002 (Graduate School of Math. Sci. Bldg.)

Gelfand type problem for two phase porous media (ENGLISH)

http://www.ms.u-tokyo.ac.jp/gcoe/index_007.html

**Peter Gordon**(Akron University)Gelfand type problem for two phase porous media (ENGLISH)

[ Abstract ]

In this talk I will introduce a generalization of well known Gelfand problem arising in a Frank-Kamenetskii theory of thermal explosion. This generalization is a natural extension of the Gelfand problem to two phase materials, where, in contrast to classical Gelfand problem which utilizes single temperature approach, the state of the system is described by two different temperatures. As a result the problem is modeled by a system of two coupled nonlinear heat equations. The new ingredient in such a generalized Gelfand problem is a presence of inter-phase heat exchange which can be viewed as a strength of coupling for the system.

I will show that similar to classical Gelfand problem the thermal explosion (blow up of solution) for generalized Gelfand problem occurs exclusively due to the absence of stationary temperature distribution, that is non-existence of solution of corresponding elliptic problem. I also will show that the presence of inter-phase heat exchange delays a thermal explosion. Moreover, in the limit of infinite heat exchange between phases the problem of thermal explosion in two phase porous media reduces to classical Gelfand problem with re-normalized constants. The latter result partially justifies a single temperature approach to two phase systems often used in a physical literature.

This is a joint work with Vitaly Moroz (Swansea University).

[ Reference URL ]In this talk I will introduce a generalization of well known Gelfand problem arising in a Frank-Kamenetskii theory of thermal explosion. This generalization is a natural extension of the Gelfand problem to two phase materials, where, in contrast to classical Gelfand problem which utilizes single temperature approach, the state of the system is described by two different temperatures. As a result the problem is modeled by a system of two coupled nonlinear heat equations. The new ingredient in such a generalized Gelfand problem is a presence of inter-phase heat exchange which can be viewed as a strength of coupling for the system.

I will show that similar to classical Gelfand problem the thermal explosion (blow up of solution) for generalized Gelfand problem occurs exclusively due to the absence of stationary temperature distribution, that is non-existence of solution of corresponding elliptic problem. I also will show that the presence of inter-phase heat exchange delays a thermal explosion. Moreover, in the limit of infinite heat exchange between phases the problem of thermal explosion in two phase porous media reduces to classical Gelfand problem with re-normalized constants. The latter result partially justifies a single temperature approach to two phase systems often used in a physical literature.

This is a joint work with Vitaly Moroz (Swansea University).

http://www.ms.u-tokyo.ac.jp/gcoe/index_007.html

#### Lectures

16:25-17:10 Room #002 (Graduate School of Math. Sci. Bldg.)

On the shape of charged drops: an isoperimetric problem with a competing non-local term (ENGLISH)

http://www.ms.u-tokyo.ac.jp/gcoe/index_007.html

**Cyrill Muratov**(New Jersey Institute of Technology)On the shape of charged drops: an isoperimetric problem with a competing non-local term (ENGLISH)

[ Abstract ]

In this talk I will give an overview of my recent work with H. Knuepfer on the analysis of a class of geometric problems in the calculus of variations. I will discuss the basic questions of existence and non-existence of energy minimizers for the isoperimetric problem with a competing non-local term. A complete answer will be given for the case of slowly decaying kernels in two space dimensions, and qualitative properties of the minimizers will be established for general Riesz kernels.

[ Reference URL ]In this talk I will give an overview of my recent work with H. Knuepfer on the analysis of a class of geometric problems in the calculus of variations. I will discuss the basic questions of existence and non-existence of energy minimizers for the isoperimetric problem with a competing non-local term. A complete answer will be given for the case of slowly decaying kernels in two space dimensions, and qualitative properties of the minimizers will be established for general Riesz kernels.

http://www.ms.u-tokyo.ac.jp/gcoe/index_007.html

### 2012/11/21

#### PDE Real Analysis Seminar

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Shape Optimization And Asymptotic For The Twisted Dirichlet Eigenvalue (ENGLISH)

**Giovanni Pisante**(Seconda Università degli Studi di Napoli)Shape Optimization And Asymptotic For The Twisted Dirichlet Eigenvalue (ENGLISH)

[ Abstract ]

Aim of the talk is to discuss some recent results obtained with G. Croce and A. Henrot on a generalization of the functional defining the first twisted eigenvalue.

We look at the set functional defined by minimizing a Rayleigh quotient involving Lebesgue norms with different exponents p and q among functions satisfying a zero boundary condition as well as a zero mean condition of order q.

First under suitable conditions on p and q, that ensure the existence of a minimizing function, we investigate the validity of an isoperimetric type inequality of the Reyleigh-Faber-Krahn type.

Then we study the limit of the functional for p and q tending to 1 and to infinity and discuss the relation with the limits of the second eigenvalues of the p-laplacian operator.

Aim of the talk is to discuss some recent results obtained with G. Croce and A. Henrot on a generalization of the functional defining the first twisted eigenvalue.

We look at the set functional defined by minimizing a Rayleigh quotient involving Lebesgue norms with different exponents p and q among functions satisfying a zero boundary condition as well as a zero mean condition of order q.

First under suitable conditions on p and q, that ensure the existence of a minimizing function, we investigate the validity of an isoperimetric type inequality of the Reyleigh-Faber-Krahn type.

Then we study the limit of the functional for p and q tending to 1 and to infinity and discuss the relation with the limits of the second eigenvalues of the p-laplacian operator.

#### Classical Analysis

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

Beyond the fundamental group (ENGLISH)

**Philip Boalch**(ENS-DMA & CNRS Paris)Beyond the fundamental group (ENGLISH)

[ Abstract ]

Moduli spaces of representations of the fundamental group of a Riemann surface have been studied from numerous points of view and appear in many parts of mathematics and theoretical physics. They form an interesting class of symplectic manifolds, they often have Kahler or hyperkahler metrics (in which case they are diffeomorphic to spaces of Higgs bundles, i.e. Hitchin integrable systems), and they admit nonlinear actions of braid groups and mapping class groups with fascinating dynamical properties. The aim of this talk is to describe some aspects of this story and sketch their extension to the context of the "wild fundamental group", which naturally appears when one considers {\\em meromorphic} connections on Riemann surfaces. In particular some new examples of hyperkahler manifolds appear in this way, some of which are familiar from classical work on the Painleve equations.

Moduli spaces of representations of the fundamental group of a Riemann surface have been studied from numerous points of view and appear in many parts of mathematics and theoretical physics. They form an interesting class of symplectic manifolds, they often have Kahler or hyperkahler metrics (in which case they are diffeomorphic to spaces of Higgs bundles, i.e. Hitchin integrable systems), and they admit nonlinear actions of braid groups and mapping class groups with fascinating dynamical properties. The aim of this talk is to describe some aspects of this story and sketch their extension to the context of the "wild fundamental group", which naturally appears when one considers {\\em meromorphic} connections on Riemann surfaces. In particular some new examples of hyperkahler manifolds appear in this way, some of which are familiar from classical work on the Painleve equations.

### 2012/11/20

#### Tuesday Seminar on Topology

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

3 dimensional hyperbolic geometry and cluster algebras (JAPANESE)

**Kentaro Nagao**(Nagoya University)3 dimensional hyperbolic geometry and cluster algebras (JAPANESE)

[ Abstract ]

The cluster algebra was discovered by Fomin-Zelevinsky in 2000.

Recently, the structures of cluster algebras are recovered in

many areas including the theory of quantum groups, low

dimensional topology, discrete integrable systems, Donaldson-Thomas

theory, and string theory and there is dynamic development in the

research on these subjects. In this talk I introduce a relation between

3 dimensional hyperbolic geometry and cluster algebras motivated

by some duality in string theory.

The cluster algebra was discovered by Fomin-Zelevinsky in 2000.

Recently, the structures of cluster algebras are recovered in

many areas including the theory of quantum groups, low

dimensional topology, discrete integrable systems, Donaldson-Thomas

theory, and string theory and there is dynamic development in the

research on these subjects. In this talk I introduce a relation between

3 dimensional hyperbolic geometry and cluster algebras motivated

by some duality in string theory.

#### Lie Groups and Representation Theory

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

On the geometry of discontinuous subgroups acting on some homogeneous spaces (ENGLISH)

**Ali Baklouti**(Sfax University)On the geometry of discontinuous subgroups acting on some homogeneous spaces (ENGLISH)

[ Abstract ]

Let $G$ be a Lie group, $H$ a closed subgroup of $G$ and \\Gamma$ a discontinuous subgroup for the homogeneous space $G/H$. I first introduce the deformation space ${\\mathcal{T}}^{K_o}(\\Gamma, G, H)$ of the action of $\\Gamma$ on $G/H$ in the sense of Kobayashi and some of its refined versions, namely the Clifford--Klein space of deformations of the form ${\\mathcal{X}}=\\Gamma \\backslash G/H$. The deformation space ${\\mathcal{T}}^{G_o}(\\Gamma, G,H)$ of marked $(G,H)$-structures on ${\\mathcal{X}}$ in the sense of Goldman is also introduced. As an important motivation, I will explain the connection between the spaces ${\\mathcal{T}}^{K_o}(\\Gamma, G, H)$ and ${\\mathcal{T}}^{G_o}(\\Gamma, G, H)$ and study some of their topological features, namely the rigidity in the sense of Selberg--Weil--Kobayashi and the stability in the sense of Kobayashi--Nasrin. The latter appears to be of major interest to write down the connection explicitly.

Let $G$ be a Lie group, $H$ a closed subgroup of $G$ and \\Gamma$ a discontinuous subgroup for the homogeneous space $G/H$. I first introduce the deformation space ${\\mathcal{T}}^{K_o}(\\Gamma, G, H)$ of the action of $\\Gamma$ on $G/H$ in the sense of Kobayashi and some of its refined versions, namely the Clifford--Klein space of deformations of the form ${\\mathcal{X}}=\\Gamma \\backslash G/H$. The deformation space ${\\mathcal{T}}^{G_o}(\\Gamma, G,H)$ of marked $(G,H)$-structures on ${\\mathcal{X}}$ in the sense of Goldman is also introduced. As an important motivation, I will explain the connection between the spaces ${\\mathcal{T}}^{K_o}(\\Gamma, G, H)$ and ${\\mathcal{T}}^{G_o}(\\Gamma, G, H)$ and study some of their topological features, namely the rigidity in the sense of Selberg--Weil--Kobayashi and the stability in the sense of Kobayashi--Nasrin. The latter appears to be of major interest to write down the connection explicitly.

### 2012/11/19

#### Lectures

16:45-17:45 Room #126 (Graduate School of Math. Sci. Bldg.)

Invariant measure of the stochastic Allen-Cahn equation: the regime of small noise and large system size (ENGLISH)

**Hendrik Weber**(University of Warwick)Invariant measure of the stochastic Allen-Cahn equation: the regime of small noise and large system size (ENGLISH)

[ Abstract ]

We study the invariant measure of the one-dimensional stochastic Allen-Cahn equation for a small noise strength and a large but finite system. We endow the system with inhomogeneous Dirichlet boundary conditions that enforce at least one transition from -1 to 1. We are interested in the competition between the ``energy'' that should be minimized due to the small noise strength and the ``entropy'' that is induced by the large system size.

Our methods handle system sizes that are exponential with respect to the inverse noise strength, up to the ``critical'' exponential size predicted by the heuristics. We capture the competition between energy and entropy through upper and lower bounds on the probability of extra transitions between $\\pm 1$. These bounds are sharp on the exponential scale and imply in particular that the probability of having one and only one transition from -1 to +1 is exponentially close to one. In addition, we show that the position of the transition layer is uniformly distributed over the system on scales larger than the logarithm of the inverse noise strength.

Our arguments rely on local large deviation bounds, the strong Markov property, the symmetry of the potential, and measure-preserving reflections.

This is a joint work with Felix Otto and Maria Westdickenberg.

We study the invariant measure of the one-dimensional stochastic Allen-Cahn equation for a small noise strength and a large but finite system. We endow the system with inhomogeneous Dirichlet boundary conditions that enforce at least one transition from -1 to 1. We are interested in the competition between the ``energy'' that should be minimized due to the small noise strength and the ``entropy'' that is induced by the large system size.

Our methods handle system sizes that are exponential with respect to the inverse noise strength, up to the ``critical'' exponential size predicted by the heuristics. We capture the competition between energy and entropy through upper and lower bounds on the probability of extra transitions between $\\pm 1$. These bounds are sharp on the exponential scale and imply in particular that the probability of having one and only one transition from -1 to +1 is exponentially close to one. In addition, we show that the position of the transition layer is uniformly distributed over the system on scales larger than the logarithm of the inverse noise strength.

Our arguments rely on local large deviation bounds, the strong Markov property, the symmetry of the potential, and measure-preserving reflections.

This is a joint work with Felix Otto and Maria Westdickenberg.

#### Algebraic Geometry Seminar

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

Stability conditions and birational geometry (JAPANESE)

**Yukinobu Toda**(IPMU)Stability conditions and birational geometry (JAPANESE)

[ Abstract ]

I propose a conjecture which claims that MMP for a smooth projective variety is realized as a variation of Bridgeland moduli spaces of semistable objects in the derived category of coherent sheaves. I will discuss the surface case and extremal contractions for 3-folds. In the former case, the conjecture is completely solved. In the latter case, I will construct the perverse t-structure associated to the extremal contraction, and construct a candidate of the desired stability condition as a double tilting of the perverse heart.

I propose a conjecture which claims that MMP for a smooth projective variety is realized as a variation of Bridgeland moduli spaces of semistable objects in the derived category of coherent sheaves. I will discuss the surface case and extremal contractions for 3-folds. In the former case, the conjecture is completely solved. In the latter case, I will construct the perverse t-structure associated to the extremal contraction, and construct a candidate of the desired stability condition as a double tilting of the perverse heart.

#### GCOE Seminars

15:30-17:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Linearisable mappings (ENGLISH)

**Alfred Ramani**(Ecole Polytechnique)Linearisable mappings (ENGLISH)

[ Abstract ]

We present a series of results on linearisable second-order mappings.

Three distinct families of such mappings do exist: projective, mappings of Gambier type and mappings which we have dubbed "of third kind".

Our starting point are the linearisable mappings belonging to the QRT family. We show how they can be linearised and how in some cases their explicit solution can be constructed. We discuss also the growth property of these mappings, a property intimately related to linearisability.

In the second part of the talk we address the question of the extension of these mappings to a non-autonomous form.

We show that the QRT invariant can also be extended (to a quantity which depends explicitly on the independent variable). Using this non-autonomous form we show that it is possible to construct the explicit solution of all third-kind mappings. We discuss also the relation of mappings of the third kind to Gambier-type mappings. We show that a large subclass of third-kind mappings can be considered as the discrete derivative of Gambier-type ones.

We present a series of results on linearisable second-order mappings.

Three distinct families of such mappings do exist: projective, mappings of Gambier type and mappings which we have dubbed "of third kind".

Our starting point are the linearisable mappings belonging to the QRT family. We show how they can be linearised and how in some cases their explicit solution can be constructed. We discuss also the growth property of these mappings, a property intimately related to linearisability.

In the second part of the talk we address the question of the extension of these mappings to a non-autonomous form.

We show that the QRT invariant can also be extended (to a quantity which depends explicitly on the independent variable). Using this non-autonomous form we show that it is possible to construct the explicit solution of all third-kind mappings. We discuss also the relation of mappings of the third kind to Gambier-type mappings. We show that a large subclass of third-kind mappings can be considered as the discrete derivative of Gambier-type ones.

#### Seminar on Geometric Complex Analysis

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

On a degenerate family of Riemann surfaces of genus two over an elliptic curve (JAPANESE)

**Yohei Komori**(Waseda University)On a degenerate family of Riemann surfaces of genus two over an elliptic curve (JAPANESE)

[ Abstract ]

We construct a degenerate family of Riemann surfaces of genus two constructed as double branched covering surfaces of a fixed torus. We determine its singular fibers and holomorphic sections.

We construct a degenerate family of Riemann surfaces of genus two constructed as double branched covering surfaces of a fixed torus. We determine its singular fibers and holomorphic sections.

### 2012/11/16

#### Colloquium

16:30-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Integral invariants in complex differential geometry (JAPANESE)

**Akito FUTAKI**(University of Tokyo)Integral invariants in complex differential geometry (JAPANESE)

#### GCOE Seminars

16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Subfactors with small Jones index (ENGLISH)

**Dietmar Bisch**(Vanderbilt University)Subfactors with small Jones index (ENGLISH)

### 2012/11/14

#### Number Theory Seminar

18:00-19:00 Room #002 (Graduate School of Math. Sci. Bldg.)

De Rham-Witt complexes with coefficients and rigid cohomology

(ENGLISH)

**Pierre Berthelot**(Université de Rennes 1)De Rham-Witt complexes with coefficients and rigid cohomology

(ENGLISH)

[ Abstract ]

For a smooth scheme over a perfect field of characteristic p, we will explain a generalization of the classical comparison theorem between crystalline cohomology and de Rham-Witt cohomology to the case of cohomologies with coefficients in a p-torsion free crystal. This provides in particular a description of the rigid cohomology of a proper singular scheme in terms of a de Rham-Witt complex built from a closed immersion into a smooth scheme.

For a smooth scheme over a perfect field of characteristic p, we will explain a generalization of the classical comparison theorem between crystalline cohomology and de Rham-Witt cohomology to the case of cohomologies with coefficients in a p-torsion free crystal. This provides in particular a description of the rigid cohomology of a proper singular scheme in terms of a de Rham-Witt complex built from a closed immersion into a smooth scheme.

#### Geometry Colloquium

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

Construction of coassociative submanifolds (JAPANESE)

**Kotaro Kawai**(Tohoku University)Construction of coassociative submanifolds (JAPANESE)

[ Abstract ]

The notion of coassociative submanifolds is defined as the special class of the minimal submanifolds in G_2 manifolds. In this talk, we introduce the method to construct coassociative submanifolds by using the symmetry of the Lie group action. As an application, we give explicit examples in the 7-dimensional Euclidean space and in the anti-self-dual bundle over the 4-sphere.

The notion of coassociative submanifolds is defined as the special class of the minimal submanifolds in G_2 manifolds. In this talk, we introduce the method to construct coassociative submanifolds by using the symmetry of the Lie group action. As an application, we give explicit examples in the 7-dimensional Euclidean space and in the anti-self-dual bundle over the 4-sphere.

### 2012/11/13

#### Tuesday Seminar on Topology

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

The virtual fibering theorem and sutured manifold hierarchies (JAPANESE)

**Takahiro Kitayama**(RIMS, Kyoto University,JSPS PD)The virtual fibering theorem and sutured manifold hierarchies (JAPANESE)

[ Abstract ]

In 2007 Agol showed that every irreducible 3-manifold whose fundamental

group is nontrivial and virtually residually finite rationally solvable

(RFRS) is virtually fibered. In the proof he used the theory of

least-weight taut normal surfaces introduced and developed by Oertel and

Tollefson-Wang. We give another proof using complexities of sutured

manifolds. This is a joint work with Stefan Friedl (University of

Cologne).

In 2007 Agol showed that every irreducible 3-manifold whose fundamental

group is nontrivial and virtually residually finite rationally solvable

(RFRS) is virtually fibered. In the proof he used the theory of

least-weight taut normal surfaces introduced and developed by Oertel and

Tollefson-Wang. We give another proof using complexities of sutured

manifolds. This is a joint work with Stefan Friedl (University of

Cologne).

#### 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)

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