## Seminar information archive

Seminar information archive ～05/21｜Today's seminar 05/22 | Future seminars 05/23～

### 2012/05/07

#### Seminar on Geometric Complex Analysis

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

The second metric variation of the total $Q$-curvature in conformal geometry (JAPANESE)

**Yoshihiko Matsumoto**(University of Tokyo)The second metric variation of the total $Q$-curvature in conformal geometry (JAPANESE)

[ Abstract ]

Branson's $Q$-curvature of even-dimensional compact conformal manifolds integrates to a global conformal invariant called the total $Q$-curvature. While it is topological in two dimensions and is essentially the Weyl action in four dimensions, in the higher dimensional cases its geometric meaning remains mysterious. Graham and Hirachi have shown that the first metric variation of the total $Q$-curvature coincides with the Fefferman-Graham obstruction tensor. In this talk, the second variational formula will be presented, and it will be made explicit especially for conformally Einstein manifolds. The positivity of the second variation will be discussed in connection with the smallest eigenvalue of the Lichnerowicz Laplacian.

Branson's $Q$-curvature of even-dimensional compact conformal manifolds integrates to a global conformal invariant called the total $Q$-curvature. While it is topological in two dimensions and is essentially the Weyl action in four dimensions, in the higher dimensional cases its geometric meaning remains mysterious. Graham and Hirachi have shown that the first metric variation of the total $Q$-curvature coincides with the Fefferman-Graham obstruction tensor. In this talk, the second variational formula will be presented, and it will be made explicit especially for conformally Einstein manifolds. The positivity of the second variation will be discussed in connection with the smallest eigenvalue of the Lichnerowicz Laplacian.

#### Algebraic Geometry Seminar

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

Algebro-geometric characterization of Cayley polytopes (JAPANESE)

**Atsushi Ito**(University of Tokyo)Algebro-geometric characterization of Cayley polytopes (JAPANESE)

[ Abstract ]

A lattice polytope is called a Cayley polytope if it is "small" in some

sense.

In this talk, I will explain an algebro-geometric characterization of

Cayley polytopes

by considering whether or not the corresponding polarized toric

varieties are covered by lines, planes, etc.

We can apply this characterization to the study of Seshadri constants,

which are invariants measuring the positivity of ample line bundles.

That is, we can obtain an explicit description of a polarized toric

variety whose Seshadri constant is one.

A lattice polytope is called a Cayley polytope if it is "small" in some

sense.

In this talk, I will explain an algebro-geometric characterization of

Cayley polytopes

by considering whether or not the corresponding polarized toric

varieties are covered by lines, planes, etc.

We can apply this characterization to the study of Seshadri constants,

which are invariants measuring the positivity of ample line bundles.

That is, we can obtain an explicit description of a polarized toric

variety whose Seshadri constant is one.

#### GCOE Seminars

14:30-16:00 Room #370 (Graduate School of Math. Sci. Bldg.)

Distributional behaviors of time-averaged observables in anomalous diffusions (subdiffusion and superdiffusion) (ENGLISH)

**Takuma Akimoto**(Keio university, Global environmental leaders program)Distributional behaviors of time-averaged observables in anomalous diffusions (subdiffusion and superdiffusion) (ENGLISH)

[ Abstract ]

In anomalous diffusions attributed to a power-law distribution,

time-averaged observables such as diffusion coefficient and velocity of drift are intrinsically random. Anomalous diffusion is ubiquitous phenomenon not only in material science but also in biological transports, which is characterized by a non-linear growth of the mean square displacement (MSD).

(subdiffusion: sublinear growth, super diffusion: superlinear growth).

It has been known that there are three different mechanisms generating subdiffusion. One of them is a power-law distribution in the trapping-time distribution. Such anomalous diffusion is modeled by the continuous time random walk (CTRW). In CTRW, the time-averaged MSD grows linearly with time whereas the ensemble-averaged MSD does not. Using renewal theory, I show that diffusion coefficients obtained by single trajectories converge in distribution. The distribution is the Mittag-Leffler (or inverse Levy) distribution [1,2].

In superdiffusion, there are three different mechanisms. One stems from positive correlations in random walks; the second from persistent motions in random walks, called Levy walk; the third from very long jumps in random walks, called Levy flight.

If the persistent time distribution obeys a power law with divergent mean in Levy walks, the MSD grows as t^2 whereas the mean of positions is zero. When an external bias is added in Levy walks, the response to bias (velocity of drift) appears in the distribution, which is what we term a distributional response [3]. The distribution is the generalized arcsine distribution.

These distributional behaviors open a new window to dealing with the average (ensemble or time average) in single particle tracking experiments.

[1] Y. He, S. Burov, R. Metzler, and E. Barkai, Phys. Rev. Lett. 101, 058101 (2008).

[2] T. Miyaguchi and T. Akimoto, Phys. Rev. E 83, 031926 (2011).

[3] T. Akimoto, Phys. Rev. Lett. 108, 164101 (2012)

In anomalous diffusions attributed to a power-law distribution,

time-averaged observables such as diffusion coefficient and velocity of drift are intrinsically random. Anomalous diffusion is ubiquitous phenomenon not only in material science but also in biological transports, which is characterized by a non-linear growth of the mean square displacement (MSD).

(subdiffusion: sublinear growth, super diffusion: superlinear growth).

It has been known that there are three different mechanisms generating subdiffusion. One of them is a power-law distribution in the trapping-time distribution. Such anomalous diffusion is modeled by the continuous time random walk (CTRW). In CTRW, the time-averaged MSD grows linearly with time whereas the ensemble-averaged MSD does not. Using renewal theory, I show that diffusion coefficients obtained by single trajectories converge in distribution. The distribution is the Mittag-Leffler (or inverse Levy) distribution [1,2].

In superdiffusion, there are three different mechanisms. One stems from positive correlations in random walks; the second from persistent motions in random walks, called Levy walk; the third from very long jumps in random walks, called Levy flight.

If the persistent time distribution obeys a power law with divergent mean in Levy walks, the MSD grows as t^2 whereas the mean of positions is zero. When an external bias is added in Levy walks, the response to bias (velocity of drift) appears in the distribution, which is what we term a distributional response [3]. The distribution is the generalized arcsine distribution.

These distributional behaviors open a new window to dealing with the average (ensemble or time average) in single particle tracking experiments.

[1] Y. He, S. Burov, R. Metzler, and E. Barkai, Phys. Rev. Lett. 101, 058101 (2008).

[2] T. Miyaguchi and T. Akimoto, Phys. Rev. E 83, 031926 (2011).

[3] T. Akimoto, Phys. Rev. Lett. 108, 164101 (2012)

### 2012/05/02

#### Operator Algebra Seminars

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

A measurable group theoretic solution to von Neumann's Problem (after Gaboriau and Lyons) (JAPANESE)

**Yuhei Suzuki**(Univ. Tokyo)A measurable group theoretic solution to von Neumann's Problem (after Gaboriau and Lyons) (JAPANESE)

### 2012/05/01

#### Tuesday Seminar on Topology

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

Minimal models, formality and hard Lefschetz property of

solvmanifolds with local systems (JAPANESE)

**Hisashi Kasuya**(The University of Tokyo)Minimal models, formality and hard Lefschetz property of

solvmanifolds with local systems (JAPANESE)

### 2012/04/27

#### Seminar on Probability and Statistics

15:00-16:10 Room #006 (Graduate School of Math. Sci. Bldg.)

Convergence conditions on step sizes in temporal difference learning (JAPANESE)

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2012/02.html

**NOMURA, Ryosuke**(Graduate school of Mathematical Sciences, Univ. of Tokyo)Convergence conditions on step sizes in temporal difference learning (JAPANESE)

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2012/02.html

### 2012/04/25

#### Operator Algebra Seminars

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Bost-Connes system and class field theory (JAPANESE)

**Takuya Takeishi**(Univ. Tokyo)Bost-Connes system and class field theory (JAPANESE)

### 2012/04/24

#### Numerical Analysis Seminar

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

Quantum mechanics and numerical analysis (JAPANESE)

[ Reference URL ]

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

**Hideaki Ishikawa**(Semiconductor Leading Edge Technologies, Inc.)Quantum mechanics and numerical analysis (JAPANESE)

[ Reference URL ]

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

#### Tuesday Seminar on Topology

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

Combinatorial Heegaard Floer homology (ENGLISH)

**Dylan Thurston**(Columbia University)Combinatorial Heegaard Floer homology (ENGLISH)

[ Abstract ]

Heegaard Floer homology is a powerful invariant of 3- and 4-manifolds.

In 4 dimensions, Heegaard Floer homology (together with the

Seiberg-Witten and Donaldson equations, which are conjecturally

equivalent), provides essentially the only technique for

distinguishing smooth 4-manifolds. In 3 dimensions, it provides much

geometric information, like the simplest representatives of a given

homology class.

In this talk we will focus on recent progress in making Heegaard Floer

homology more computable, including a complete algorithm for computing

it for knots.

Heegaard Floer homology is a powerful invariant of 3- and 4-manifolds.

In 4 dimensions, Heegaard Floer homology (together with the

Seiberg-Witten and Donaldson equations, which are conjecturally

equivalent), provides essentially the only technique for

distinguishing smooth 4-manifolds. In 3 dimensions, it provides much

geometric information, like the simplest representatives of a given

homology class.

In this talk we will focus on recent progress in making Heegaard Floer

homology more computable, including a complete algorithm for computing

it for knots.

### 2012/04/23

#### Algebraic Geometry Seminar

17:10-18:40 Room #122 (Graduate School of Math. Sci. Bldg.)

Motivic integration and wild group actions (JAPANESE)

**Takehiko Yasuda**(Osaka University)Motivic integration and wild group actions (JAPANESE)

[ Abstract ]

The cohomological McKay correspondence proved by Batyrev is the equality of an orbifold invariant

and a stringy invariant. The former is an invariant of a smooth variety with a finite group action and the latter is

an invariant of its quotient variety. Denef and Loeser gave an alternative proof of it which uses the motivic integration theory developped by themselves.

Then I pushed forward with their study by generalizing the motivic integration to

Deligne-Mumford stacks and reformulating the cohomological McKay correspondence from the viewpoint of

the birational geometry of stacks.

However all of these are about tame group actions (the order of a group is not divisible by the characteristic of the base field),

and the wild (= not tame) case has remained unexplored.

In this talk, I will explain my attempt to examine the simplest situation of the wild case. Namely linear actions of a cyclic group

of order equal to the characteristic of the base field are treated. A remarkable new phenomenon is that the space of generalized

arcs is a fibration over an infinite dimensional space with infinite dimensional fibers, where the base space is the space of

Artin-Schreier extensions of $k((t))$, the field of Laurent series.

The cohomological McKay correspondence proved by Batyrev is the equality of an orbifold invariant

and a stringy invariant. The former is an invariant of a smooth variety with a finite group action and the latter is

an invariant of its quotient variety. Denef and Loeser gave an alternative proof of it which uses the motivic integration theory developped by themselves.

Then I pushed forward with their study by generalizing the motivic integration to

Deligne-Mumford stacks and reformulating the cohomological McKay correspondence from the viewpoint of

the birational geometry of stacks.

However all of these are about tame group actions (the order of a group is not divisible by the characteristic of the base field),

and the wild (= not tame) case has remained unexplored.

In this talk, I will explain my attempt to examine the simplest situation of the wild case. Namely linear actions of a cyclic group

of order equal to the characteristic of the base field are treated. A remarkable new phenomenon is that the space of generalized

arcs is a fibration over an infinite dimensional space with infinite dimensional fibers, where the base space is the space of

Artin-Schreier extensions of $k((t))$, the field of Laurent series.

### 2012/04/21

#### Harmonic Analysis Komaba Seminar

13:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Dyadic, classical and martingale harmonic analysis (JAPANESE)

A_\\\\infty constants between BMO and weighted BMO (JAPANESE)

**Yutaka, Terasawa**(Graduate School of Mathematical Sciences, University of Tokyo) 13:30-15:00Dyadic, classical and martingale harmonic analysis (JAPANESE)

[ Abstract ]

**Yohei Tsutsui**(Waseda University) 15:30-17:00A_\\\\infty constants between BMO and weighted BMO (JAPANESE)

[ Abstract ]

### 2012/04/20

#### Seminar on Probability and Statistics

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

On the asymptotic mixed normality of the pre-averaged Hayashi-Yoshida

estimator with random and nonsynchronous sampling (JAPANESE)

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2012/01.html

**KOIKE, Yuta**(Graduate school of Mathematical Sciences, Univ. of Tokyo)On the asymptotic mixed normality of the pre-averaged Hayashi-Yoshida

estimator with random and nonsynchronous sampling (JAPANESE)

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2012/01.html

### 2012/04/18

#### Number Theory Seminar

16:40-17:40 Room #056 (Graduate School of Math. Sci. Bldg.)

Explicit constructions of rational points on elliptic curves (ENGLISH)

**Alan Lauder**(University of Oxford)Explicit constructions of rational points on elliptic curves (ENGLISH)

[ Abstract ]

I will present an algorithm for computing certain special

values of p-adic L-functions, and discuss an application to

the efficient construction of rational points on elliptic curves.

I will present an algorithm for computing certain special

values of p-adic L-functions, and discuss an application to

the efficient construction of rational points on elliptic curves.

#### Operator Algebra Seminars

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Classification of Group Actions on Factors (after Masuda) (JAPANESE)

**Koichi Shimada**(Univ. Tokyo)Classification of Group Actions on Factors (after Masuda) (JAPANESE)

### 2012/04/17

#### Tuesday Seminar on Topology

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

Pseudo-Anosov mapping classes with small dilatation (ENGLISH)

**Eriko Hironaka**(Florida State University)Pseudo-Anosov mapping classes with small dilatation (ENGLISH)

[ Abstract ]

A mapping class is a homeomorphism of an oriented surface

to itself modulo isotopy. It is pseudo-Anosov if the lengths of essential

simple closed curves under iterations of the map have exponential growth

rate. The growth rate, an algebraic integer of degree bounded with

respect to the topology of the surface, is called the dilatation of the

mapping class. In this talk we will discuss the minimization problem

for dilatations of pseudo-Anosov mapping classes, and give two general

constructions of pseudo-Anosov mapping classes with small dilatation.

A mapping class is a homeomorphism of an oriented surface

to itself modulo isotopy. It is pseudo-Anosov if the lengths of essential

simple closed curves under iterations of the map have exponential growth

rate. The growth rate, an algebraic integer of degree bounded with

respect to the topology of the surface, is called the dilatation of the

mapping class. In this talk we will discuss the minimization problem

for dilatations of pseudo-Anosov mapping classes, and give two general

constructions of pseudo-Anosov mapping classes with small dilatation.

### 2012/04/16

#### Algebraic Geometry Seminar

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

Toric degenerations of minuscule Schubert varieties and mirror symmetry (JAPANESE)

**Makoto Miura**(University of Tokyo)Toric degenerations of minuscule Schubert varieties and mirror symmetry (JAPANESE)

[ Abstract ]

Minuscule Schubert varieties admit the flat degenerations to projective

Hibi toric varieties, whose combinatorial structure is explicitly

described by finite posets. In this talk, I will explain these toric

degenerations and discuss the mirror symmetry for complete intersection

Calabi-Yau varieties in Gorenstein minuscule Schubert varieties.

Minuscule Schubert varieties admit the flat degenerations to projective

Hibi toric varieties, whose combinatorial structure is explicitly

described by finite posets. In this talk, I will explain these toric

degenerations and discuss the mirror symmetry for complete intersection

Calabi-Yau varieties in Gorenstein minuscule Schubert varieties.

#### Seminar on Geometric Complex Analysis

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

Fekete configuration, quantitative equidistribution and wanderting critical orbits in non-archimedean dynamics

(JAPANESE)

**Yusuke Okuyama**(Kyoto Institute of Technology)Fekete configuration, quantitative equidistribution and wanderting critical orbits in non-archimedean dynamics

(JAPANESE)

### 2012/04/14

#### Monthly Seminar on Arithmetic of Automorphic Forms

13:30-16:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Explicit formula for the formal degree of the discrete series representations of GL_m(D). (JAPANESE)

Moments of the derivatives of the Riemann zeta function (JAPANESE)

**Kazutoshi Kariyama**(Onomichi city university) 13:30-14:30Explicit formula for the formal degree of the discrete series representations of GL_m(D). (JAPANESE)

**Keijyu Souno**(Math.-Sci., Tokyo Univ.) 15:00-16:00Moments of the derivatives of the Riemann zeta function (JAPANESE)

[ Abstract ]

In my talk, we consider the integral moments of the derivatives of the Riemann zeta function on the critical line. We give certain lower bounds for these moments under the assumption of the Riemann hypothesis.

In my talk, we consider the integral moments of the derivatives of the Riemann zeta function on the critical line. We give certain lower bounds for these moments under the assumption of the Riemann hypothesis.

### 2012/04/13

#### Seminar on Probability and Statistics

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

Asymptotic properties of MCMC for cumulative link model (JAPANESE)

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2012/00.html

**KAMATANI, Kengo**(Graduate School of Engineering Science, Osaka University)Asymptotic properties of MCMC for cumulative link model (JAPANESE)

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~kengok/statseminar/2012/00.html

### 2012/04/11

#### Operator Algebra Seminars

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Mathematical Aspects of Fractional Quantum Hall Effect (ENGLISH)

**Shweta Sharma**(Univ. Paris Sud)Mathematical Aspects of Fractional Quantum Hall Effect (ENGLISH)

#### Number Theory Seminar

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

Around the Mordell-Lang conjecture in positive characteristic (ENGLISH)

**Damian Rossler**(CNRS, Universite de Toulouse)Around the Mordell-Lang conjecture in positive characteristic (ENGLISH)

[ Abstract ]

Let V be a subvariety of an abelian variety A over C and let G\\subseteq A(C) be a subgroup. The classical Mordell-Lang conjecture predicts that if V is of general type and G\\otimesQ is finite dimensional, then V\\cap G is not Zariski dense in V. This statement contains the Mordell conjecture as well as the Manin-Mumford conjecture (for curves). The positive characteristic analog of the Mordell-Lang conjecture makes sense, when A is supposed to have no subquotient, which is defined over a finite field. This positive characteristic analog was proven in 1996 by E. Hrushovski using model-theoretic methods. We shall discuss the prehistory and context of this proof. We shall also discuss the proof (due to the speaker) of the fact that in positive characteristic, the Manin-Mumford conjecture implies the Mordell-Lang conjecture (whereas this seems far from true in characteristic 0).

Let V be a subvariety of an abelian variety A over C and let G\\subseteq A(C) be a subgroup. The classical Mordell-Lang conjecture predicts that if V is of general type and G\\otimesQ is finite dimensional, then V\\cap G is not Zariski dense in V. This statement contains the Mordell conjecture as well as the Manin-Mumford conjecture (for curves). The positive characteristic analog of the Mordell-Lang conjecture makes sense, when A is supposed to have no subquotient, which is defined over a finite field. This positive characteristic analog was proven in 1996 by E. Hrushovski using model-theoretic methods. We shall discuss the prehistory and context of this proof. We shall also discuss the proof (due to the speaker) of the fact that in positive characteristic, the Manin-Mumford conjecture implies the Mordell-Lang conjecture (whereas this seems far from true in characteristic 0).

### 2012/04/10

#### Tuesday Seminar on Topology

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

On homology of symplectic derivation Lie algebras of

the free associative algebra and the free Lie algebra (JAPANESE)

**Takuya Sakasai**(The University of Tokyo)On homology of symplectic derivation Lie algebras of

the free associative algebra and the free Lie algebra (JAPANESE)

[ Abstract ]

We discuss homology of symplectic derivation Lie algebras of

the free associative algebra and the free Lie algebra

with particular stress on their abelianizations (degree 1 part).

Then, by using a theorem of Kontsevich,

we give some applications to rational cohomology of the moduli spaces of

Riemann surfaces and metric graphs.

This is a joint work with Shigeyuki Morita and Masaaki Suzuki.

We discuss homology of symplectic derivation Lie algebras of

the free associative algebra and the free Lie algebra

with particular stress on their abelianizations (degree 1 part).

Then, by using a theorem of Kontsevich,

we give some applications to rational cohomology of the moduli spaces of

Riemann surfaces and metric graphs.

This is a joint work with Shigeyuki Morita and Masaaki Suzuki.

### 2012/04/09

#### Algebraic Geometry Seminar

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

On mirror symmetry for weighted Calabi-Yau hypersurfaces (JAPANESE)

**Kazushi Ueda**(Osaka University)On mirror symmetry for weighted Calabi-Yau hypersurfaces (JAPANESE)

[ Abstract ]

In the talk, I will discuss relation between homological mirror symmetry for weighted projective spaces, their Calabi-Yau hypersurfaces, and weighted homogeneous singularities.

If the time permits, I will also discuss an application to monodromy of hypergeometric functions.

In the talk, I will discuss relation between homological mirror symmetry for weighted projective spaces, their Calabi-Yau hypersurfaces, and weighted homogeneous singularities.

If the time permits, I will also discuss an application to monodromy of hypergeometric functions.

#### Seminar on Geometric Complex Analysis

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

Effective estimate on the number of deformation types of families of canonically polarized manifolds over curves

(JAPANESE)

**Shigeharu TAKAYAMA**(University of Tokyo)Effective estimate on the number of deformation types of families of canonically polarized manifolds over curves

(JAPANESE)

### 2012/04/04

#### PDE Real Analysis Seminar

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

Multi linear formulation of differential geometry and matrix regularizations (ENGLISH)

**Jens Hoppe**(Sogang University / KTH Royal Institute of Technology)Multi linear formulation of differential geometry and matrix regularizations (ENGLISH)

[ Abstract ]

We prove that many aspects of the differential geometry of embedded Riemannian manifolds can be formulated in terms of multi linear algebraic structures on the space of smooth functions. In particular, we find algebraic expressions for Weingarten's formula, the Ricci curvature and the Codazzi-Mainardi equations.

For matrix analogues of embedded surfaces we define discrete curvatures and Euler characteristics, and a non-commutative Gauss–Bonnet theorem is shown to follow. We derive simple expressions for the discrete Gauss curvature in terms of matrices representing the embedding coordinates, and a large class of explicit examples is provided.

We prove that many aspects of the differential geometry of embedded Riemannian manifolds can be formulated in terms of multi linear algebraic structures on the space of smooth functions. In particular, we find algebraic expressions for Weingarten's formula, the Ricci curvature and the Codazzi-Mainardi equations.

For matrix analogues of embedded surfaces we define discrete curvatures and Euler characteristics, and a non-commutative Gauss–Bonnet theorem is shown to follow. We derive simple expressions for the discrete Gauss curvature in terms of matrices representing the embedding coordinates, and a large class of explicit examples is provided.

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