## Seminar information archive

Seminar information archive ～11/15｜Today's seminar 11/16 | Future seminars 11/17～

### 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 ]

http://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 ]

http://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 ]

http://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 ]

http://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.

### 2012/04/03

#### 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.

### 2012/03/23

#### Operator Algebra Seminars

16:30-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Higher Rank Graph $C^*$-algebras (ENGLISH)

**Alex Kumjian**(University of Nevada, Reno)Higher Rank Graph $C^*$-algebras (ENGLISH)

#### Lectures

10:30-11:30 Room #117 (Graduate School of Math. Sci. Bldg.)

Cell decomposition of homotopy Deligne-Mumford. (ENGLISH)

**R. Penner**(Aarhus/Caltech)Cell decomposition of homotopy Deligne-Mumford. (ENGLISH)

[ Abstract ]

A long-standing problem has been to extend the ideal cell decomposition of Riemann's moduli space to its compactification by stable curves. In joint work with Doug LaFountain, we have solved this problem with an explicit generalization of fatgraphs. The solution immediately provides a construction of odd-degree cycles, which are conjectured to be non-trivial, thus addressing yet another long-standing issue.

A long-standing problem has been to extend the ideal cell decomposition of Riemann's moduli space to its compactification by stable curves. In joint work with Doug LaFountain, we have solved this problem with an explicit generalization of fatgraphs. The solution immediately provides a construction of odd-degree cycles, which are conjectured to be non-trivial, thus addressing yet another long-standing issue.

### 2012/03/21

#### Lectures

10:15-12:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Geochemical structure of biological macromolecules (ENGLISH)

**R. Penner**(Aarhus/Caltech)Geochemical structure of biological macromolecules (ENGLISH)

[ Abstract ]

This first of two lectures will explain the basic combinatorial and geometrical structures of both protein and RNA. It is intended to set the stage of subsequent discussions for an audience with mathematical background.

This first of two lectures will explain the basic combinatorial and geometrical structures of both protein and RNA. It is intended to set the stage of subsequent discussions for an audience with mathematical background.

#### Lectures

15:15-17:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Moduli space techniques in computational biology

(ENGLISH)

**R. Penner**(Aarhus/Caltech)Moduli space techniques in computational biology

(ENGLISH)

[ Abstract ]

Basic fatgraph models of RNA and protein will be discussed, where edges are associated with base pairs in the former case and with hydrogen bonds between backbone atoms in the latter. For RNA, this leads to new methods described by context-free grammars of RNA folding prediction including certain classes of pseudo-knots. For protein, beyond these discrete invariants lie continuous ones which associate a rotation of

3-dimensional space to each hydrogen bond linking a pair of peptide units. Histograms of these rotations over the entire database of proteins exhibit a small number of "peptide unit legos" which can be used to advantage for the protein folding problem.

Basic fatgraph models of RNA and protein will be discussed, where edges are associated with base pairs in the former case and with hydrogen bonds between backbone atoms in the latter. For RNA, this leads to new methods described by context-free grammars of RNA folding prediction including certain classes of pseudo-knots. For protein, beyond these discrete invariants lie continuous ones which associate a rotation of

3-dimensional space to each hydrogen bond linking a pair of peptide units. Histograms of these rotations over the entire database of proteins exhibit a small number of "peptide unit legos" which can be used to advantage for the protein folding problem.

#### PDE Real Analysis Seminar

10:00-11:00 Room #056 (Graduate School of Math. Sci. Bldg.)

The asymptotic behaviors of the solutions of Poisson-Boltzmann type of equations (ENGLISH)

**Chiun-Chang Lee**(National Taiwan University)The asymptotic behaviors of the solutions of Poisson-Boltzmann type of equations (ENGLISH)

[ Abstract ]

Understanding the existence of electrical double layers around particles in the colloidal dispersion (system) is a crucial phenomenon of the colloid science. The Poisson-Boltzmann (PB) equation is one of the most widely used models to describe the equilibrium phenomenon of an electrical double layer in colloidal systems. This motivates us to study the asymptotic behavior for the boundary layer of the solutions of the PB equation. In this talk, we introduce the precise asymptotic formulas for the slope of the boundary layers with the exact leading order term and the second-order term. In particular, these formulas show that the mean curvature of the boundary exactly appears in the second-order term. This part is my personal work.

On the other hand, to study how the ionic concentrations and ionic valences affect the difference between the boundary and interior potentials in an electrolyte solution, we also introduce a modified PB equation - New Poisson-Boltzmann (PB_n) equation - joint works with Prof. Tai-Chia Lin and Chun Liu and YunKyong Hyon. We give a specific formula showing the influence of these crucial physical quantities on the potential difference in an electrolyte solution. This cannot be found in the PB equation.

Understanding the existence of electrical double layers around particles in the colloidal dispersion (system) is a crucial phenomenon of the colloid science. The Poisson-Boltzmann (PB) equation is one of the most widely used models to describe the equilibrium phenomenon of an electrical double layer in colloidal systems. This motivates us to study the asymptotic behavior for the boundary layer of the solutions of the PB equation. In this talk, we introduce the precise asymptotic formulas for the slope of the boundary layers with the exact leading order term and the second-order term. In particular, these formulas show that the mean curvature of the boundary exactly appears in the second-order term. This part is my personal work.

On the other hand, to study how the ionic concentrations and ionic valences affect the difference between the boundary and interior potentials in an electrolyte solution, we also introduce a modified PB equation - New Poisson-Boltzmann (PB_n) equation - joint works with Prof. Tai-Chia Lin and Chun Liu and YunKyong Hyon. We give a specific formula showing the influence of these crucial physical quantities on the potential difference in an electrolyte solution. This cannot be found in the PB equation.

### 2012/03/16

#### Colloquium

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

On Complex Fluids (ENGLISH)

**Chun LIU**(University of Tokyo / Penn State University)On Complex Fluids (ENGLISH)

[ Abstract ]

The talk is on the mathematical theories, in particular the energetic variational approaches, of anisotropic complex fluids, such as viscoelastic materials, liquid crystals and ionic fluids in proteins and bio-solutions.

Complex fluids, including mixtures and solutions, are abundant in our daily life. The complicated phenomena and properties exhibited by these materials reflects the coupling and competition between the microscopic interactions and the macroscopic dynamics. We study the underlying energetic variational structures that is common among all these multiscale-multiphysics systems.

In this talk, I will demonstrate the modeling as well as analysis and numerical issues arising from various complex fluids.

The talk is on the mathematical theories, in particular the energetic variational approaches, of anisotropic complex fluids, such as viscoelastic materials, liquid crystals and ionic fluids in proteins and bio-solutions.

Complex fluids, including mixtures and solutions, are abundant in our daily life. The complicated phenomena and properties exhibited by these materials reflects the coupling and competition between the microscopic interactions and the macroscopic dynamics. We study the underlying energetic variational structures that is common among all these multiscale-multiphysics systems.

In this talk, I will demonstrate the modeling as well as analysis and numerical issues arising from various complex fluids.

### 2012/03/14

#### PDE Real Analysis Seminar

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

Exponential convergence to equilibria for a general model in hydrodynamics (ENGLISH)

**Jürgen Saal**(Technische Universität Darmstadt)Exponential convergence to equilibria for a general model in hydrodynamics (ENGLISH)

[ Abstract ]

We present a thorough analysis of the Navier-Stokes-Nernst-Planck-Poisson equations. This system describes the dynamics of charged particles dispersed in an incompressible fluid.

In contrast to existing literature and in view of its physical relevance, we also allow for different diffusion coefficients of the charged species.

In addition, the commonly assumed electro-neutrality condition is not required by our approach.

Our aim is to present results on local and global well-posedness as well as exponential stability of equilibria. The results are obtained jointly with Dieter Bothe and Andre Fischer at the Center of Smart Interfaces at TU Darmstadt.

We present a thorough analysis of the Navier-Stokes-Nernst-Planck-Poisson equations. This system describes the dynamics of charged particles dispersed in an incompressible fluid.

In contrast to existing literature and in view of its physical relevance, we also allow for different diffusion coefficients of the charged species.

In addition, the commonly assumed electro-neutrality condition is not required by our approach.

Our aim is to present results on local and global well-posedness as well as exponential stability of equilibria. The results are obtained jointly with Dieter Bothe and Andre Fischer at the Center of Smart Interfaces at TU Darmstadt.

### 2012/03/13

#### Colloquium

15:00-16:00 Room #050 (Graduate School of Math. Sci. Bldg.)

Problems and results on Hardy's Z-function (JAPANESE)

**Aleksandar Ivic**(University of Belgrade, the Serbian Academy of Science and Arts)Problems and results on Hardy's Z-function (JAPANESE)

[ Abstract ]

The title is self-explanatory: G.H. Hardy first used the function

$Z(t)$ to show that there are infinitely many zeta-zeros on the

critical line $\\Re s = 1/2$. In recent years there is a revived

interest in this function, with many results and open problems.

The title is self-explanatory: G.H. Hardy first used the function

$Z(t)$ to show that there are infinitely many zeta-zeros on the

critical line $\\Re s = 1/2$. In recent years there is a revived

interest in this function, with many results and open problems.

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