## Seminar information archive

Seminar information archive ～11/05｜Today's seminar 11/06 | Future seminars 11/07～

### 2019/12/11

#### Seminar on Geometric Complex Analysis

16:00-17:00 Room #156 (Graduate School of Math. Sci. Bldg.)

Einstein-Weyl structures (English)

**Joel Merker**(Paris Sud)Einstein-Weyl structures (English)

[ Abstract ]

On a conformal 3D manifold with electromagnetic field, Einstein-Weyl equations are the counterpart of Einstein's classical field equations. In 1943, Elie Cartan showed, using abstract arguments, that the general solution depends on 4 functions of 2 variables. I will present families of explicit solutions depending on 9 functions of 1 variable, much beyond what was known before. Such solutions are generic in the sense that the Cotton tensor is nonzero. This is joint work with Pawel Nurowski.

On a conformal 3D manifold with electromagnetic field, Einstein-Weyl equations are the counterpart of Einstein's classical field equations. In 1943, Elie Cartan showed, using abstract arguments, that the general solution depends on 4 functions of 2 variables. I will present families of explicit solutions depending on 9 functions of 1 variable, much beyond what was known before. Such solutions are generic in the sense that the Cotton tensor is nonzero. This is joint work with Pawel Nurowski.

### 2019/12/10

#### Operator Algebra Seminars

17:00-18:30 Room #122 (Graduate School of Math. Sci. Bldg.)

On gauging symmetries of topological phases of matter (English)

**Zhenghan Wang**(Microsoft Station Q)On gauging symmetries of topological phases of matter (English)

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

q-Deformation of a continued fraction and its applications (JAPANESE)

**Takeyoshi Kogiso**(Josai University)q-Deformation of a continued fraction and its applications (JAPANESE)

[ Abstract ]

A kind of q-Deformation of continued fractions was introduced by Morier-Genoud and Ovsienko. The most important application of this q-deformation of regular (or negative) continued fraction expansion of rational number r/s is to calculate the Jones polynomial of the rational link of r/s. Moreover we can apply the q-deformation of this continued fraction to quadratic irrational number theory and combinatorics.

On the other hand, there exist another recipes for determing the Jones polynomials by using Lee-Schiffler's snake graph and by using Kogiso-Wakui's Conway-Coxeter frieze method. Therefore, another approach of the result due to Morier-Genoud and Ovsienko can be considered from the viewpoint of a Conway-Coxeter frieze and a snake graph. Furthermore, we can consider a cluster-variable transformation of continued fractions as a further generalization by using ancestoral triangles used in the Kogiso-Wakui.

A kind of q-Deformation of continued fractions was introduced by Morier-Genoud and Ovsienko. The most important application of this q-deformation of regular (or negative) continued fraction expansion of rational number r/s is to calculate the Jones polynomial of the rational link of r/s. Moreover we can apply the q-deformation of this continued fraction to quadratic irrational number theory and combinatorics.

On the other hand, there exist another recipes for determing the Jones polynomials by using Lee-Schiffler's snake graph and by using Kogiso-Wakui's Conway-Coxeter frieze method. Therefore, another approach of the result due to Morier-Genoud and Ovsienko can be considered from the viewpoint of a Conway-Coxeter frieze and a snake graph. Furthermore, we can consider a cluster-variable transformation of continued fractions as a further generalization by using ancestoral triangles used in the Kogiso-Wakui.

#### Tuesday Seminar of Analysis

16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)

Vorticity alignment vs vorticity creation at the boundary (English)

**Tobias Barker**(École Normale Supérieure)Vorticity alignment vs vorticity creation at the boundary (English)

[ Abstract ]

The Navier-Stokes are used as a model for viscous incompressible fluids such as water. The question as to whether or not the equations in three dimensions form singularities is an open Millennium prize problem. In their celebrated paper in 1993, Constantin and Fefferman showed that (in the whole plane) if the vorticity is sufficiently well aligned in regions of high vorticity then the Navier-Stokes equations remain smooth. For the half-space it is commonly assumed that viscous fluids `stick' to the boundary, which generates vorticity at the boundary. In such a setting, it is open as to whether Constantin and Fefferman's result remains to be true. In my talk I will present recent results in this direction. Joint work with Christophe Prange (CNRS, Université de Bordeaux)

The Navier-Stokes are used as a model for viscous incompressible fluids such as water. The question as to whether or not the equations in three dimensions form singularities is an open Millennium prize problem. In their celebrated paper in 1993, Constantin and Fefferman showed that (in the whole plane) if the vorticity is sufficiently well aligned in regions of high vorticity then the Navier-Stokes equations remain smooth. For the half-space it is commonly assumed that viscous fluids `stick' to the boundary, which generates vorticity at the boundary. In such a setting, it is open as to whether Constantin and Fefferman's result remains to be true. In my talk I will present recent results in this direction. Joint work with Christophe Prange (CNRS, Université de Bordeaux)

#### FMSP Lectures

17:00-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

A priori and a posteriori error estimation for solutions of ill-posed problems (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_AnatolyYagola.pdf

**Anatoly G. Yagola**(Lomonosov Moscow State University)A priori and a posteriori error estimation for solutions of ill-posed problems (ENGLISH)

[ Abstract ]

In order to calculate a priori or a posteriori error estimates for solutions of an ill-posed operator equation with an injective operator we need to describe a set of approximate solutions that contains an exact solution. After that we have to calculate a diameter of this set or maximal distance from a fixed approximate solution to any element of this set. I will describe three approaches for constructing error estimates and also their practical applications.

[ Reference URL ]In order to calculate a priori or a posteriori error estimates for solutions of an ill-posed operator equation with an injective operator we need to describe a set of approximate solutions that contains an exact solution. After that we have to calculate a diameter of this set or maximal distance from a fixed approximate solution to any element of this set. I will describe three approaches for constructing error estimates and also their practical applications.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_AnatolyYagola.pdf

### 2019/12/09

#### Numerical Analysis Seminar

16:50-18:20 Room #117 (Graduate School of Math. Sci. Bldg.)

Point-wise error estimation for the finite element solution to Poisson's equation --- new approach based on Kato-Fujita's method (Japanese)

**Xuefeng Liu**(Niigata University)Point-wise error estimation for the finite element solution to Poisson's equation --- new approach based on Kato-Fujita's method (Japanese)

[ Abstract ]

In 1950s, H. Fujita proposed a method to provide the upper and lower bounds in boundary value problems, which is based on the T*T theory of T. Kato about differential equations. Such a method can be regarded a different formulation of the hypercircle method from Prage-Synge's theorem.

Recently, the speaker extended Kato-Fujita's method to the case of the finite element solution of Poisson's equation and proposed a guaranteed point-wise error estimation. The newly proposed error estimation can be applied to problems defined over domains of general shapes along with general boundary conditions.

In 1950s, H. Fujita proposed a method to provide the upper and lower bounds in boundary value problems, which is based on the T*T theory of T. Kato about differential equations. Such a method can be regarded a different formulation of the hypercircle method from Prage-Synge's theorem.

Recently, the speaker extended Kato-Fujita's method to the case of the finite element solution of Poisson's equation and proposed a guaranteed point-wise error estimation. The newly proposed error estimation can be applied to problems defined over domains of general shapes along with general boundary conditions.

#### Seminar on Geometric Complex Analysis

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

Analytic torsions associated with the Rumin complex on contact spheres (Japanese)

**Akira Kitaoka**(The Univ. of Tokyo)Analytic torsions associated with the Rumin complex on contact spheres (Japanese)

[ Abstract ]

The Rumin complex, which is defined on contact manifolds, is a resolution of the constant sheaf of $\mathbb{R}$ given by a subquotient of the de Rham complex. In this talk, we explicitly write down all eigenvalues of the Rumin Laplacian on the standard contact spheres, and express the analytic torsion functions associated with the Rumin complex in terms of the Riemann zeta function. In particular, we find that the functions vanish at the origin and determine the analytic torsions.

The Rumin complex, which is defined on contact manifolds, is a resolution of the constant sheaf of $\mathbb{R}$ given by a subquotient of the de Rham complex. In this talk, we explicitly write down all eigenvalues of the Rumin Laplacian on the standard contact spheres, and express the analytic torsion functions associated with the Rumin complex in terms of the Riemann zeta function. In particular, we find that the functions vanish at the origin and determine the analytic torsions.

### 2019/12/05

#### Information Mathematics Seminar

16:50-18:35 Room #118 (Graduate School of Math. Sci. Bldg.)

Business development of NLP (Japanese)

**Akihito Ogino**(A.I. Squared, Inc.)Business development of NLP (Japanese)

[ Abstract ]

Explanation of AI in Business development of NLP

Explanation of AI in Business development of NLP

### 2019/12/04

#### Number Theory Seminar

17:00-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Characteristic epsilon cycles of l-adic sheaves on varieties (ENGLISH)

**Daichi Takeuchi**(University of Tokyo)Characteristic epsilon cycles of l-adic sheaves on varieties (ENGLISH)

[ Abstract ]

For l-adic sheaves on varieties over finite fields, the constant terms of the functional equations of the L-functions, called global epsilon factors, are important arithmetic invariants. When the varieties are curves, Deligne and Laumon show that they admit product formulae in terms of local epsilon factors.

In this talk, I will explain that, attaching some coefficients to irreducible components of singular supports, we can define refinements of characteristic cycles. We will see that, after taking modulo roots of unity, they give product formulae of global epsilon factors for higher dimensional varieties.

I will also explain that these results can be generalized to arbitrary perfect fields of any characteristic.

For l-adic sheaves on varieties over finite fields, the constant terms of the functional equations of the L-functions, called global epsilon factors, are important arithmetic invariants. When the varieties are curves, Deligne and Laumon show that they admit product formulae in terms of local epsilon factors.

In this talk, I will explain that, attaching some coefficients to irreducible components of singular supports, we can define refinements of characteristic cycles. We will see that, after taking modulo roots of unity, they give product formulae of global epsilon factors for higher dimensional varieties.

I will also explain that these results can be generalized to arbitrary perfect fields of any characteristic.

#### Operator Algebra Seminars

16:45-18:15 Room #126 (Graduate School of Math. Sci. Bldg.)

An infinite-dimensional index theory and the Higson-Kasparov-Trout algebra

**Doman Takata**(Univ. Tokyo)An infinite-dimensional index theory and the Higson-Kasparov-Trout algebra

### 2019/12/03

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Homotopy Gerstenhaber algebras, Courant algebroids, and Field Equations (ENGLISH)

**Anton Zeitlin**(Louisiana State University)Homotopy Gerstenhaber algebras, Courant algebroids, and Field Equations (ENGLISH)

[ Abstract ]

I will talk about the underlying homotopical structures within field equations, which emerge in string theory as conformal invariance conditions for sigma models. I will show how these, often hidden, structures emerge from the homotopy Gerstenhaber algebra associated to vertex and Courant algebroids, thus making all such equations the natural objects within vertex algebra theory.

I will talk about the underlying homotopical structures within field equations, which emerge in string theory as conformal invariance conditions for sigma models. I will show how these, often hidden, structures emerge from the homotopy Gerstenhaber algebra associated to vertex and Courant algebroids, thus making all such equations the natural objects within vertex algebra theory.

#### Algebraic Geometry Seminar

14:30-16:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Moduli of K3 surfaces via cubic 4-folds (English)

**Gavril Farkas**(Humboldt Univ. Berlin)Moduli of K3 surfaces via cubic 4-folds (English)

[ Abstract ]

In a celebrated series of papers, Mukai established structure theorems for polarized K3 surfaces of all genera g<21, with the exception of the case g=14. Using the identification between certain moduli spaces of polarized K3 surfaces of genera 14 and the moduli space of special cubic fourfolds of given discriminant, we discuss a novel approach to moduli spaces of K3 surfaces. As an application, we establish the rationality of the universal K3 surface of these genus 14,22. This is joint work with A. Verra.

In a celebrated series of papers, Mukai established structure theorems for polarized K3 surfaces of all genera g<21, with the exception of the case g=14. Using the identification between certain moduli spaces of polarized K3 surfaces of genera 14 and the moduli space of special cubic fourfolds of given discriminant, we discuss a novel approach to moduli spaces of K3 surfaces. As an application, we establish the rationality of the universal K3 surface of these genus 14,22. This is joint work with A. Verra.

### 2019/12/02

#### Seminar on Geometric Complex Analysis

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

Toward classification of Moishezon twistor spaces

**Nobuhiro Honda**(Tokyo Tech.)Toward classification of Moishezon twistor spaces

[ Abstract ]

Twistor spaces are complex 3-folds which arise from 4-dimensional conformal geometry. These spaces always have negative Kodaira dimension, and most of them are known to be non-Kahler. But there are a plenty of compact twistor spaces which are Moishezon variety. The topology of such spaces is strongly constrained, and it seems not hopeless to obtain a classification and explicit description of them. I will talk about results in such a direction, which classify such spaces under a simple assumption. No example seems to be known which does not satisfy that assumption.

Twistor spaces are complex 3-folds which arise from 4-dimensional conformal geometry. These spaces always have negative Kodaira dimension, and most of them are known to be non-Kahler. But there are a plenty of compact twistor spaces which are Moishezon variety. The topology of such spaces is strongly constrained, and it seems not hopeless to obtain a classification and explicit description of them. I will talk about results in such a direction, which classify such spaces under a simple assumption. No example seems to be known which does not satisfy that assumption.

### 2019/11/29

#### Operator Algebra Seminars

15:00-17:00 Room #126 (Graduate School of Math. Sci. Bldg.)

**Reiji Tomatsu**(Hokkaido Univ.)### 2019/11/28

#### Operator Algebra Seminars

15:00-17:00 Room #117 (Graduate School of Math. Sci. Bldg.)

**Reiji Tomatsu**(Hokkaido Univ.)#### Information Mathematics Seminar

16:50-18:35 Room #122 (Graduate School of Math. Sci. Bldg.)

AI-aided cell visualization technology (Japanese)

**Tamio Mizukami**(Nagahama Institute of Bio-Science and Technology、Frontier Pharma, Inc.)AI-aided cell visualization technology (Japanese)

[ Abstract ]

Explanation of AI-aided cell visualization technology

Explanation of AI-aided cell visualization technology

### 2019/11/27

#### Operator Algebra Seminars

17:15-18:45 Room #126 (Graduate School of Math. Sci. Bldg.)

The homotopy groups of the automorphism groups of Cuntz-Toeplitz algebras

**Taro Sogabe**(Kyoto University)The homotopy groups of the automorphism groups of Cuntz-Toeplitz algebras

#### Number Theory Seminar

17:00-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)

The relative Hodge-Tate spectral sequence (ENGLISH)

**Ahmed Abbes**(CNRS & IHÉS)The relative Hodge-Tate spectral sequence (ENGLISH)

[ Abstract ]

It is well known that the p-adic étale cohomology of a smooth and proper variety over a p-adic field admits a Hodge-Tate decomposition and that it is the abutment of a spectral sequence called Hodge-Tate; these two properties are incidentally equivalent. The Hodge-Tate decomposition was generalized in higher dimensions to Hodge-Tate local systems by Hyodo, and was studied by Faltings, Tsuji and others. But the generalization of the Hodge-Tate spectral sequence to a relative situation has not yet been considered (not even conjectured), with the exception of a special case of abelian schemes by Hyodo. This has now been done in a joint work with Michel Gros. The relative Hodge-Tate spectral sequence that we construct takes place in the Faltings topos, but its construction requires the introduction of a relative variant of this topos which is the main novelty of our work. The relative Hodge-Tate spectral sequence sheds new light on the fact that the relative p-adic étale cohomology is Hodge-Tate, but the two properties are not equivalent in general.

It is well known that the p-adic étale cohomology of a smooth and proper variety over a p-adic field admits a Hodge-Tate decomposition and that it is the abutment of a spectral sequence called Hodge-Tate; these two properties are incidentally equivalent. The Hodge-Tate decomposition was generalized in higher dimensions to Hodge-Tate local systems by Hyodo, and was studied by Faltings, Tsuji and others. But the generalization of the Hodge-Tate spectral sequence to a relative situation has not yet been considered (not even conjectured), with the exception of a special case of abelian schemes by Hyodo. This has now been done in a joint work with Michel Gros. The relative Hodge-Tate spectral sequence that we construct takes place in the Faltings topos, but its construction requires the introduction of a relative variant of this topos which is the main novelty of our work. The relative Hodge-Tate spectral sequence sheds new light on the fact that the relative p-adic étale cohomology is Hodge-Tate, but the two properties are not equivalent in general.

#### Operator Algebra Seminars

15:00-17:00 Room #126 (Graduate School of Math. Sci. Bldg.)

(日本語)

**Reiji Tomatsu**(Hokkaido Univ.)(日本語)

### 2019/11/26

#### Tuesday Seminar of Analysis

16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)

Accurate lower bounds for eigenvalues of electronic Hamiltonians (Japanese)

**ASHIDA Sohei**(Gakushuin University)Accurate lower bounds for eigenvalues of electronic Hamiltonians (Japanese)

[ Abstract ]

Electronic Hamiltonians are differential operators depending on relative positions of nuclei as parameters. When we regard an eigenvalues of an electronic Hamiltonian as a function of relative positions of nuclei, minimum points correspond to shapes of molecules. Upper bounds for eigenvalues are obtained by variational methods. However, since the physical information as minimum points does not change when a reference point of energy changes, physical information can not be obtained by variational methods only. Combining lower and upper bounds physical information is obtained.

In this talk we discuss the Weinstein-Arnszajn intermediate problem methods for lower bounds of eigenvalues based on comparison of operators. A method for lower bounds of one-electronic Hamiltonians is also introduced. Some computations for two kinds of hydrogen molecule-ion are shown.

Electronic Hamiltonians are differential operators depending on relative positions of nuclei as parameters. When we regard an eigenvalues of an electronic Hamiltonian as a function of relative positions of nuclei, minimum points correspond to shapes of molecules. Upper bounds for eigenvalues are obtained by variational methods. However, since the physical information as minimum points does not change when a reference point of energy changes, physical information can not be obtained by variational methods only. Combining lower and upper bounds physical information is obtained.

In this talk we discuss the Weinstein-Arnszajn intermediate problem methods for lower bounds of eigenvalues based on comparison of operators. A method for lower bounds of one-electronic Hamiltonians is also introduced. Some computations for two kinds of hydrogen molecule-ion are shown.

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

$2+1$-TQFTs from non-semisimple modular categories (ENGLISH)

**Marco De Renzi**(Waseda University)$2+1$-TQFTs from non-semisimple modular categories (ENGLISH)

[ Abstract ]

Non-semisimple constructions have substantially generalized the standard approach of Witten, Reshetikhin, and Turaev to quantum topology, producing powerful invariants and TQFTs with unprecedented properties. We will explain how to use the theory of

Based on a joint work with Azat Gainutdinov, Nathan Geer, Bertrand Patureau, and Ingo Runkel.

Non-semisimple constructions have substantially generalized the standard approach of Witten, Reshetikhin, and Turaev to quantum topology, producing powerful invariants and TQFTs with unprecedented properties. We will explain how to use the theory of

*modified traces*to renormalize Lyubashenko’s closed 3-manifold invariants coming from*finite twist non-degenerate unimodular ribbon categories*. Under the additional assumption of*factorizability*, our renormalized invariants extend to $2+1$-TQFTs, unlike Lyubashenko’s original ones. This general framework encompasses important examples of non-semisimple modular categories which were left out of previous non-semisimple TQFT constructions.Based on a joint work with Azat Gainutdinov, Nathan Geer, Bertrand Patureau, and Ingo Runkel.

#### Operator Algebra Seminars

15:00-17:00 Room #126 (Graduate School of Math. Sci. Bldg.)

(日本語)

**Reiji Tomatsu**(Hokkaido Univ.)(日本語)

#### PDE Real Analysis Seminar

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

A Hamiltonian approach with penalization in shape and topology optimization (English)

**Dan Tiba**(Institute of Mathematics of the Romanian Academy / Academy of Romanian Scientists)A Hamiltonian approach with penalization in shape and topology optimization (English)

[ Abstract ]

General geometric optimization problems involve boundary and topology variations. This research area has already almost fifty years of history and very rich applications in computer aided industrial design. Recently, a new representation of manifolds, using iterated Hamiltonian systems, has been introduced in arbitrary dimension and co-dimension. Combining this technique with a penalization procedure for the boundary conditions, a comprehensive approximation method for optimal design problems associated to elliptic equations, is obtained. It reduces shape and topology optimization problems to optimal control problems, in a general setting. It enters the category of fixed domain methods in variable/unknown domain problems and it has consistent advantages at the computational level. It allows "free" changes of the boundary and/or the topology, during the iterations. This methodology, based on iterated Hamiltonian systems and implicit parametrizations, was also applied to nonlinear programming problems in arbitrary dimension.

General geometric optimization problems involve boundary and topology variations. This research area has already almost fifty years of history and very rich applications in computer aided industrial design. Recently, a new representation of manifolds, using iterated Hamiltonian systems, has been introduced in arbitrary dimension and co-dimension. Combining this technique with a penalization procedure for the boundary conditions, a comprehensive approximation method for optimal design problems associated to elliptic equations, is obtained. It reduces shape and topology optimization problems to optimal control problems, in a general setting. It enters the category of fixed domain methods in variable/unknown domain problems and it has consistent advantages at the computational level. It allows "free" changes of the boundary and/or the topology, during the iterations. This methodology, based on iterated Hamiltonian systems and implicit parametrizations, was also applied to nonlinear programming problems in arbitrary dimension.

### 2019/11/25

#### Operator Algebra Seminars

15:00-17:00 Room #128 (Graduate School of Math. Sci. Bldg.)

(日本語)

**Reiji Tomatsu**(Hokkaido Univ.)(日本語)

### 2019/11/22

#### Discrete mathematical modelling seminar

17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

The Hopf algebra structure of coloured non-commutative symmetric functions

**Adam Doliwa**(University of Warmia and Mazury)The Hopf algebra structure of coloured non-commutative symmetric functions

[ Abstract ]

The Hopf algebra of symmetric functions (Sym), especially its Schur function basis, plays an important role in the theory of KP hierarchy. The Hopf algebra of non-commutative symmetric functions (NSym) was introduced by Gelfand, Krob, Lascoux, Leclerc, Retakh and Thibon. In my talk I would like to present its "A-coloured" version NSym_A and its graded dual - the Hopf algebra QSym_A of coloured quasi-symmetric functions. It turns out that these two algebras are both non-commutative and non-cocommutative (for |A|>1), and their product and coproduct operations allow for simple combinatorial meaning. I will also show how the structure of the poset of sentences over alphabet A (A-coloured compositions) gives rise to a description of the corresponding coloured version of the ribbon Schur basis of NSym_A.

The Hopf algebra of symmetric functions (Sym), especially its Schur function basis, plays an important role in the theory of KP hierarchy. The Hopf algebra of non-commutative symmetric functions (NSym) was introduced by Gelfand, Krob, Lascoux, Leclerc, Retakh and Thibon. In my talk I would like to present its "A-coloured" version NSym_A and its graded dual - the Hopf algebra QSym_A of coloured quasi-symmetric functions. It turns out that these two algebras are both non-commutative and non-cocommutative (for |A|>1), and their product and coproduct operations allow for simple combinatorial meaning. I will also show how the structure of the poset of sentences over alphabet A (A-coloured compositions) gives rise to a description of the corresponding coloured version of the ribbon Schur basis of NSym_A.

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