## Seminar information archive

Seminar information archive ～02/26｜Today's seminar 02/27 | Future seminars 02/28～

#### Seminar on Probability and Statistics

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

Effective PCA for high-dimensional, non-Gaussian data under power spiked model (JAPANESE)

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2012/11.html

**YATA, Kazuyoshi**(Institute of Mathematics, University of Tsukuba)Effective PCA for high-dimensional, non-Gaussian data under power spiked model (JAPANESE)

[ Abstract ]

In this talk, we introduce a general spiked model called the power spiked model in high-dimensional settings. We first consider asymptotic properties of the conventional estimator of eigenvalues under the power spiked model. We give several conditions on the dimension $p$, the sample size $n$ and the high-dimensional noise structure in order to hold several consistency properties of the estimator. We show that the estimator is affected by the noise structure, directly, so that the estimator becomes inconsistent for such cases. In order to overcome such difficulties in a high-dimensional situation, we develop new PCAs called the noise-reduction methodology and the cross-data-matrix methodology under the power spiked model. This is a joint work with Prof. Aoshima (University of Tsukuba).

[ Reference URL ]In this talk, we introduce a general spiked model called the power spiked model in high-dimensional settings. We first consider asymptotic properties of the conventional estimator of eigenvalues under the power spiked model. We give several conditions on the dimension $p$, the sample size $n$ and the high-dimensional noise structure in order to hold several consistency properties of the estimator. We show that the estimator is affected by the noise structure, directly, so that the estimator becomes inconsistent for such cases. In order to overcome such difficulties in a high-dimensional situation, we develop new PCAs called the noise-reduction methodology and the cross-data-matrix methodology under the power spiked model. This is a joint work with Prof. Aoshima (University of Tsukuba).

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2012/11.html

### 2012/11/29

#### Lie Groups and Representation Theory

16:30-17:30 Room #122 (Graduate School of Math. Sci. Bldg.)

On a relation between certain character values of symmetric groups (JAPANESE)

**Masaki Watanabe**(the University of Tokyo)On a relation between certain character values of symmetric groups (JAPANESE)

[ Abstract ]

We present a relation of new kind between character values of

symmetric groups which explains a curious phenomenon in character

tables of symmetric groups. Similar relations for characters of

Brauer and walled Brauer algebras and projective characters of

symmetric groups are also presented.

We present a relation of new kind between character values of

symmetric groups which explains a curious phenomenon in character

tables of symmetric groups. Similar relations for characters of

Brauer and walled Brauer algebras and projective characters of

symmetric groups are also presented.

#### GCOE lecture series

10:00-12:10 Room #123 (Graduate School of Math. Sci. Bldg.)

Sobolev maps with values into the circle (ENGLISH)

**Haim Brezis**(Rutgers University / Technion)Sobolev maps with values into the circle (ENGLISH)

[ Abstract ]

Sobolev functions with values into R are very well understood and play an immense role in many branches of Mathematics. By contrast, the theory of Sobolev maps with values into the unit circle is still under construction. Such maps occur e.g. in the asymptotic analysis of the Ginzburg-Landau model. The reason one is interested in Sobolev maps, rather than smooth maps is to allow singularities such as x/|x| in 2D or line singularities 3D which appear in physical problems. Our focus in these lectures is not the Ginzburg-Landau equation per se, but rather the intrinsic study of the function space W^{1,p} of maps from a smooth domain in R^N taking their values into the unit circle. Such classes of maps have an amazingly rich structure. Geometrical and Topological effects are already noticeable in this simple framework, since S^1 has nontrivial topology. Moreover the fact that the target space is the circle (as opposed to higher-dimensional manifolds) offers the option to introduce a lifting. We'll see that "optimal liftings" are in one-to-one correspondence with minimal connections (resp. minimal surfaces) spanned by the topological singularities of u.

I will also discuss the question of uniqueness of lifting . A key ingredient in some of the proofs is a formula (due to myself, Bourgain and Mironescu) which provides an original way of approximating Sobolev norms (or the total variation) by nonlocal functionals. Nonconvex versions of these functionals raise very challenging questions recently tackled together with H.-M. Nguyen. Comparable functionals also occur in Image Processing and suggest exciting interactions with this field.

Sobolev functions with values into R are very well understood and play an immense role in many branches of Mathematics. By contrast, the theory of Sobolev maps with values into the unit circle is still under construction. Such maps occur e.g. in the asymptotic analysis of the Ginzburg-Landau model. The reason one is interested in Sobolev maps, rather than smooth maps is to allow singularities such as x/|x| in 2D or line singularities 3D which appear in physical problems. Our focus in these lectures is not the Ginzburg-Landau equation per se, but rather the intrinsic study of the function space W^{1,p} of maps from a smooth domain in R^N taking their values into the unit circle. Such classes of maps have an amazingly rich structure. Geometrical and Topological effects are already noticeable in this simple framework, since S^1 has nontrivial topology. Moreover the fact that the target space is the circle (as opposed to higher-dimensional manifolds) offers the option to introduce a lifting. We'll see that "optimal liftings" are in one-to-one correspondence with minimal connections (resp. minimal surfaces) spanned by the topological singularities of u.

I will also discuss the question of uniqueness of lifting . A key ingredient in some of the proofs is a formula (due to myself, Bourgain and Mironescu) which provides an original way of approximating Sobolev norms (or the total variation) by nonlocal functionals. Nonconvex versions of these functionals raise very challenging questions recently tackled together with H.-M. Nguyen. Comparable functionals also occur in Image Processing and suggest exciting interactions with this field.

### 2012/11/28

#### Geometry Colloquium

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

Ricci curvature and angles (JAPANESE)

**Shouhei Honda**(Kyushu University)Ricci curvature and angles (JAPANESE)

[ Abstract ]

Let X be the Gromov-Hausdorff limit space of a sequence of Riemannian manifolds with a lower Ricci curvature bound. In this talk we will give the definition of angles between geodesics on X. We apply this to prove there is a weakly twice differentiable structure on X and prove there is a unique Levi-Civita connection allowing us to define the Hessian of a twice differentiable function.

Let X be the Gromov-Hausdorff limit space of a sequence of Riemannian manifolds with a lower Ricci curvature bound. In this talk we will give the definition of angles between geodesics on X. We apply this to prove there is a weakly twice differentiable structure on X and prove there is a unique Levi-Civita connection allowing us to define the Hessian of a twice differentiable function.

#### Operator Algebra Seminars

16:30-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Fundamental group of simple $C^*$-algebras with unique trace (JAPANESE)

**Norio Nawata**(Chiba University)Fundamental group of simple $C^*$-algebras with unique trace (JAPANESE)

#### Lectures

10:45-11:45 Room #002 (Graduate School of Math. Sci. Bldg.)

Pattern formation in the hyperbolic plane (ENGLISH)

**Pascal Chossat**(CNRS / University of Nice)Pattern formation in the hyperbolic plane (ENGLISH)

[ Abstract ]

Initially motivated by a model for the visual perception of textures by the cortex, the problem of pattern formation in the hyperbolic plane, or equivalently the Poincaré disc D, shows some similar but mostly quite different features from the same problem posed on the Euclidean plane. The hyperbolic structure induces a large variety of possible periodic patterns and even the bifurcation of "hyperbolic" traveling waves. We call these patterns "H-planforms". I shall show how H-planforms are determined by the means of equivariant bifurcation theory and Helgason-Fourier analysis in D. However the question of their observability is still open. The talk will be illustrated with pictures of H-planforms that have been computed using non trivial algorithms based on harmonic analysis in D.

Initially motivated by a model for the visual perception of textures by the cortex, the problem of pattern formation in the hyperbolic plane, or equivalently the Poincaré disc D, shows some similar but mostly quite different features from the same problem posed on the Euclidean plane. The hyperbolic structure induces a large variety of possible periodic patterns and even the bifurcation of "hyperbolic" traveling waves. We call these patterns "H-planforms". I shall show how H-planforms are determined by the means of equivariant bifurcation theory and Helgason-Fourier analysis in D. However the question of their observability is still open. The talk will be illustrated with pictures of H-planforms that have been computed using non trivial algorithms based on harmonic analysis in D.

#### GCOE lecture series

13:30-14:30 Room #123 (Graduate School of Math. Sci. Bldg.)

How Poincare became my hero (ENGLISH)

**Haim Brezis**(Rutgers University / Technion)How Poincare became my hero (ENGLISH)

[ Abstract ]

I recently discovered little-known texts of Poincare which include fundamental results on PDEs together with prophetic insights into their future impact on various branches of modern mathematics.

I recently discovered little-known texts of Poincare which include fundamental results on PDEs together with prophetic insights into their future impact on various branches of modern mathematics.

#### GCOE lecture series

14:50-17:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Can you hear the degree of a map from the circle into itself? An intriguing story which is not yet finished (ENGLISH)

**Haim Brezis**(Rutgers University / Technion)Can you hear the degree of a map from the circle into itself? An intriguing story which is not yet finished (ENGLISH)

[ Abstract ]

A few years ago - following a suggestion by I. M. Gelfand - I discovered an intriguing connection between the topological degree of a map from the circle into itself and its Fourier coefficients. This relation is easily justified when the map is smooth. However, the situation turns out to be much more delicate if one assumes only continuity, or even Holder continuity. I will present recent developments and open problems. The initial motivation for this direction of research came from the analysis of the Ginzburg-Landau model.

A few years ago - following a suggestion by I. M. Gelfand - I discovered an intriguing connection between the topological degree of a map from the circle into itself and its Fourier coefficients. This relation is easily justified when the map is smooth. However, the situation turns out to be much more delicate if one assumes only continuity, or even Holder continuity. I will present recent developments and open problems. The initial motivation for this direction of research came from the analysis of the Ginzburg-Landau model.

### 2012/11/27

#### Tuesday Seminar on Topology

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

On a finite aspect of characteristic classes of foliations (JAPANESE)

**Hiraku Nozawa**(JSPS-IHES fellow)On a finite aspect of characteristic classes of foliations (JAPANESE)

[ Abstract ]

Characteristic classes of foliations are not bounded due to Thurston.

In this talk, we will explain finiteness of characteristic classes for

foliations with certain transverse structures (e.g. transverse

conformally flat structure) and its relation to unboundedness and

rigidity of foliations.

(This talk is based on a joint work with Jesús Antonio

Álvarez López at University of Santiago de Compostela,

which is available as arXiv:1205.3375.)

Characteristic classes of foliations are not bounded due to Thurston.

In this talk, we will explain finiteness of characteristic classes for

foliations with certain transverse structures (e.g. transverse

conformally flat structure) and its relation to unboundedness and

rigidity of foliations.

(This talk is based on a joint work with Jesús Antonio

Álvarez López at University of Santiago de Compostela,

which is available as arXiv:1205.3375.)

#### Lie Groups and Representation Theory

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

Convergence of Kahler to real polarizations on flag manifolds (JAPANESE)

**Hiroshi Konno**(the University of Tokyo)Convergence of Kahler to real polarizations on flag manifolds (JAPANESE)

[ Abstract ]

In this talk we will discuss geometric quantization of a flag manifold. In particular, we construct a family of complex structures on a flag manifold that converge 'at the quantum level' to the real polarization coming from the Gelfand-Cetlin integrable system.

Our construction is based on a toric degeneration of flag varieties and a deformation of K¥"ahler structure on toric varieties by symplectic potentials.

This is a joint work with Mark Hamilton.

In this talk we will discuss geometric quantization of a flag manifold. In particular, we construct a family of complex structures on a flag manifold that converge 'at the quantum level' to the real polarization coming from the Gelfand-Cetlin integrable system.

Our construction is based on a toric degeneration of flag varieties and a deformation of K¥"ahler structure on toric varieties by symplectic potentials.

This is a joint work with Mark Hamilton.

### 2012/11/26

#### Algebraic Geometry Seminar

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

A configuration of rational curves on the superspecial K3 surface (JAPANESE)

**Toshiyuki Katsura**(Hosei University)A configuration of rational curves on the superspecial K3 surface (JAPANESE)

#### Seminar on Geometric Complex Analysis

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

Quaternionic CR geometry (JAPANESE)

**Shin Nayatani**(Nagoya University)Quaternionic CR geometry (JAPANESE)

### 2012/11/22

#### Lectures

13:30-14:15 Room #002 (Graduate School of Math. Sci. Bldg.)

A multiple scale pattern formation cascade in reaction-diffusion systems of activator-inhibitor type (ENGLISH)

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

**Danielle Hilhorst**(CNRS / Univ. Paris-Sud)A multiple scale pattern formation cascade in reaction-diffusion systems of activator-inhibitor type (ENGLISH)

[ Abstract ]

A family of singular limits of reaction-diffusion systems of activator-inhibitor type in which stable stationary sharp-interface patterns may form is investigated. For concreteness, the analysis is performed for the FitzHugh-Nagumo model on a suitably rescaled bounded domain in $\\R^N$, with $N \\geq 2$. It is proved that when the system is sufficiently close to the limit the dynamics starting from the appropriate smooth initial data breaks down into five distinct stages on well-separated time scales, each of which can be approximated by a suitable reduced problem. The analysis allows to follow fully the progressive refinement of spatiotemporal patterns forming in the systems under consideration and provides a framework for understanding the pattern formation scenarios in a large class of physical, chemical, and biological systems modeled by the class of reaction-diffusion equations, which we consider. This is joint work with Marie Henry and Cyrill Muratov.

[ Reference URL ]A family of singular limits of reaction-diffusion systems of activator-inhibitor type in which stable stationary sharp-interface patterns may form is investigated. For concreteness, the analysis is performed for the FitzHugh-Nagumo model on a suitably rescaled bounded domain in $\\R^N$, with $N \\geq 2$. It is proved that when the system is sufficiently close to the limit the dynamics starting from the appropriate smooth initial data breaks down into five distinct stages on well-separated time scales, each of which can be approximated by a suitable reduced problem. The analysis allows to follow fully the progressive refinement of spatiotemporal patterns forming in the systems under consideration and provides a framework for understanding the pattern formation scenarios in a large class of physical, chemical, and biological systems modeled by the class of reaction-diffusion equations, which we consider. This is joint work with Marie Henry and Cyrill Muratov.

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

#### Lectures

14:25-15:10 Room #002 (Graduate School of Math. Sci. Bldg.)

Formal asymptotic limit of a diffuse interface tumor-growth model (ENGLISH)

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

**Thanh Nam Ngyuen**(University of Paris-Sud)Formal asymptotic limit of a diffuse interface tumor-growth model (ENGLISH)

[ Abstract ]

We consider a diffuse interface tumor-growth model, which has the form of a phase-field system. We discuss the singular limit of this problem. More precisely, we formally prove that as the reaction coefficient tends to zero, the solution converges to the solution of a free boundary problem.

This is a joint work with Danielle Hilhorst, Johannes Kampmann and Kristoffer G. van der Zee.

[ Reference URL ]We consider a diffuse interface tumor-growth model, which has the form of a phase-field system. We discuss the singular limit of this problem. More precisely, we formally prove that as the reaction coefficient tends to zero, the solution converges to the solution of a free boundary problem.

This is a joint work with Danielle Hilhorst, Johannes Kampmann and Kristoffer G. van der Zee.

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

#### Lectures

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

Gelfand type problem for two phase porous media (ENGLISH)

https://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).

https://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)

https://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.

https://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)

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