## Seminar information archive

Seminar information archive ～02/15｜Today's seminar 02/16 | Future seminars 02/17～

#### Lie Groups and Representation Theory

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

Affine Springer fibers of type A and combinatorics of diagonal

coinvariants

Affine Springer fibers of type A and combinatorics of diagonal

coinvariants (JAPANESE)

**Tatsuyuki Hikita**(Kyoto University)Affine Springer fibers of type A and combinatorics of diagonal

coinvariants

Affine Springer fibers of type A and combinatorics of diagonal

coinvariants (JAPANESE)

[ Abstract ]

We introduce certain filtrations on the homology of

affine Springer fibers of type A and give combinatorial formulas for the bigraded Frobenius series of the associated graded modules.

The results are essentially given by generalizations of the symmetric function introduced by Haglund, Haiman, Loehr, Remmel, and Ulyanov which is conjectured to coincide with the bigraded Frobenius series of the ring of diagonal coinvariants.

We introduce certain filtrations on the homology of

affine Springer fibers of type A and give combinatorial formulas for the bigraded Frobenius series of the associated graded modules.

The results are essentially given by generalizations of the symmetric function introduced by Haglund, Haiman, Loehr, Remmel, and Ulyanov which is conjectured to coincide with the bigraded Frobenius series of the ring of diagonal coinvariants.

### 2012/12/10

#### Algebraic Geometry Seminar

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

A hyperbolic metric and stability conditions on K3 surfaces with $¥rho=1$ (JAPANESE)

**Kotaro Kawatani**(Nagoya University)A hyperbolic metric and stability conditions on K3 surfaces with $¥rho=1$ (JAPANESE)

[ Abstract ]

We introduce a hyperbolic metric on the (normalized) space of stability conditions on projective K3 surfaces $X$ with Picard rank 1. Furthermore we demonstrate how this hyperbolic metric is helpful for us by discussing two or three topics.

We introduce a hyperbolic metric on the (normalized) space of stability conditions on projective K3 surfaces $X$ with Picard rank 1. Furthermore we demonstrate how this hyperbolic metric is helpful for us by discussing two or three topics.

#### Seminar on Geometric Complex Analysis

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

A Dirichlet space on ends of tree and Dirichlet forms with a nodewise orthogonal property (JAPANESE)

**Hiroshi Kaneko**(Tokyo University of Science)A Dirichlet space on ends of tree and Dirichlet forms with a nodewise orthogonal property (JAPANESE)

### 2012/12/07

#### Seminar on Probability and Statistics

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

Conditional Independence and Linear Algebra (JAPANESE)

[ Reference URL ]

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2012/12.html

**TANAKA, Kentaro**(Tokyo Institute of Technology)Conditional Independence and Linear Algebra (JAPANESE)

[ Reference URL ]

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2012/12.html

### 2012/12/05

#### PDE Real Analysis Seminar

10:30-11:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Convergence to Equilibrium of Bounded Solutions with Application of Lojasiewicz-Simon's inequality (ENGLISH)

**Jie Jiang**(Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences)Convergence to Equilibrium of Bounded Solutions with Application of Lojasiewicz-Simon's inequality (ENGLISH)

[ Abstract ]

In this talk, we present the application of Lojasiewicz-Simon's inequality to the study on convergence of bounded global solutions to some evolution equations. We take a semi-linear parabolic initial-boundary problem as an example. With the help of Lojasiewicz-Simon's inequality we prove that the bounded global solution will converge to an equilibrium as time goes to infinity provided the nonlinear term is analytic in the unknown function. We also present the application of Lojasiewicz-Simon's inequality to the asymptotic behavior studies on phase-field models with Cattaneo law and chemotaxis models with volume-filling effect.

In this talk, we present the application of Lojasiewicz-Simon's inequality to the study on convergence of bounded global solutions to some evolution equations. We take a semi-linear parabolic initial-boundary problem as an example. With the help of Lojasiewicz-Simon's inequality we prove that the bounded global solution will converge to an equilibrium as time goes to infinity provided the nonlinear term is analytic in the unknown function. We also present the application of Lojasiewicz-Simon's inequality to the asymptotic behavior studies on phase-field models with Cattaneo law and chemotaxis models with volume-filling effect.

#### Operator Algebra Seminars

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

Construction of two dimensional QFT through Longo-Witten endomorphisms (JAPANESE)

**Yoh Tanimoto**(Univ. Goettingen)Construction of two dimensional QFT through Longo-Witten endomorphisms (JAPANESE)

#### Geometry Colloquium

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

Maximal torus actions on complex manifolds (JAPANESE)

**Hiroaki Ishida**(Osaka City University Advanced Mathematical Institute)Maximal torus actions on complex manifolds (JAPANESE)

[ Abstract ]

We say that an effective action of a compact torus $T$ on a connected manifold $M$ is maximal if there is an orbit of dimension $2\\dim T-\\dim M$. In this talk, we give a one-to-one correspondence between the family of connected closed complex manifolds with maximal torus actions and the family of certain combinatorial objects, which is a generalization of the correspondence between complete nonsingular toric varieties and nonsingular complete fans. As an application, we construct a lot of concrete examples of non-K\\"{a}hler manifolds with maximal torus actions.

We say that an effective action of a compact torus $T$ on a connected manifold $M$ is maximal if there is an orbit of dimension $2\\dim T-\\dim M$. In this talk, we give a one-to-one correspondence between the family of connected closed complex manifolds with maximal torus actions and the family of certain combinatorial objects, which is a generalization of the correspondence between complete nonsingular toric varieties and nonsingular complete fans. As an application, we construct a lot of concrete examples of non-K\\"{a}hler manifolds with maximal torus actions.

#### Classical Analysis

16:00-17:30 Room #270 (Graduate School of Math. Sci. Bldg.)

On the Riemann-Hilbert approach to the Malgrange divisor: $P_I^2$ case (ENGLISH)

**Andrei Kapaev**(SISSA, Trieste, Italy)On the Riemann-Hilbert approach to the Malgrange divisor: $P_I^2$ case (ENGLISH)

[ Abstract ]

Equation $P_I^2$ is the second member in the hierarchy of ODEs associated with the classical Painlev\\’e first equation $P_I$ and can be solved via the Riemann-Hilbert (RH) problem approach. It is known also that solutions of equation $P_I^2$ as the functions of $x$ depending on the parameter $t$ can be used to construct a 4-parameter family of isomonodromic solutions to the KdV equation. Given the monodromy data, the set of points $(x,t)$, where the above mentioned RH problem is not solvable, is called the Malgrange divisor. The function $x=a(t)$, which parametrizes locally the Malgrange divisor, satisfies a nonlinear ODE which admits a Lax pair representation and can be also studied using an RH problem. We discuss the relations between these two kinds of the RH problems and the properties of their $t$-large genus 1 asymptotic solutions.

Equation $P_I^2$ is the second member in the hierarchy of ODEs associated with the classical Painlev\\’e first equation $P_I$ and can be solved via the Riemann-Hilbert (RH) problem approach. It is known also that solutions of equation $P_I^2$ as the functions of $x$ depending on the parameter $t$ can be used to construct a 4-parameter family of isomonodromic solutions to the KdV equation. Given the monodromy data, the set of points $(x,t)$, where the above mentioned RH problem is not solvable, is called the Malgrange divisor. The function $x=a(t)$, which parametrizes locally the Malgrange divisor, satisfies a nonlinear ODE which admits a Lax pair representation and can be also studied using an RH problem. We discuss the relations between these two kinds of the RH problems and the properties of their $t$-large genus 1 asymptotic solutions.

### 2012/12/04

#### Numerical Analysis Seminar

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

Tsunami simulation of Hakata Bay using the viscous shallow-water equations (JAPANESE)

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

**Hiroshi Kanayama**(Kyushu University)Tsunami simulation of Hakata Bay using the viscous shallow-water equations (JAPANESE)

[ Abstract ]

The tsunami caused by the great East Japan earthquake gave serious damage in the coastal areas of the Tohoku district. Numerical simulation is used for damage prediction as disaster measures to these tsunami hazards. Generally in the numerical simulation about the tsunami propagation to the coast from an open sea, shallow-water equations are used. This research focuses on viscous shallow-water equations and attempts to generate a computational method using finite element techniques based on the previous investigations of Kanayama and Ohtsuka (1978). First, the viscous shallow-water equation system is derived from the Navier-Stokes equations, based on the assumption of hydrostatic pressure in the direction of gravity. Next the numerical scheme is shown. Then, tsunami simulations of Hakata Bay and Tohoku-Oki are shown using the approach. Finally, a stability condition in L2 sense for the numerical scheme of a linearized viscous shallow-water problem is introduced from Kanayama and Ushijima (1988-1989) and its actual effectiveness is discussed from the view point of practical computation. This presentation will be done in Japanese.

[ Reference URL ]The tsunami caused by the great East Japan earthquake gave serious damage in the coastal areas of the Tohoku district. Numerical simulation is used for damage prediction as disaster measures to these tsunami hazards. Generally in the numerical simulation about the tsunami propagation to the coast from an open sea, shallow-water equations are used. This research focuses on viscous shallow-water equations and attempts to generate a computational method using finite element techniques based on the previous investigations of Kanayama and Ohtsuka (1978). First, the viscous shallow-water equation system is derived from the Navier-Stokes equations, based on the assumption of hydrostatic pressure in the direction of gravity. Next the numerical scheme is shown. Then, tsunami simulations of Hakata Bay and Tohoku-Oki are shown using the approach. Finally, a stability condition in L2 sense for the numerical scheme of a linearized viscous shallow-water problem is introduced from Kanayama and Ushijima (1988-1989) and its actual effectiveness is discussed from the view point of practical computation. This presentation will be done in Japanese.

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

#### Tuesday Seminar on Topology

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

Conformal field theory for C2-cofinite vertex algebras (JAPANESE)

**Yoshitake Hashimoto**(Tokyo City University)Conformal field theory for C2-cofinite vertex algebras (JAPANESE)

[ Abstract ]

This is a jount work with Akihiro Tsuchiya (Kavli IPMU).

We consider sheaves of covacua and conformal blocks over parameter spaces of n-pointed Riemann surfaces

for a vertex algebra of which the category of modules is not necessarily semi-simple.

We assume the C2-cofiniteness condition for vertex algebras.

We define "tensor product" of two modules over a C2-cofinite vertex algebra.

This is a jount work with Akihiro Tsuchiya (Kavli IPMU).

We consider sheaves of covacua and conformal blocks over parameter spaces of n-pointed Riemann surfaces

for a vertex algebra of which the category of modules is not necessarily semi-simple.

We assume the C2-cofiniteness condition for vertex algebras.

We define "tensor product" of two modules over a C2-cofinite vertex algebra.

#### Tuesday Seminar of Analysis

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

Evolution of smooth shapes and the KP hierarchy (ENGLISH)

Group of diffeomorphisms of the unit circle and sub-Riemannian geometry (ENGLISH)

**Alexander Vasiliev**(Department of Mathematics, University of Bergen, Norway) 16:30-17:30Evolution of smooth shapes and the KP hierarchy (ENGLISH)

[ Abstract ]

We consider a homotopic evolution in the space of smooth

shapes starting from the unit circle. Based on the Loewner-Kufarev

equation we give a Hamiltonian formulation of this evolution and

provide conservation laws. The symmetries of the evolution are given

by the Virasoro algebra. The 'positive' Virasoro generators span the

holomorphic part of the complexified vector bundle over the space of

conformal embeddings of the unit disk into the complex plane and

smooth on the boundary. In the covariant formulation they are

conserved along the Hamiltonian flow. The 'negative' Virasoro

generators can be recovered by an iterative method making use of the

canonical Poisson structure. We study an embedding of the

Loewner-Kufarev trajectories into the Segal-Wilson Grassmannian,

construct the tau-function, the Baker-Akhiezer function, and finally,

give a class of solutions to the KP hierarchy, which are invariant on

Loewner-Kufarev trajectories.

We consider a homotopic evolution in the space of smooth

shapes starting from the unit circle. Based on the Loewner-Kufarev

equation we give a Hamiltonian formulation of this evolution and

provide conservation laws. The symmetries of the evolution are given

by the Virasoro algebra. The 'positive' Virasoro generators span the

holomorphic part of the complexified vector bundle over the space of

conformal embeddings of the unit disk into the complex plane and

smooth on the boundary. In the covariant formulation they are

conserved along the Hamiltonian flow. The 'negative' Virasoro

generators can be recovered by an iterative method making use of the

canonical Poisson structure. We study an embedding of the

Loewner-Kufarev trajectories into the Segal-Wilson Grassmannian,

construct the tau-function, the Baker-Akhiezer function, and finally,

give a class of solutions to the KP hierarchy, which are invariant on

Loewner-Kufarev trajectories.

**Irina Markina**(Department of Mathematics, University of Bergen, Norway) 17:30-18:30Group of diffeomorphisms of the unit circle and sub-Riemannian geometry (ENGLISH)

[ Abstract ]

We consider the group of sense-preserving diffeomorphisms of the unit

circle and its central extension - the Virasoro-Bott group as

sub-Riemannian manifolds. Shortly, a sub-Riemannian manifold is a

smooth manifold M with a given sub-bundle D of the tangent bundle, and

with a metric defined on the sub-bundle D. The different sub-bundles

on considered groups are related to some spaces of normalized

univalent functions. We present formulas for geodesics for different

choices of metrics. The geodesic equations are generalizations of

Camassa-Holm, Huter-Saxton, KdV, and other known non-linear PDEs. We

show that any two points in these groups can be connected by a curve

tangent to the chosen sub-bundle. We also discuss the similarities and

peculiarities of the structure of sub-Riemannian geodesics on infinite

and finite dimensional manifolds.

We consider the group of sense-preserving diffeomorphisms of the unit

circle and its central extension - the Virasoro-Bott group as

sub-Riemannian manifolds. Shortly, a sub-Riemannian manifold is a

smooth manifold M with a given sub-bundle D of the tangent bundle, and

with a metric defined on the sub-bundle D. The different sub-bundles

on considered groups are related to some spaces of normalized

univalent functions. We present formulas for geodesics for different

choices of metrics. The geodesic equations are generalizations of

Camassa-Holm, Huter-Saxton, KdV, and other known non-linear PDEs. We

show that any two points in these groups can be connected by a curve

tangent to the chosen sub-bundle. We also discuss the similarities and

peculiarities of the structure of sub-Riemannian geodesics on infinite

and finite dimensional manifolds.

### 2012/12/03

#### Seminar on Geometric Complex Analysis

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

On the geometric meaning of the maximal number

of exceptional values of Gauss maps for immersed surfaces in space forms

(JAPANESE)

**Yu Kawakami**(Yamaguchi University)On the geometric meaning of the maximal number

of exceptional values of Gauss maps for immersed surfaces in space forms

(JAPANESE)

### 2012/12/01

#### Infinite Analysis Seminar Tokyo

13:30-15:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Manin matrices and quantum integrable systems (ENGLISH)

**Alexey Silantyev**(Univ. Tokyo)Manin matrices and quantum integrable systems (ENGLISH)

[ Abstract ]

Manin matrices (known also as right quantum matrices) is a class of

matrices with non-commutative entries. The natural generalization of the

usual determinant for these matrices is so-called column determinant.

Manin matrices, their determinants and minors have the most part of the

properties possessed by the usual number matrices. Manin matrices arise

from the RLL-relations and help to find quantum analogues of Poisson

commuting traces of powers of Lax operators and to establish relations

between different types of quantum commuting families. The RLL-relations

also give us q-analogues of Manin matrices in the case of trigonometric

R-matrix (which define commutation relations for the quantum affine

algebra).

Manin matrices (known also as right quantum matrices) is a class of

matrices with non-commutative entries. The natural generalization of the

usual determinant for these matrices is so-called column determinant.

Manin matrices, their determinants and minors have the most part of the

properties possessed by the usual number matrices. Manin matrices arise

from the RLL-relations and help to find quantum analogues of Poisson

commuting traces of powers of Lax operators and to establish relations

between different types of quantum commuting families. The RLL-relations

also give us q-analogues of Manin matrices in the case of trigonometric

R-matrix (which define commutation relations for the quantum affine

algebra).

### 2012/11/30

#### Colloquium

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

What do Siegel Eisenstein series know about all modular forms? (ENGLISH)

**Siegfried BOECHERER**(University of Tokyo)What do Siegel Eisenstein series know about all modular forms? (ENGLISH)

[ Abstract ]

Eisenstein series came up in C.L.Siegel's famous work on quadratic forms. The main properties of such Eisensetin series such as analytic continuation and explict form of Fourier expansion are well understood. Nowadays, we use Eisenstein series of higher rank symplectic groups and their restrictions to study properties of all modular forms. I will try to survey the use of “pullbacks of Eisenstein series”: Basis problem, L-functions, p-adic properties, rationality and integrality questions.

Eisenstein series came up in C.L.Siegel's famous work on quadratic forms. The main properties of such Eisensetin series such as analytic continuation and explict form of Fourier expansion are well understood. Nowadays, we use Eisenstein series of higher rank symplectic groups and their restrictions to study properties of all modular forms. I will try to survey the use of “pullbacks of Eisenstein series”: Basis problem, L-functions, p-adic properties, rationality and integrality questions.

#### Mathematical Biology Seminar

14:30-15:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Targeting control in the presence of uncertainty (ENGLISH)

**Michael Tildesley**( Infectious Disease Epidemiology (Modelling) at the University of Warwick)Targeting control in the presence of uncertainty (ENGLISH)

[ Abstract ]

The availability of epidemiological data in the early stages of an outbreak of an infectious disease is vital to enable modellers to make accurate predictions regarding the likely spread of disease and preferred intervention strategies. However, in some countries, epidemic data are not available whilst necessary demographic data are only available at an aggregate scale. Here we investigate the ability of models of livestock infectious diseases to predict epidemic spread and optimal control policies in the event of uncertainty. We focus on investigating predictions in the presence of uncertainty regarding contact networks, demographic data and epidemiological parameters. Our results indicate that mathematical models could be utilized in regions where individual farm-level data are not available, to allow predictive analyses to be carried out regarding the likely spread of disease. This method can also be used for contingency planning in collaboration with policy makers to determine preferred control strategies in the event of a future outbreak of infectious disease in livestock.

The availability of epidemiological data in the early stages of an outbreak of an infectious disease is vital to enable modellers to make accurate predictions regarding the likely spread of disease and preferred intervention strategies. However, in some countries, epidemic data are not available whilst necessary demographic data are only available at an aggregate scale. Here we investigate the ability of models of livestock infectious diseases to predict epidemic spread and optimal control policies in the event of uncertainty. We focus on investigating predictions in the presence of uncertainty regarding contact networks, demographic data and epidemiological parameters. Our results indicate that mathematical models could be utilized in regions where individual farm-level data are not available, to allow predictive analyses to be carried out regarding the likely spread of disease. This method can also be used for contingency planning in collaboration with policy makers to determine preferred control strategies in the event of a future outbreak of infectious disease in livestock.

#### 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.

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