# MSJ Memoirs

# MSJ Memoirs

### Editorial Board

S. Naito(Editor-in-chief, Tokyo Tech)

H. Endo(Managing Editor, Tokyo Tech)

G. Akagi(Tohoku), A. Atsuji(Keio), K. Bannai(Keio), H. Endo(Tokyo Tech), O. Fujino(Kyoto),

T.
Ito(Kyoto), J. Kamimoto(Kyushu), K. Kato(TUS), H. Konno(Meiji), M. Kubo(Nagoya),

H. Masuda(Kyushu),
S. Naito(Tokyo Tech), S. Seirin-Lee(Kyoto),

T. Shioya(Tohoku), K. Takemura(Ochanomizu), K.
Yokoyama(Rikkyo)

The series MSJ Memoirs is devoted to the publications of lecture notes, graduate textbooks and long research papers* in pure and applied mathematics. In principle, two to three volumes are published each year by the Mathematical Society of Japan.

* long research papers are limited to contributions by MSJ members

Submission: Each volume should be an integrated monograph. Proceedings of conferences or collections of independent papers are not accepted. The author(s) can submit the article to one of the editors in the form of hard copy. When the article is accepted, the author(s) is (are) requested to send a camera-ready manuscript. For further technical conditions, please contact one of the editors.

Subscription/Orders: Each volume can be purchased separately. Orders from inside Japan can be made directly to the Mathematical Society of Japan. From Volume 15 onward, the distribution of the series outside Japan is conducted exclusively through World Scientific Publishing Company. For details, see the website:

Mathematical Society of Japan Memoirs http://www.worldscientific.com/series/msjm

From Volume 1 through Volume 14, orders from overseas should be made to one of the following two companies:

MARUZEN CO., LTDInternational Division P.O.Box 5050 Tokyo International Tokyo 100-3191, Japan Tel : 81 3 3273 3234 FAX : 81 3 3278 9256 E-mail : export (at) maruzen.co.jp |
JAPAN PUBLICATIONS TRADING CO., LTDP.O.Box 5030 Tokyo International Tokyo 100-3191, Japan FAX : 81 3 3292 0410 E-mail : serials (at) jptco.co.jp |

### List of Titles

#### Vol.41

- Author: Tomoki Nakanishi
- Title: Cluster Algebras and Scattering Diagrams

The theme of this monograph is the relation between cluster algebras and scattering diagrams. Cluster algebras were introduced by Fomin and Zelevinsky around 2000 as an algebraic and combinatorial structure originated in Lie theory. Recently, Gross, Hacking, Keel, and Kontsevich solved several important conjectures in cluster algebra theory by the scattering diagram method introduced in the homological mirror symmetry. This monograph is the first comprehensive exposition of this important development. The text consists of three parts. Part I is a first step guide to the theory of cluster algebras for readers without any knowledge on cluster algebras. Part II is the main part of the monograph, where we focus on the column sign-coherence of C-matrices and the Laurent positivity for cluster patterns, both of which were conjectured by Fomin and Zelevinsky and proved by Gross, Hacking, Keel, and Kontsevich based on the scattering diagram method. Part III is a self-contained exposition of several fundamental properties of cluster scattering diagrams with emphasis on the roles of the dilogarithm elements and the pentagon relation. As a specific feature of this monograph, each part is written without explicitly relying on the other parts. Thus, readers can start reading from any part depending on their interest and knowledge.

2023, 279p, ISBN: 978-4-86497-105-8

#### Vol.40

- Author: Yuji Odaka and Yoshiki Oshima
- Title: Collapsing K3 Surfaces, Tropical Geometry and Moduli Compactifications of Satake, Morgan-Shalen Type

This research monograph mainly discusses a canonical and explicit compactification of the moduli spaces of abelian varieties, K3 surfaces and compact hyperKähler varieties. For that, we use two theories of compactification---Satake compactifications for locally symmetric spaces in terms of the Lie theory, and Morgan-Shalen compactifications of complex varieties in terms of valuations. We show they coincide for Shimura varieties. The obtained compactifications are no longer varieties but we provide geometric meanings to them.

We partially prove that the boundary parametrizes collapsed limits of the Ricci-flat Kähler metrics. Such limits also coincide with a posteriori defined “tropicalized version” or equivalently the dual graphs of degenerations of original varieties. From differential geometric perspective, this work provides a moduli-theoretic framework for the limiting behavior of Ricci-flat Kähler metrics. From Lie theoretic perspective, this work provides a geometric meaning to the Satake compactification associated to adjoint representations, which are not the same as the Baily-Borel compactifications. Applying our theory to the case of one parameter maximal degeneration of K3 surfaces, we obtain proofs of conjectures of Gross-Wilson and Kontsevich-Soibelman.

We formulate general conjectures on the limits of Ricci-flat Kähler metrics in the above framework and partially prove them, but they largely remain open.

2021, 165p, ISBN: 978-4-86497-104-1

#### Vol.39

- Author: Masaharu Taniguchi
- Title: Traveling Front Solutions in Reaction-Diffusion Equations

The study on traveling fronts in reaction-diffusion equations is the first step to understand various kinds of propagation phenomena in reaction-diffusion models in natural science. One dimensional traveling fronts have been studied from the 1970s, and multidimensional ones have been studied from around 2005. This volume is a text book for graduate students to start their studies on traveling fronts. Using the phase plane analysis, we study the existence of traveling fronts in several kinds of reaction-diffusion equations. For a nonlinear reaction term, a bistable one is a typical one. For a bistable reaction-diffusion equation, we study the existence and stability of two-dimensional V-form fronts, and we also study pyramidal traveling fronts in three or higher space dimensions. The cross section of a pyramidal traveling front forms a convex polygon. It is known that the limit of a pyramidal traveling front gives a new multidimensional traveling front. For the study the multidimensional traveling front, studying properties of pyramidal traveling fronts plays an important role. In this volume, we study the existence, uniqueness and stability of a pyramidal traveling front as clearly as possible for further studies by graduate students. For a help of their studies, we briefly explain and prove the well-posedness of reaction-diffusion equations and the Schauder estimates and the maximum principles of solutions.

2021, 170p, ISBN: 978-4-86497-097-6

#### Vol.38

- Author: Alex Casella, Dominic Tate and Stephan Tillmann
- Title: Moduli spaces of real projective structures on surfaces

This book is an excellent first encounter with the burgeoning field of real projective manifolds. It gives a comprehensive introduction to the theory of real projective structures on surfaces and their moduli spaces. A central theme is an attractive parameterisation of moduli space discovered by Fock and Goncharov that allows the explicit description or analysis of many key features. These include a natural Poisson structure, the effect of projective duality, holonomy representations and the geometry of ends, to name but a few.

This book is written with two kinds of readers in mind: those who would like to learn about real projective surfaces or manifolds, and those who have a passing knowledge thereof but are interested in the geometric underpinnings of Fock and Goncharov’s parameterisation of moduli space of certain real projective structures.

The material is accessible to any mathematician interested in these topics. It is presented in a self-contained manner with minimal prerequisites. Applications of Fock and Goncharov’s parameterisation of moduli space presented in this book include new proofs of results by Teichmüller (1939) concerning hyperbolic structures, by Goldman (1990) concerning closed surfaces, and by Marquis (2010) concerning structures of finite area.

2020, 122p, ISBN: 978-4-86497-096-9

#### Vol.37

- Author: Kazuki Hiroe, Hiroshi Kawakami, Akane Nakamura and Hidetaka Sakai
- Title: 4-dimensional Painlevé-type equations

The Painlevé equations were discovered as nonlinear ordinary differential equations that define new special functions, and their importance has long been recognized. Since the 1990s, there have been many studies on various generalizations of the Painlevé equations such as discretizations, higher dimensional analogues, quantizations, and so on. The aim of this book is to provide a unified approach to understand higher dimensional analogues of the Painlevé equations from the viewpoint of the deformation theory of linear ordinary differential equations. Especially, a detailed study will be given when the phase spaces of their Hamiltonian systems are four dimensional. More specifically, starting from the classification of the Fuchsian equations with four accessory parameters, we construct a degeneration scheme of linear equations by considering confluences of singular points. Then we write down the Hamiltonians of the Painlevé-type equations associated with these resulting linear equations. The following topics are explained together with examples: spectral types of linear equations, a method to calculate the Hamiltonians, confluences of singularities and degenerations of the Painlevé-type equations, the correspondence between linear equations or their spectral types through the Laplace transform. In addition, Appendix 1 discusses symmetries of moduli spaces of linear equations. As its application, it is shown that the equations obtained in this book constitute a complete list of 4-dimensional Painlevé-type equations corresponding to unramified linear equations. Appendix 2 gives a list of the 4-dimensional Painlevé-type equations corresponding to ramified linear equations.

2018, 172p, ISBN: 978-4-86497-087-7

#### Vol.36

- Author: Soichiro Katayama
- Title: Global solutions and the asymptotic behavior for nonlinear wave equations with small initial data

In the study of the Cauchy problem for nonlinear wave equations with small initial data, the case where the nonlinearity has the critical power is of special interest. In this case, depending on the structure of the nonlinearity, one may observe global existence and finite time blow-up of solutions. In 80’s, Klainerman introduced a sufficient condition, called the null condition, for the small data global existence in the critical case. Recently, weaker sufficient conditions are also studied.

This volume offers a comprehensive survey of the theory of nonlinear wave equations, including the classical local existence theorem, the global existence in the supercritical case, the finite time blow-up and the lifespan estimate in the critical case, and the global existence under the null condition in two and three space dimensions. The main tool here is the so-called vector field method. This volume also contains recent progress in the small data global existence under some conditions weaker than the null condition, and it is shown that a wide variety of the asymptotic behavior is observed under such weaker conditions.

This volume is written not only for researchers, but also for graduate students who are interested in nonlinear wave equations. The exposition is intended to be self-contained and a complete proof is given for each theorem.

2017, 298p, ISBN: 978-4-86497-054-9

#### Vol.35

- Author: Osamu Fujino
- Title: Foundations of the minimal model program

Around 1980, Shigefumi Mori initiated a new theory, which is now known as the minimal model program or Mori theory, for higher-dimensional algebraic varieties. This theory has developed into a powerful tool with applications to diverse questions in algebraic geometry and related fields.

One of the main purposes of this book is to establish the fundamental theorems of the minimal model program, that is, various Kodaira type vanishing theorems, the cone and contraction theorem, and so on, for quasi-log schemes. The notion of quasi-log schemes was introduced by Florin Ambro and is now indispensable for the study of semi-log canonical pairs from the cohomological point of view. By the recent developments of the minimal model program, we know that the appropriate singularities to permit on the varieties at the boundaries of moduli spaces are semi-log canonical. In order to achieve this goal, we generalize Kollár's injectivity, torsion-free, and vanishing theorems for reducible varieties by using the theory of mixed Hodge structures on cohomology with compact support. We also review many important classical Kodaira type vanishing theorems in detail and explain the basic results of the minimal model program for the reader's convenience.

2017, 289p, ISBN: 978-4-86497-045-7

#### Vol.34

- Author: Martin T. Barlow, Tibor Jordán and Andrzej Zuk
- Title: Discrete Geometric Analysis

This is a volume of lecture notes based on three series of lectures given by visiting professors of RIMS, Kyoto University during the year-long project "Discrete Geometric Analysis", which took place in the Japanese academic year 2012-13. The aim of the project was to make comprehensive research on topics related to discreteness in geometry, analysis and optimization.

Discrete geometric analysis is a hybrid field of several traditional disciplines, including graph theory, geometry, discrete group theory, and probability. The name of the area was coined by Toshikazu Sunada, and since being introduced, it has been extending and making new interactions with many other fields.

This volume consists of three chapters:

(I) Loop Erased Walks and Uniform Spanning Trees, by Martin T. Barlow.

(II) Combinatorial Rigidity: Graphs and Matroids in the Theory of Rigid Frameworks, by Tibor Jordán.

(III) Analysis and Geometry on Groups, by Andrzej Zuk.

The lecture notes are useful surveys that provide an introduction to the history and recent progress in the areas covered. They will also help researchers who work in related interdisciplinary fields to gain an understanding of the material from the viewpoint of discrete geometric analysis.

2016, 157p, ISBN: 978-4-86497-035-8

#### Vol.33

- Author: Masaki Maruyama with collaboration of T. Abe and M. Inaba
- Title: Moduli spaces of stable sheaves on schemes

restriction theorems, boundedness and the GIT construction

The notion of stability for algebraic vector bundles on curves was originally introduced by Mumford, and moduli spaces of semi-stable vector bundles were studied intensively by Indian mathematicians. The notion of stability for algebraic sheaves was generalized to higher dimensional varieties. The study of moduli spaces of algebraic sheaves not only on curves but also on higher dimensional algebraic varieties has attracted much interest for decades and its importance has been increasing not only in algebraic geometry but also in related fields as differential geometry, mathematical physics.

Masaki Maruyama is one of the pioneers in the theory of algebraic vector bundles on higher dimensional algebraic varieties. This book is a posthumous publication of his manuscript. It starts with basic concepts such as stability of sheaves, Harder-Narasimhan filtration and generalities on boundedness of sheaves. It then presents fundamental theorems on semi-stable sheaves : restriction theorems of semi-stable sheaves, boundedness of semi-stable sheaves, tensor products of semi-stable sheaves. Finally, after constructing quote-schemes, it explains the construction of the moduli space of semi-stable sheaves. The theorems are stated in a general setting and the proofs are rigorous.

2016, 154p, ISBN: 978-4-86497-034-1

#### Vol.32

- Author: Hiroshi Isozaki and Yaroslav Kurylev
- Title: Introduction to spectral theory and inverse problem on asymptotically hyperbolic manifolds

This manuscript is devoted to a rigorous and detailed exposition of the spectral theory and associated forward and inverse scattering problems for the Laplace-Beltrami operators on asymptotically hyperbolic manifolds. Based upon the classical stationary scattering theory in ℝn, the key point of the approach is the generalized Fourier transform, which serves as the basic tool to introduce and analyse the time-dependent wave operators and the S-matrix. The crucial role is played by the characterization of the space of the scattering solutions for the Helmholtz equations utilizing a properly defined Besov-type space. After developing the scattering theory, we describe, for some cases, the inverse scattering on the asymptotically hyperbolic manifolds by adopting, for the considered case, the boundary control method for inverse problems.

The manuscript is aimed at graduate students and young mathematicians interested in spectral and scattering theories, analysis on hyperbolic manifolds and theory of inverse problems. We try to make it self-consistent and, to a large extent, not dependent on the existing treatises on these topics. To our best knowledge, it is the first comprehensive description of these theories in the context of the asymptotically hyperbolic manifolds.

2014, 251p, ISBN: 978-4-86497-021-1

#### Vol.31

- Author: Satoshi Takanobu
- Title: Bohr-Jessen Limit Theorem, Revisited

This book is a self-contained exposition on the Bohr-Jessen limit theorem. This limit theorem, which is concerned with the behavior of the Riemann zeta function ζ(s) on the line Res=σ, where 1/2<σ≤1, was found by Bohr-Jessen in the early 1930s. After Bohr-Jessen, alternative proofs were given by Jessen-Wintner, Borchsenius-Jessen, Laurinčikas, Matsumoto and others. They dealt with this within the framework of probability theory. Their formulation, originated by Jessen-Wintner, is standard nowadays. The present book proposes a new approach for the formulation to refine their works. By this method, the whole story of the proof of the Bohr-Jessen limit theorem will now become clearer, so that the reader must be able to understand the essence of the proof in depth but without difficulty.

2013, 216p, ISBN: 978-4-86497-019-8

#### Vol.30

- Author: Tatsuo Nishitani
- Title: Cauchy Problem for Noneffectively Hyperbolic Operators

At a double characteristic point of a differential operator with real characteristics, the linearization of the Hamilton vector field of the principal symbol is called the Hamilton map and according to either the Hamilton map has non-zero real eigenvalues or not, the operator is said to be effectively hyperbolic or noneffectively hyperbolic.

For noneffectively hyperbolic operators, it was proved in the late of 1970s that for the Cauchy problem to be C∞ well posed the subprincipal symbol has to be real and bounded, in modulus, by the sum of modulus of pure imaginary eigenvalues of the Hamilton map.

It has been recognized that what is crucial to the C∞ well-posedness is not only the Hamilton map but also the behavior of orbits of the Hamilton flow near the double characteristic manifold and the Hamilton map itself is not enough to determine completely the behavior of orbits of the flow. Strikingly enough, if there is an orbit of the Hamilton flow which lands tangentially on the double characteristic manifold then the Cauchy problem is not C∞ well posed even though the Levi condition is satisfied, only well posed in much smaller function spaces, the Gevrey class of order 1≤s<5 and not well posed in the Gevrey class of order s>5.

In this lecture, we provide a general picture of the Cauchy problem for noneffectively hyperbolic operators, from the view point that the Hamilton map and the geometry of orbits of the Hamilton flow completely characterizes the well/not well-posedness of the Cauchy problem, exposing well/not well-posed results of the Cauchy problem with detailed proofs.

2013, 170p, ISBN: 978-4-86497-018-1

#### Vol.29

- Author: Takeshi Hirai, Akihito Hora and Etsuko Hirai
- Title: Projective representations and spin characters of complex reflection groups G(m,p,n) and G(m,p,∞)

This volume consists of one expository paper and two research papers: 1. T. Hirai, A. Hora and E. Hirai, Introductory expositions on projective representations of groups (referred as [E]); 2. T. Hirai, E. Hirai and A. Hora, Projective representations and spin characters of complex reflection groups G(m,p,n) and G(m,p,∞), I; 3. T. Hirai, A. Hora and E. Hirai, Projective representations and spin characters of complex reflection groups G(m,p,n) and G(m,p,∞), II, Case of generalized symmetric groups.

Since Schur's trilogy on 1904 and so on, many mathematicians studied projective representations of groups and algebras, and also of their characters. Nevertheless, to invite mathematicians to this interesting and important areas, the paper [E] collects introductory expositions, with a historical plotting, for the theory of projective representations of groups and their characters. The paper [I] treats general theory for projective (or spin) representations and spin characters of complex reflection groups G(m,p,n) and G(m,p,∞)=limn→∞G(m,p,n), and clarifies the intimate relations between mother groups, G(m,1,n), G(m,1,∞)(p=1), called generalized symmetric groups, and their child groups, G(m,p,n), G(m,p,∞)(p|m,p>1). Also we treat explicitly a case of spin type in connection with the case of non-spin type (i.e. of linear representations). A detailed and general account on the so-called Vershik-Kerov theory on limits of characters is added. The paper [II] treats spin irreducible representations and spin characters of generalized symmetric groups (mother groups) for other spin types.

2013, 272p, ISBN: 978-4-86497-017-4

#### Vol.28

- Author: Toshio Oshima
- Title: Fractional calculus of Weyl algebra and Fuchsian differential equations

In this book we give a unified interpretation of confluences, contiguity relations and Katz's middle convolutions for linear ordinary differential equations with polynomial coefficients and their generalization to partial differential equations. The integral representations and series expansions of their solutions are also within our interpretation. As an application to Fuchsian differential equations on the Riemann sphere, we construct a universal model of Fuchsian differential equations with a given spectral type, in particular, we construct a single ordinary differential equation without apparent singularities corresponding to any rigid local system on the Riemann sphere, whose existence was an open problem presented by N. Katz. Furthermore we obtain fundamental properties of the solutions of the rigid Fuchsian differential equations such as their connection coefficients and the necessary and sufficient condition for the irreducibility of their monodromy groups. We give many examples calculated by our fractional calculus.

2012, 203p, ISBN: 978-4-86497-016-7

#### Vol.27

- Author: Suhyoung Choi
- Title: Geometric Structures on 2-Orbifolds: Exploration of Discrete Symmetry

This book exposes the connection between the low-dimensional orbifold theory and geometry that was first discovered by Thurston in 1970s providing a key tool in his proof of the hyperbolization of Haken 3-manifolds. Our main aims are to explain most of the topology of orbifolds but to explain the geometric structure theory only for 2-dimensional orbifolds, including their Teichmüller (Fricke) spaces. We tried to collect the theory of orbifolds scattered in various literatures for our purposes. Here, we set out to write down the traditional approach to orbifolds using charts, and we include the categorical definition using groupoids. We will also maintain a collection of illustrative MathematicaTM packages at our homepages.

2012, 182p, ISBN: 978-4-931469-68-6

#### Vol.26

- Author: Shinya Nishibata and Masahiro Suzuki
- Title: Hierarchy of semiconductor equations: relaxation limits with initial layers for large initial data

This volume provides a recent study of mathematical research on semiconductor equations. With recent developments in semiconductor technology, several mathematical models have been established to analyze and to simulate the behavior of electron flow in semiconductor devices. Among them, a hydrodynamic, an energy-transport and a drift-diffusion models are frequently used for the device simulation with the suitable choice, depending on the purpose of the device usage. Hence, it is interesting and important not only in mathematics but also in engineering to study a model hierarchy, relations among these models. The model hierarchy has been formally understood by relaxation limits letting the physical parameters, called relaxation times, tend to zero. The main concern of this volume is the mathematical justification of the relaxation limits. Precisely, we show that the time global solution for the hydrodynamic model converges to that for the energy-transport model as a momentum relaxation time tends to zero. Moreover, it is shown that the solution for the energy-transport model converges to that for the drift-diffusion model as an energy relaxation time tends to zero. For beginners' help, this volume also presents the physical background of the semiconductor devices, the derivation of the models, and the basic mathematical results such as the unique existence of time local solutions.

2011, 109p, ISBN: 978-4-931469-66-2

#### Vol.25

- Author: Hiroshi Sugita
- Title: Monte Carlo method, random number, and pseudorandom number

Although the Monte Carlo method is used in so many fields, its mathematical foundation has been weak until now because of the fundamental problem that a computer cannot generate random numbers. This book presents a strong mathematical formulation of the Monte Carlo method which is based on the theory of random number by Kolmogorov and others and that of pseudorandom number by Blum and others. As a result, we see that the Monte Carlo method may not need random numbers and pseudorandom numbers may suffice. In particular, for the Monte Carlo integration, there exist pseudorandom numbers which serve as complete substitutes for random numbers.

2011, 133p, ISBN: 978-4-931469-65-5

#### Vol.24

- Author: Taro Asuke
- Title: Godbillon-Vey class of transversely holomorphic foliations

This volume provides a study of the Godbillon-Vey class and other real secondary characteristic classes of transversely holomorphic foliations. One of the main tools in the study is complex secondary characteristic classes. Intended to be self-contained and introductory, this volume contains a brief survey of the theory of secondary characteristic classes of transversely holomorphic foliations. A construction of secondary characteristic classes of families of such foliations is also included. By means of these classes, new proofs of the rigidity of the Godbillon-Vey class in the category of transversely holomorphic foliations are given.

2010, 130p, ISBN: 978-4-931469-61-7

#### Vol.23

- Author: Armen Sergeev
- Title: Kähler geometry of loop spaces

In this book we study three important classes of infinite-dimensional Kähler manifolds --- loop spaces of compact Lie groups, Teichmüller spaces of complex structures on loop spaces, and Grassmannians of Hilbert spaces. Each of these manifolds has a rich Kähler geometry, considered in the first part of the book, and may be considered as a universal object in a category, containing all its finite-dimensional counterparts.

On the other hand, these manifolds are closely related to string theory. This motivates our interest in their geometric quantization presented in the second part of the book together with a brief survey of geometric quantization of finite-dimensional Kähler manifolds.

The book is provided with an introductory chapter containing basic notions on infinite-dimensional Frechet manifolds and Frechet Lie groups. It can also serve as an accessible introduction to Kähler geometry of infinite-dimensional complex manifolds with special attention to the aforementioned three particular classes.

It may be interesting for mathematicians working with infinite-dimensional complex manifolds and physicists dealing with string theory.

2010, 212p, ISBN: 978-4-931469-60-0

#### Vol.22

- Author: Michael Ruzhansky and James Smith
- Title: Dispersive and Strichartz estimates for hyperbolic equations with constant coefficients

In this work dispersive and Strichartz estimates for solutions to general strictly hyperbolic partial differential equations with constant coefficients with lower order terms are considered. The global time decay estimates of Lp−Lq norms of propagators are analysed in detail and it is described how the time decay rates depend on the geometry of the problem. For these purposes, the frequency space is separated in several zones each giving a certain decay rate. Geometric conditions on characteristics responsible for the particular decay are presented.

A comprehensive analysis is carried out for strictly hyperbolic equations of high orders with lower order terms of a general form. Most of the analysis also applies to equations with are pseudo-differential in the space variables. We also show how the obtained estimates apply to solutions to hyperbolic systems with constant coefficients. The applications of the obtained results include the time decay estimates for the solutions to the Fokker-Planck equation and for the solutions of semilinear hyperbolic equations.

2010, 147p, ISBN: 978-4-931469-57-0

#### Vol.21

- Author: Gautami Bhowmik, Kohji Matsumoto and Hirofumi Tsumura (Eds.)
- Title: Algebraic and Analytic Aspects of Zeta Functions and L-functions

This volume contains lectures presented at the French-Japanese Winter School on Zeta and L-functions, held at Muira, Japan, 2008. The main aim of the School was to study various aspects of zeta and L-functions with special emphasis on recent developments. A series of detailed lectures were given by experts in topics that include height zeta-functions, spherical functions and Igusa zeta-functions, multiple zeta values and multiple zeta-functions, classes of Euler products of zeta-functions, and L-functions associated with modular forms. This volume should be helpful to future generations in their study of the fascinating theory of zeta and L-functions.

2010, 183p, ISBN: 978-4-931469-56-3

#### Vol.20

- Author: Danny Calegari
- Title: scl

This book is a comprehensive introduction to the theory of stable commutator length, an important subfield of quantitative topology, with substantial connections to 2-manifolds, dynamics, geometric group theory, bouded cohomology, symplectic topology, and many other subjects. We use constructive methods whenever possible, and focus on fundamental and explicit examples. We give a self-contained presentation of several foundational results in the theory, including Bavard's Duality Theorem, the Spectral Gap Theorem, the Rationality Theorem, and the Central Limit Theorem. The contents should be accessible to any mathematician interested in these subjects, and are presented with a minimal number of prerequisites, but with a view to applications in many areas of mathematics.

2009, 217p, ISBN: 978-4-931469-53-2

#### Vol.19

- Author: Joseph Najnudel, Bernard Roynette and Marc Yor
- Title: A Global View of Brownian Penalisations

The present volume is an expository monograph on Brownian penalisation, an important new notion the authors introduced to the theory of Wiener measure and Markov processes. It will serve as a concise guidebook for students and researchers who study probability theory, stochastic processes and mathematical finance.

2009, 137p, ISBN: 978-4-931469-52-5

#### Vol.18

- Author: Yasutaka Ihara
- Title: On Congruence Monodromy Problems

It is now well-known that the group SL2(Z[1p]) and the system of modular curves over Fp2 are ``closely related", and that the latter provided first ``examples" of curves over finite fields having many rational points. However, the ``three basic relationships", which really justify the former to be called the arithmetic fundamental group of the latter, still do not seem to be so commonly known.

This book consists of two parts; a reproduction of the author's unpublished Lecture Notes (1968,69), and Author's Notes (2008). The former starts with explicit three main conjectural relationships for more general cases and gives various approaches towards their proofs. Though remained formally unpublished, these Lecture Notes had been widely circulated and have stimulated researches in various directions. The main conjectures themselves have also been proved since then. The Author's Notes (2008) gives detailed explanations of these developments, together with open problems.

2008, 230p, ISBN: 978-4-931469-50-1

#### Vol.17

- Author: Arkady Berenstein, David Kazhdan, Cédric Lecouvey, Masato Okado, Anne Schilling, Taichiro Takagi and Alexander Veselov
- Title: Combinatorial Aspect of Integrable Systems

This volume is a collection of six papers based on the expository lectures of the workshop ``Combinatorial Aspect of Integrable Systems" held at RIMS during July 26--30, 2004, as a part of the Project Research 2004 ``Method of Algebraic Analysis in Integrable Systems".

The topics range over crystal bases of quantum groups, its algebro-geometric analogue known as geometric crystal, generalizations of Robinson-Schensted type correspondence, fermionic formula related to Bethe ansatz, applications of crystal bases to soliton celluar automata, Yang-Baxter maps, and integrable discrete dynamics.

All the papers are friendly written with many illustrative examples and intimately related to each other. This volume will serve as a good guide for researchers and graduate students who are interested in this fascinating subject.

2007, 167p, ISBN: 978-4-931469-37-2

#### Vol.16

- Author: Brian H. Bowditch
- Title: A course on geometric group theory

This volume is intended as a self-contained introduction to the basic notions of geometric group theory, the main ideas being illustrated with various examples and exercises. One goal is to establish the foundations of the theory of hyperbolic groups. There is a brief discussion of classical hyperbolic geometry, with a view to motivating and illustrating this.

The notes are based on a course given by the author at the Tokyo Institute of Technology, intended for fourth year undergraduates and graduate students, and could form the basis of a similar course elsewhere. Many references to more sophisticated material are given, and the work concludes with a discussion of various areas of recent and current research.

2006, 104p, ISBN: 4-931469-35-3

#### Vol.15

- Author: Valery Alexeev and Viacheslav V. Nikulin
- Title: Del Pezzo and K3 surfaces

The present volume is a self-contained exposition on the complete classification of singular del Pezzo surfaces of index one or two. The method of the classification used here depends on the intriguing interplay between del Pezzo surfaces and K3 surfaces, between geometry of exceptional divisors and the theory of hyperbolic lattices.

The topics involved contain hot issues of research in algebraic geometry, group theory and mathematical physics.

This book, written by two leading researchers of the subjects, is not only a beautiful and accessible survey on del Pezzo surfaces and K3 surfaces, but also an excellent introduction to the general theory of Q-Fano varieties.

2006, 149p, ISBN:4-931469-34-5

#### Vol.14

- Author: Noboru Nakayama
- Title: Zariski-decomposition and Abundance

Dr. Noboru Nakayama, the author of this book, studies the birational classification of algebraic varieties and of compact complex manifolds. This book is a collection of his works on the numerical aspects of divisors of algebraic varieties.

The notion of Zariski-decomposition introduced by Oscar Zariski is a powerful tool in the study of open surfaces. In the higher dimensional generalization, we encounter interesting phenomena on the numerical aspects of divisors. The author treats the higher dimensional Zariski-decomposition systematically.

The abundance conjecture predicts that the numerical Kodaira dimension of a minimal variety coincides with the usual Kodaira dimension. The Kodaira dimension is an invariant of the canonical divisor of a variety. The numerical analogue used to be defined only for nef divisors, but it is now extended to arbitrary divisors in this book. Explained in details are many important results on the numerical Kodaira dimension related to the abundance, to the addition theorem for fiber spaces, and to the deformation invariance.

2004, 277p, ISBN:4-931469-31-0

#### Vol.13

- Author: Shigeaki Koike
- Title: A beginner's guide to the theory of viscosity solutions

The notion of viscosity solutions was first introduced by M. G. Crandall and P.-L. Lions in 1981 to study first-order partial differential equations of nondivergence form, typically, Hamilton-Jacobi equations. Later, the study of viscosity solutions was extended to second-order elliptic/parabolic equations. It has turned out by many researchers that the viscosity solution theory is a powerful tool to investigate fully nonlinear second-order (degenerate) elliptic/parabolic equations arising in optimal control problems, differential games, mean curvature flow, phase transitions, mathematical finance, conservation laws, variational problems, etc. This text is an introduction to the viscosity solution theory as indicated by the title.

After a brief history of weak solutions, it presents several uniqueness (comparison principle) and existence results, which are main issues. For further topics, it chooses generalized boundary value problems and regularity results. In Appendix, which is the hardest part, it provides proofs of several important propositions.

Dr. Koike's current mathematical interests still lie in the viscosity solution theory and its applications.

2004, 132p, ISBN:4-931469-28-0, Not in stock

#### Vol.12

- Author: Yves André

(with appendices by F. Kato and N. Tsuzuki) - Title: Period mappings and differential equations. Form C to C_p

Tohoku-Hokkaido lectures in Arithmetic Geometry

The theorey of period mappings has played a central role in nineteen-century mathematics as a fertile place of interaction between algebraic and differential geometry, differential equations, and group theory, from Gauss and Riemann to Klein and Poincaré. This text is an introduction to the p-adic counterpart of this theory, which is much more recent and still mysterious. It should be of interest both to some complex geometers and to some arithmetic geometers.

Starting with an introduction to p-adic analytic geometry (in the sense of Berkovich), it then presents the Rapoport-Zink theory of period mappings, emphasizing the relation with Picard-Fuchs differential equtions. a new theory of fundamental groups, orbifolds, and uniformizing equations (in the p-adic context) accounts for the group-theoretic aspects of these period mappings. The books ends with a theory of p-adictriangle groups.

Dr. André's current mathematical interests lie in arithmetic geometry and in the theory of motives.

2003, 246p, ISBN4-931469-22-1, Not in stock

#### Vol.11

- Author: John R. Stembridge, Jean-Yves Thibon and Marc A. A. van Leeuwen
- Title: Interaction of combinatorics and representation theory

This volume consisting of two research papers and one survey paper is a good guide to look into a new emerging field, which stems from the interaction of combinatorics and representation theory.

Dr. John Stembrige is famous for his study on combinatorics in Lie algebra representations, Coxeter/Weyl groups, and other topics. Also he is the author of the Maple package software ``SF'' (Schur functions), ``coxeter/weyl'', and ``posets''.

Dr. Jean-Yves Thibon is one of the most active researchers in this field and is famous for many collaborated works with Alain Lascoux and Bernard Leclerc and other famous researchers.

Dr. Marc van Leeuwen is famous in the field of manipulation of Young tableaux and its related topics. He is one of the authors of the software package ``LiE'' for Lie group computation.

2001, 145p, ISBN4-931469-14-0, Not in stock

#### Vol.10

- Author: Yuri G. Prokhorov
- Title: Lectures on complements on log surfaces

Dr. Yuri Prokhorov, the author of this book, is an expert in birational geometry in the field of algebraic geometry. This book is the first significant expository lecture for ``complements''; this notion was introduced by Vyacheslav Shokurov quite recently and is important in understanding singularities of a pair consisting of an algebraic variety and a divisor on it. There is currently much ongoing research on this subject, a very active area in algebraic geometry.

This book helps the reader to understand the ``complement'' concept and provides the basic knowledge about the singularities of a pair. The author gives a simple proof of the boundedness of the complements for two dimensional pairs under some restrictive condition, where this boundedness has been conjectured by Shokurov for every dimension. This book contains information and encouragement necessary to attack the problem of the higher dimensional case.

2001, 130p, ISBN4-931469-12-4, Not in stock

#### Vol.9

- Author: Peter Orlik and Hiroaki Terao
- Title: Arrangements and hypergeometric integrals

An affine arrangement of hyperplanes is a finite collection of one-codimensional affine linear spaces in Cn. P. Orlik and H. Terao are leading specialists in the theory of arrangements and the co-authors of the well-known book ``Arrangements of Hyperplanes''. In this monograph, they give an introductory survey which also contains the recent progress in the theory of hypergeometric functions. The main argument is done from the arrangement-theoretic point of view. This will be a nice text for a student to begin the study of hypergeometric functions.

2001, 112p, ISBN4-931469-10-8, Not in stock

#### Vol.8

- Author: Eric M. Opdam
- Title: Lecture notes on Dunkl operators for real and complex reflection groups

Eric M. Opdam studied a generalization of the system of differential equations satisfies by the Harish-Chandra spherical functions, and with Gerrit Heckman established the theory of Heckman-Opdam hypergeometric functions by the use of a trigonometric extension of Dunkl operators.

In this note he introduces this theory, and includes a recent result on the harmonic analysis of the hypergeometric functions and also an application of Dunkl operators to the study of reflection groups.

2000, 90p, ISBN4-931469-08-6, Not in stock

#### Vol.7

- Author: Vladimir Georgiev
- Title: Semilinear hyperbolic equations

Most of the standard theorems of global in time existence for solutions of the nonlinear evolution equations in mathematical physics depend heavily upon estimates for the solution's total energy. Typically, to prove the global existence of a smooth solution, one argues that a certain amount of energy would necessarily be dissipated in the development of a singularity, which is limited by virtue of small data assumptions so far, except for some semilinear evolution equations with good sign.

Under the small data assumption, the main observation is devoted to the investigation of the dissipative mechanism of linearized equations, which is described by the decay estimate of solutions mathematically. V. Georgiev is one of the most excellent mathematicians who created outstanding a priori estimates about hyperbolic equations in mathematical physics, which yield solutions of the corresponding nonlinear hyperbolic equations under small data assumption.

The aim of this lecture note is to explain how to derive sharp a priori estimates which enable us to prove a global in time existence of solutions to semilinear wave equation and non-linear Klein-Gordon equation.

The core of the lecture note is Section 8, which is devoted to Fourier transform on manifolds with constant negative curvature. Combining this with the interpolation method and psudodifferential operator approach enables us to obtain better Lp weighted a priori estimates.

Key words: semilinear wave equation, Fourier transform on hyperboloid, Sobolev spaces on hyperboloid, Klein - Gordon equation

2000, 209p, ISBN4-931469-07-8, Not in stock

#### Vol.6

- Author: Kong De-xing
- Title: Cauchy problem for quasilinear hyperbolic systems

This book is concerned with Cauchy problem for quasilinear hyperbolic systems. By introducing the concepts weak linear degeneracy and matching condition, we give a systematic presentation on the global existence, the large time behaviour and the blow-up phenomenon, particularly, the life span of C1 solutions to the Cauchy problem with small and decaying initial data. Some successful applications of our general theory are given to the quasilinear canonical system related to the Monge-Amp`ere equation, the system of nonlinear three-wave interaction in plasma physics, the nonlinear wave equation with higher order dissipation, the system of one-dimensional gas dynamics with nonlinear dissipation, the system of motion of an elastic string, the system of plane elastic waves for hyperelastic materials and so on.

Key words and phrases: Quasilinear hyperbolic system, Cauchy problem, C1 solution, blow-up, life span.

2000, 213p, ISBN4-931469-06-X

#### Vol.5

- Author: Daryl Cooper, Craig D. Hodgson and Steven P. Kerckhoff
- Title: Three-dimensional orbifolds and cone-manifolds

This volume provides an excellent introduction of the statement and main ideas in the proof of the orbifold theorem announced by Thurston in late 1981. It is based on the authors' lecture series entitled ``Geometric Structures on 3-Dimensional Orbifolds" which was featured in the third MSJ Regional Workshop on ``Cone-Manifolds and Hyperbolic Geometry" held on July 1-10, 1998, at Tokyo Institute of Technology. The orbifold theorem shows the existence of geometric structures on many 3-orbifolds and on 3-manifolds with symmetry. The authors develop the basic properties of orbifolds and cone-manifolds, extends many ideas from the differential geometry to the setting of cone-manifolds and outlines a proof of the orbifold theorem.

2000, 170p, ISBN4-931469-05-1, Not in stock

#### Vol.4

- Author: Atsushi Matsuo and Kiyokazu Nagatomo
- Title: Axioms for a vertex algebra and the locality of quantum fields

Dr. A. Matsuo has been working on various mathematical structures related to two-dimensional conformal field theory. He is famous for his study on the Knizhnik-Zamolodchkov equation and its analogues. He is recently interested in searching for examples of vertex algebras having interesting symmetries.

Dr. K. Nagatomo is working on the theory of vertex oeprator algebras and related topics. His interests include applications of the representation theory of infinite dimensional algebras to completely integrable systems. He dedicates this paper to Dr. Matsuo's daughter who was born a few days ago.

1999, 110p, ISBN4-931469-04-3, Not in stock

#### Vol.3

- Author: Tomotada Ohtsuki
- Title: Combinatorial quantum method in 3-dimensional topology

This book is based on a series of lectures by the author in the workshop "Combinatorial Quantum Method in 3-dimensional Topology" held in Oiwake Seminar House of Waseda University in the end of September, 1996.

After the discovery of the Jones polynomial at the middle of 1980's, many new invariants of knots and 3-manifolds, what we call quantum invariants, have been found. At the present we have two key words to understand quantum invariants of knots; "the Kontsevich invariant" and "Vassiliev invariants". Correspondingly we have also two notions for 3-manifold invariants; "The LMO invariant" and "finite type invariants". The aim of this book is to explain about construction and basic properties of these invariants and how to understand quantum invariants via these invariants.

1999, 83p, ISBN4-931469-03-5, Not in stock

#### Vol.2

- Author: Masako Takahashi, Mitsuhiro Okada and Mariangiola Dezani-Ciancaglini (Eds.)
- Title: Theories of types and proofs

This is an excellent collection of refereed articles on theories of types and proofs. The articles are written by noted experts in the area. In addition to the value of the individual articles, the collection is notable for covering a range of related topics. The collection begins with useful primer on the subject that will make the subsequent articles more accessible to potential readers. Following the primer, there are good articles on traditional topics in type assignment systems. These are followed by explanations of applications to program analysis and a series of articles on application to logic. The collection includes articles on intuitionistic logic, a standard use of type-theoretic notions, and concludes with an article on linear logic.

1998, 295p, ISBN4-931469-02-7

#### Vol.1

- Author: Ivan Cherednik, Peter J. Forrester and Denis Uglov
- Title: Quantum many-body problems and representation theory

Dr. I. Cherednik is famous for introducing the double affine Hecke algebras, which is the main topics in his article ``Lectures on affine Knizhnik-Zamolodchikov equations, . . .''. This focuses on the equivalence of the affine Knizhnik-Zamolodchikov equations and the quantum many-body problems. It also serves as an introduction to the new theory of the spherical and the hypergeometric functions based on the affine and the double affine Hecke algebras.

Dr. P. J. Forrester is an expert in random matrix theory and Coulomb systems. Dr. Forrester has also been a pioneer in the application of Jack symmetric functions to statistical physics. The article ``Random Matrices, Log-Gases and the Calogero-Sutherland Model'' deals precisely with these three areas.

Dr. D. Uglov is actively working in the quantum many-body problems and the related representation theory. The article ``Symmetric functions and the Yangian decomposition . . .'' is an exposition of his recent works on these topics.

1998, 241p, ISBN4-931469-01-9, Not in stock