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## Research Theme

**Shin-ichi DOI **

**Partial Differential Equations **

Partial differential equations have their origins in various fields such as mathematical physics, differential geometry, and technology. Among them I am particularly interested in the partial differential equations that describe wave propagation phenomena: hyperbolic equations and dispersive equations. A typical example of the former is the wave equation, and that of the latter is the Schroedinger equation. For many years I have studied basic problems for these equations: existence and uniqueness of solutions, structure of singularities of solutions, asymptotic behavior of solutions, and spectral properties. Recently I make efforts to understand how the singularities of solutions for Schroedinger equations or, more generally, dispersive equations propagate. The center of this problem is to determine when and how the singularities of solutions for the dispersive equations can be described by the asymptotic behavior of solutions for the associated canonical equations.

### Ichiro ENOKI

**Complex Differential Geometry **

A complex manifold is, locally, the world build out of open subsets of complex Euclid spaces and holomorphic functions on them. If two holomorphic functions are defined on a connected set and coincide on an open subset, then they coincide on the whole. Complex manifolds inherit this kind of property from holomorphic functions. That is, they are stiff and hard in a sense. It seems to me that complex manifolds are not metallically hard but have common warm feeling with wood or bamboo, which have grain and gnarl. Analytic continuations, as you learned in the course on the function theory of complex variables, is analogous to the process of growth of plants. Instead of considering whole holomorphic functions, a class of complex manifold can be build out of polynomials. This is the world of complex algebraic manifolds, the most fertile area in the world of complex manifolds. To complex algebraic manifolds, since they are algebraically defined, algebraic methods are of course useful to study them. In certain cases, however, transcendental methods (the word "transcendental" means only "not algebraic") are powerful. For example, one of the simplest proof for the fundamental theorem of algebra is given by the function theory of one complex variable. These two methods have been competing with each other since the very beginning of the history of the study of complex manifolds. This competing seems to me the prime mover of the development of the theory of complex manifolds. Comparing the world of complex manifold to the earth, the world of complex algebraic manifold is to compare to continents, and the boundary to continental shells. The reason I wanted to begin to study complex manifolds was I heard the Kodaira embedding theorem, which characterizes complex algebraic manifolds in the whole complex manifolds. The place I begin to study is, however, something like the North Pole or the Mariana Trench. Now the center of my interest is in the study of complex algebraic manifolds by transcendental methods. (Thus I have reached land but I found this was a jungle.)

### Akio FUJIWARA

**Mathematical Engineering **

"What is information?'' Having this naive yet profound question in mind, I have been studying mainly on quantum information theory, noncommutative statistics, information geometry, and algorithmic randomness theory. Quantum information theory is a quantum extension of classical information theory. Since Shor's invention of a factorizing algorithm on a quantum computer, it is one of the most exciting research field in information science. Among diverse branches of research subjects, I am interested in quantum channel coding theory, especially in the additivity problem of the quantum channel capacity. One may conceive of quantum theory as a noncommutative extension of probability theory. Likewise, noncommutative statistics is a quantum extension of classical statistics. It aims to find the optimal strategy for identifying an unknown quantum object from a statistical point of view. My interest includes quantum state/channel estimation, and quantum hypothesis testing. Probability theory is usually regarded as a branch of analysis. Yet it is also possible to investigate the space of probability measures from a differential geometrical point of view. Information geometry deals with a pair of affine connections which are mutually dual (conjugate) with respect to a Riemannian metric on a statistical manifold. It is well known that geometrical methods provide a useful guiding principle as well as insightful intuition in classical statistics. I am extending such a geometrical structure to a quantum regime, not just formally but admitting a fruitful operational interpretation. Finally, I am now delving into algorithmic randomness theory from an information geometrical point of view. My dream is to reformulate thermal/statistical physics in terms of algorithmic information theory.

### Masaaki FUKASAWA

**Mathematical Statistics, Probability, Mathematical Finance **

I am mainly interested in asymptotic distributions of stochastic processes.

It is often that we encounter an equation describing a natural or social phenomenon which is so complicated that we can derive essentially nothing from its precise form. By taking a limit in an appropriate way, we are however, sometimes able to get a simple asymptotic solution which is enough to solve practical problems underlying the equation. For example, we do not need to solve the Schroedinger equation in order to calculate ballistics, even though the quantum mechanics is behind all the dynamics. This is because by letting Planck's constant, which is very small, converge to 0, we actually come back to the classical world of the Newton mechanics.

My interest is to formulate this kind of perturbation theories in terms of limit theorems with mathematical proof. Recently I have been considered problems in mathematical finance to give, say, the proof of the validity of a singular perturbation expansion of a derivative price, and asymptotically optimal discrete-hedging strategies under high-frequency trading.

Similarly to that physics has provided mathematics new ideas, financial engineering has recently raised many problems stimulating mathematicians. They are complicated as usual and so, an asymptotic analysis is a promising approach. It is magical; take a limit, solve the entangled and then, open a treasure.

### Ryushi GOTO

**Geometry **

My research interest is mostly in complex and differential geometry, which are closely related with algebraic geometry and theoretical physics. My own research started with special geometric structures such as Calabi-Yau, hyperKaehler, G2 and Spin(7) structures. These four structures exactly correspond to special holonomy groups which give rise to Ricci-flat Einstein metrics on manifolds. It is intriguing that these moduli spaces are smooth manifolds on which local Torelli type theorem holds. In order to understand these phenomena, I introduce a notion of geometric structures defined by a system of closed differential forms and establish a criterion of unobstructed deformations of structures. When we apply this approach to Calabi-Yau, hyperKaehler, G2 and Spin(7) structures, we obtain a unified construction of these moduli spaces. At present I also explore other interesting geometric structures and their moduli spaces.

### Yasuhiro HARA

**Topology **

The field of my study is topology and, especially, I study the theory of transformation groups. The Borsuk-Ulam theorem is one of famous theorems about transformation groups. This theorem is often taken up as an application in elementary lectures about the homology theory. The content of the theorem is as follows: for every continuous map from the n-dimensional sphere to the n-dimensional Euclidian space, there exists a point such that the map takes the same value at the point and at the antipodal point. A famous application of this theorem is the following. ''On the earth, there is a point such that the temperature and humidity at the point are the same as those at the antipodal point.'' We consider a free action of a group of order two on the n-dimensional sphere to prove the Borsuk-Ulam theorem. Then for any equivariant map (any continuous map which preserves the structure of the group action) from the sphere to itself, the degree of the map is odd. By using this fact, we obtain the Borsuk-Ulam theorem. In the case of the Borsuk-Ulam theorem, we consider spheres and free actions of a group of order two. Actually, when we consider other manifolds and actions of other groups, there are some restrictions of homotopy types of equivariant maps. I study such restrictions of homotopy types of equivariant maps by using the cohomology theory, and I study relationships between homotopy types of equivariant maps and topological invariants.

### Nakao HAYASHI

**Partial Differential Equations **

I am interested in asymptotic behavior in time of solutions to nonlinear dispersive equations (1 D nonlinear Schreodinger, Benjamin-Ono, Korteweg-de Vries, modified Korteweg-de Vries, derivative nonlinear Schreodinger equations) and nonlinear dissipative equations Complex Landau-Ginzburg equations, Korteweg-de Vries equation on a half line, Damped wave equations with a critical nonlinearity). These equations have important physical applications. Exact solutions of the cubic nonlinear Schreodinger equations and Korteweg-de Vries can be obtained by using the inverse scattering method. Our aim is to study asymptotic properties of these nonlinear equations with general setting through the functional analysis. We also study nonlinear Schreodinger equations in general space dimensions with a critical nonlinearity of order 1+2/n and the Hartree equation, which is considered as a critical case and the inverse scattering method does not work. On 1995, Pavel I. Naumkin and I started to study the large time behavior of small solutions of the initial value problem for the non-linear dispersive equations and we obtained asymptotic behavior in time of solutions and existence of modified scattering states to nonlinear Schreodinger with critical and subcritical nonlinearities. It is known that the usual scattering states in L2 do not exist in these equations. Recently, E.I.Kaikina and I are studying nonlinear dissipative equations (including Korteweg-de Vries) on a half line and some results concerning asymptotic behavior in time of solutions are obtained.

### Tomoyoshi IBUKIYAMA

**Number Theory **

My speciality is Number Theory. Integers are very mysterious objects. For example, even for a number with more than several hundred digits, difference only by one makes that number a prime number or a composite number, a big difference as a property of integers. So if you want to know something about integers, the approximate numbers used in the usual scale of the world are not enough. You need really precise data for those. In order to study the usual rational integers, you must investigate all algebraic integers such as the square root of two. There exists a meromorphic function called Riemann zeta function which reflects the distribution of integers and primes. Its values at integers have relation with volumes of various domains. These algebraic, analytic, and geometric theories are all needed to study integers. On the other hand, let us consider the situation that some important quantity in some area of mathematics is written by an integer. Then it means that this number has a very rigid meaning which cannot be replaced by any other quantity so easily. Besides, it sometimes happens in mathematics that invariants which come from apparently completely different origins are written by the same integers. This seems a kind of miracle, but as far as it is for integers, such thing cannot happen just by a coincidence, and there should exist a deep reason behind the phenomena. Among various areas of number theory, I am particularly studying automorphic forms which are functions invariant by an action of a certain discrete subgroup. In that area too, there are functions which are quite different but still have the same numerical properties. Indeed I had several experiences of finding astonishing relations between these. Number theory has a good tradition in mathematics in Japan since Teiji Takagi proved the famous class field theory. It is one of the areas which are difficult and highly sophisticated, and there are many conjectures which everyone can believe but cannot prove at all. But the objects of number theory are integers. So you can always give examples by concrete numbers at least in the last stage of the theory and you can even find something new by making an numerical experiment. This implies that sometimes amateur spirits can be very useful. In my study, I am always aiming to write everything explicitly. In this sense, theories and numerical experiments have always a good connection and a lot of ideas are waiting to be tested by computer calculation. Everyday I am expecting to be excited by a new discovery by experiment, since I believe this kind of excitement is more or less the origin and motivation of our study on mathematics.

### Takao IOHARA

**Nonlinear Partial Differential Equations **

My research interest is concerned with nonlinear partial differential equations appearing in fluid mechanics. The current research topic is the equations of the motion of viscous incompressible fluid which has free moving surface. The motion of viscous incompressible fluid is governed by the Navier-Stokes equations, which are not easy to solve because of their nonlinearity. The free moving surface adds another nonlinearity to the problem and the study of it needs more elaborate technique than the problems on fixed domain.

### Kazunori KIKUCHI

#### Differential Topology

I have been studying topology of smooth four-dimensional manifolds, in particular interested in homology genera, representations of diffeomorphism groups to intersection forms, and branched coverings. Let me give a simple explanation of what interests me the most, or homology genera. The homology genus of a smooth four-dimensional manifold M is a map associating to each two-dimensional integral homology class [x] of M the minimal genus g of smooth surfaces in M that represent [x]. For simplicity, reducing the dimensions of M and [x] to the halves of them respectively, consider as a two-dimensional manifold the surface of a doughnut, or torus T, and a one-dimensional integral homology class [y] of T. Draw a meridian and a longitude on T as on the terrestrial sphere, and let [m] and [l] denote the homology classes of T represented by the meridian and the longitude respectively. It turns out that [y] = a[m] + b[l] for some integers a and b, and that [y] is represented by a circle immersed on T with only double points. Naturally interesting then is the following question: what is the minimal number n of the double points of such immersions representing [y]? Easy experiments would tell you that, for example, n = 0 when (a,b) = (1,0) or (0,1) and n = 1 when (a,b) = (2,0) or (0,2). In fact, it is proved with topological methods that n = d ? 1, where d is the greatest common divisor of a and b. It is the minimal number n for T and [y] that corresponds to the minimal genus g for M and [x]. The study on the minimal genus g does not seem to proceed with only topological methods; it sometimes requires methods from differential geometry, in particular methods with gauge theory from physics; though more difficult, it is more interesting to me. I have been tackling the problem on the minimal genus g with such a topological way of thinking as to see things as if they were visible even though invisible.

### Norihito KOISO

**Differential Geometry**

I'm interested in "beautiful" curves and surfaces, and research on their properties and motion. For example, what a shape a piano wire will take when you bend it? what a shape a soap film will take when you bounds it with a wire? These are very classic problem in mathematics, but there are still many interesting open problems. Let's make a cylinder by a soap film with two circles boundary. If you blow air in the interior of the cylinder, the cylinder will swell out. When the quantity of air is small, we know the shape: it is a rotational surface. However, when the quantity of air is large, we don't know the shape: it is not rotationally symmetric. We can see the shape by physical or computer experimentation, but we know almost nothing mathematically. Let's throw a bent piano wire. We know that the piano wire will become straight line finally, but we don't know the shapes it takes on the way. I'm researching on such a problems as differential geometry with help of analysis.

### Gen KOMATSU

**Analysis, Several Complex Variables via PDE Method**

Roughly speaking, there are two methods in Mathematics, i.e. Analysis and Algebra (and these are for instance used in Geometry). My roots are in Analysis. The method of (Mathematical) Analysis, also called the PDE Method (Method of Partial Differential Equations), is the Modern Calculus (such as integration by parts, taking limits, estimating from above, etc.) based on Lebesgue's Integration Theory and Functional Analysis (infinite dimensional linear algebra combined with topology). Contrary to the fact that Algebra has a great general theory which most graduate students must study, Analysis has no such satisfactory general theory, while there are ample special topics instead. These special topics often come from Differential Geometry and/or physical phenomena. If for instance coefficients of a linear partial differential equation are randomly chosen, then there is a remote possibility of reaching deep substance. Thus PDE researchers usually seek materials in Geometry or Physics. My materials are in Function Theory of Several Complex Variables; but I am mainly concerned with integral kernels such as the Bergman kernel, and thus there is not much difference from the PDE researchers who investigate the heat kernel and/or the Green function. It is indeed interesting to derive qualitative properties via the PDE method, where mathematics of inequalities is used, but it is difficult to continue living forever in the world of inequalities; natural desire to mathematics of equalities comes out. If you encounter a theory saying that a solution is obtained in principle by such and such a method, then you will naturally wish to do computation to get a concrete answer. However, even when a proof contains an algorithm, it is seldom the case where such an algorithm can be efficiently used. Usually, it is necessary to employ a simple and direct method which really fits the problem, and, to pursue such a method often leads to a deep understanding of the nature of the problem. After all, I am working on the problem of extracting differential geometric information of the boundary of a complex domain (explicitly) from invariant singular integral kernels (such as the Bergman kernel) appearing in Function Theory of Several Complex Variables. Special functions are also my favorite (various special functions appear in the vicinity of the Bergman kernel). Function Theory of One Complex Variable is another favorite (there appears a simple but new structure that is similar to the invariant theory in Several Complex Variables).

### Kazuhiro KONNO

**Complex Algebraic Geometry**

Algebraic Geometry is a branch of Mathematics studying, by means of algebraic methods, the geometry of figures defined by simultaneous algebraic equations in several variables. You may say that you are not familiar with Algebraic Geometry. But you already know many beautiful plane curves such as an ellipse, a parabola and a hyperbola; they are in fact our jewels --- algebraic varieties. As you learned in high school, various problems on the geometry of plane curves, e.g., how two curves intersect or contact, can be solved by considering simultaneous equations. Studying figures in such a way is nothing but the algebraic geometry. Algebraic equations, however, are not so simple; it is not known so far even how to solve simultaneous quadratic equations, whilst the method for linear equations are well established as you learned in the course of Linear Algebra, and many beautiful algebraic varieties are usually given by quadratic equations. Because in general we cannot draw figures of varieties on the black board, unlike ellipses or parabolas, it requires such and such training in order to be able to touch and feel them. For example, my favorite algebraic surfaces are 4 dimensional objects and, therefore, cannot be realized in our 3 dimensional space. If you are interested in meeting them in reality, the best way is to start and enjoy learning Algebraic Geometry.

### Toshiki MABUCHI

**Complex Geometry Geometry**

Kaehler manifolds and projective algebraic manifolds are my prime research interests. We study these algebraic geometric objects from differential geometric viewpoints. Related to the moduli spaces of such manifolds, stability concepts play a very important role. Let me give an example of a theme I have been working on. The Hitchin-Kobayashi correspondence for vector bundles, established by Kobayashi, Donaldson and Uhlenbeck-Yau, states that an indecomposable holomorphic vector bundle is stable in the sense of Mumford-Takemoto if and only if the vector bundle admits a Hermitian-Einstein metric. I am working on its various manifold analogues by focusing on Donaldson-Tian-Yau's Conjecture.

### Shinichiroh MATSUO

**Differential Geometry, Geometric Analysis**

I am studying Donaldson theory in 4-manifold topology from the viewpoint of geometric analysis.

Donaldson theory considers the interplay between 4-manifolds and the moduli spaces of anti-self-dual connections on them. Anti-self-dual connections are solutions of the anti-self-dual equations, the partial differential equations of geometric origin. Then, my standpoint is to investigate the moduli spaces by using both geometric studies of 4-dimensional manifolds and analytical techniques of the anti-self-dual equations. I am espacially interested in the case when the moduli spces are of infinite dimension. For example, I studied the anti-self-dual equations on the cylinder without considering any boundary conditions.

From a broader perspective, I have always been fascinated with three things: Infinity, Spaces, and Randomness. What I have been studying is infinite dimensional spaces, which is a hybrid of Infinity and Spaces. Next I want Randomness to go on the stage.

Speaking casually, I love mathematics, because she always says, Let there be light.

### Hideyuki MIURA

**Partial Differential Equations**

I am concerned with the fundamental problems for the initial value problems on the nonlinear partial differential equations in the fluid mechanics, in particular, incompressible Navier-Stokes equations.

My recent research interest is the asymptotic behaviour of the solution near the singular point. Since the equations in the fluid mechanics often have nonlinearity, it seems to be difficult to establish general theory for these equations. Instead, we develop the theory for each equations and then it is important to apply characteristic property for the analysis of these equations. I am also interested in various estimates related with the research of these equations and the fundamental solutions for linear partial differential equations.

### Hideki MIYACHI

**Hyperbolic Geometry , Teichmüller Theory**

My research interests are in Teichmuller theory and geometric structures (especially Hyperbolic geometry) on manifolds, and their applications. I am recently interested in the mathematical phenomena which happen when geometric structures on surfaces (manifolds) are degenerating.

For instance, one of our problems is how the geometrical quantities behave under degenerations. Teichmüller theory is studied and applied in various fields including Complex analysis, Topology, Differential geometry and Algebraic geometry.

### Takehiko MORITA

**Ergodic Theory**

I specialize in ergodic theory. To be more precise, I am studying statistical behavior of dynamical systems via thermodynamic formalism and its applications.

Ergodic theory is a branch of mathematics that studies dynamical systems with measurable structure and related problems. Its origins can be found in the work of Boltzmann in the 1880s which is concerned with the so called Ergodic Hypothesis. Roughly speaking the hypothesis was introduced in order to guarantee that the system considered is ergodic i.e. the space averages and the long time averages of the physical observables coincide. Unfortunately, it turns out that dynamical systems are not always ergodic in general. Because of such a background, the ergodic problem (= the problem to determine a given dynamical system is ergodic or not) has been one of the important subjects since the theory came into existence. In nowadays ergodic theory has grown to be a huge branch and has applications not only to statistical mechanics, probability, and dynamical systems but also to number theory, differential geometry, functional analysis, and so on.

### Tomonori MORIYAMA

**Number Theory**

I am interested in automorphic forms of several variables. A classical automorpchic (modular) form of one variable is a holomorphic function on the upper half plane having certain symmetry. Such functions appear in various branches of mathematics, say notably number theory, and have been investigated by many mathematicians.

There is a family of minifolds called Riemannian symmetric spaces, which is a higher-dimensional generalization of the upper half plane. The set of isometries of a Riemannian symmetric space forms a Lie group G. Roughly speaking, an automorphic form of several variables is a function on a Riemannian symmetric space satisfying the relative invariance under an "arithmetic" subgroup of G and certain differential equations arising from the Lie group G. Studies on automorphic forms of several variables started from C. L. Siegel's works in 1930s and have been developed through interaction with mathematics of the day.

Currently I am working on two themes: (i) the zeta functions attached to automorphic forms and (ii) explicit constructions of automorphic forms, by employing representation theory of reductive groups over local fields. One of the joy in studying this area is to discover a surprisingly simple structure among seemingly complicated objects.

### Tatsuo NISHITANI

**Partial Differential Equations**

When you throw a stone into a pond then a ripple on the water propagates in all directions. The propagation is governed by a partial differential equation, the wave equation. Maxwell proposed partial differential equations (Maxwell equations) which the electric field vector and the magnetic field vector satisfy and he deduced that theses two fields propagate with the speed of light in vacuum analyzing the equations . Partial differential equations which govern phenomena that a small change occurred in a space fulfilled by some material propagates in all directions with finite speed are called hyperbolic partial differential equations, as the wave equation or Maxwell equations. I am mainly interested in characterizing hyperbolic partial differential equations and in studying properties of solutions to hyperbolic equations. Hyperbolic equations which remain hyperbolic for any choice of lower order terms enjoy a beautiful structure which closely related to the spectrum of the representative of the Hessian (of the principal part) with respect to the standard symplectic form. In these studies, you may wonder, the inequalities are much important than the equalities. I quote a favorite phrase of mine from the Introduction by Peter D. Lax in the collected works (vol II) of Jean Leray (mathematicians who made great contributions to hyperbolic partial differential equations).

"Since a priori estimates lie at the heart of most his arguments, many of Leray's papers contain symphonies of inequalities; sometimes the orchestration is heavy, but the melody is always clearly audible".

### Tadashi OCHIAI

**Number theory, Arithmetic Geometry**

When I was a student, I thought I knew number theory, geometry, but algebraic geometry was unfamiliar for me. One may say algebra and geometry are different fields, but you know the theory of quadratic curves and are aware of efficiency of algebraic methods for solving geometric problems. The field called algebraic geometry lies on such a line. When I was studying the theory of quadratic curves, I wondered, ''Why do they only treat special equations like quadratics? There are many other equations. But how can they be treated?". When I discovered the answer might lie on this field, I decided to enter this field. The easiest non-trivial equation has the form such as "the second power of y = an equation of x of degree three", which defines the so called "an elliptic curve". The theory of elliptic curve was one of the greatest achievements of nineteenth century and keeps developing today. Recently, the famous Fermat's conjecture has been solved using this theory. The theory of quadratic and elliptic curves involve only two variables x, y. It is natural to think of the equations with many variables. In fact the algebraic geometrists are expanding the theory, curves to surfaces and higher dimensional cases these days. Two dimensional version of elliptic curves are called K3 surfaces, which can be treated only using the theory of linear algebra(!) thanks to the Torelli's theorem. These days, the 3-dimensional versions, which is called Calabi-Yau threefolds are fascinating for algebraic geometrists like me. Somehow theoretical physicists are also interested in this field. To study Calabi-Yaus by specializing these to ones with a fiber structure (on which field, I'm now working) might be one method, but I have been thinking that a new theory is needed. These days, many intriguing new theories have appeared and one may find more!

### Hiroyuki OGAWA

**Number Theory**

I have an interest in periodic objects. Expanding rational numbers into decimal numbers is delightful. The decimal number expansion becomes the repeat of a sequence of some integers. I have an appetite for continued fraction expansions, never get tired to calculate it, and want to find continued fractions with sufficiently long period. It is on the way to Gauss' class number one conjecture. Recently, I am studying iteration of rational functions. For a rational function g(x) with rational coefficients, a complex number z with g(g(...g(z)...))=z is called a periodic point on g(x) and is an algebraic number. I expect that number theoretical properties which such an algebraic number z has is described by the rational function g(x). This does not seem to work out anytime, but one can find many rational functions g(x) that describe the Galois group, the class number, the class group, and so on of a periodic point of g(x). I think that this should be surely useful, and calculate like these every day.

### Keiji OGUISO

**Algebraic Geometry**

My speciality is algebraic geometry. I am interested in K3 surfaces, elliptic surfaces and their higher dimensional analogue, Calabi-Yau manifolds in wider sense and fiber spaces. These objects play important roles in the classification theory. They also naturally relate with many of mathematics such as lattice theory, number theory, group theory, complex geometry, complex dynamics and so on, via period and symmetry. Such interaction makes study of Calabi-Yau manifolds richer and more exciting, and attracts me very much. We have a complete description of K3 surfaces and elliptic surfaces. However, if one would study them from a fresh right view, then one could find many unexpected, beautiful properties. For instance, mysterious relation between finite symplectic automorphism groups of K3 surfaces and the Mathieu group of degree 23 (Mukai), very impressive construction of K3 surface automorphism with Siegel disk via Salem number (McMullen), surprising role of the theory of Mordell-Weil lattice in sphere packing problem(Shioda) and so on. Though less impressive than above mentioned three results, I was very exciting when I found the fact that K3 surfaces with infinite automorphism group are always dense in any non-trivial projective small deformation of any projective K3 surface, and when I with Shioda could complete explicit description of the Mordell-Weil lattices of rational elliptic surfaces. In the last two years, I was particularly interested in hyperkaehler manifolds (the most faithful generalization of K3 surfaces among Calabi-Yau manifolds in wider sense). If a hyperkaehler manifold admits a fibration over a normal projective variety, then it is necessarily Lagrangian (Matsushita). The general fibers are abelian varieties and if in addition it admits a bimeromorphic section, then the hyperkaehler manifold is also projective. It is conjectured that the base space is always a projective space. I classified general singular fibers (in the sense of complex geometry) with Hwang in a more general context, and I determined the Picard number of the generic fiber (in the sense of scheme) and derived the rank formula of the Mordell-Weil group, when it admits at least one holomorphic section over the projective space. Now, among other things, I am also interested in applications of these basic tools.### Koji OHNO

**Algebraic Geometry**

When I was a student, I thought I knew number theory, geometry, but algebraic geometry was unfamiliar for me. One may say algebra and geometry are different fields, but you know the theory of quadratic curves and are aware of efficiency of algebraic methods for solving geometric problems. The field called algebraic geometry lies on such a line. When I was studying the theory of quadratic curves, I wondered, ''Why do they only treat special equations like quadratics? There are many other equations. But how can they be treated?". When I discovered the answer might lie on this field, I decided to enter this field. The easiest non-trivial equation has the form such as "the second power of y = an equation of x of degree three", which defines the so called "an elliptic curve". The theory of elliptic curve was one of the greatest achievements of nineteenth century and keeps developing today. Recently, the famous Fermat's conjecture has been solved using this theory. The theory of quadratic and elliptic curves involve only two variables x, y. It is natural to think of the equations with many variables. In fact the algebraic geometrists are expanding the theory, curves to surfaces and higher dimensional cases these days. Two dimensional version of elliptic curves are called K3 surfaces, which can be treated only using the theory of linear algebra(!) thanks to the Torelli's theorem. These days, the 3-dimensional versions, which is called Calabi-Yau threefolds are fascinating for algebraic geometrists like me. Somehow theoretical physicists are also interested in this field. To study Calabi-Yaus by specializing these to ones with a fiber structure (on which field, I'm now working) might be one method, but I have been thinking that a new theory is needed. These days, many intriguing new theories have appeared and one may find more!

### Ken'ichi OHSHIKA

**Topology**

I have been studying 3-manifolds and discrete groups. Although 3-manifold topology has a long tradition of research, which started with the pioneering work of Poincaré back in the 19th century, it is still one of the most active fields in topology. In the 1980's, Thurston published a famous conjecture called the geometrisation conjecture, stating that all compact 3-manifolds would be decomposed canonically into geometric pieces each of which has a locally homogeneous metric. Recently Perelman claimed that he has succeeded in solving this conjecture. If his claim is true, then the research of 3-manifolds is reduced to that of hyperbolic ones, which have metrics of constant sectional curvature -1. I am studying hyperbolic 3-manifolds from the viewpoint of Kleinian groups which have been an important topic in complex analysis. Kleinian groups are typical examples of discrete groups in Lie groups. More generally, it is in vogue to study groups as geometric objects regarding them as discrete groups by endowing them with the word metric, and I am also interested in this field. In particular, such things as hyperbolic groups invented by Gromov or isometric group actions on R-trees are closely related to the study of Kleinian groups. More general objects called automatic groups, whose operations are governed by automata, are also important objects in geometric group theory. Although geometric group theory is a relatively new field, it is promised to flourish in the near future.

### Hiroshi SUGITA

**Probability Theory **

I specialized in Probability theory. In particular, I am interested in infinite dimensional stochastic analysis, Monte-Carlo method, and probabilistic number theory. Here I write about the Monte-Carlo method. One of the advanced features of the modern probability theory is that it can deal with "infinite number of random variables". It was E. Borel who first formulated "infinite number of coin tosses" on the Lebesgue probability space, i.e., a probability space consisting of [0,1)-interval and the Lebesgue measure. It is a remarkable fact that all of useful objects in probability theory can be constructed upon these "infinite number of coin tosses". This fact is essential in the Monte-Carlo method. Indeed, in the Monte-Carlo method, we first construct our target random variable S as a function of coin tosses. Then we compute a sample of S by plugging a sample sequence of coin tosses --- , which is computed by a pseudo-random generator, --- into the function. Now, a serious problem arises: How do we realize a pseudo-random generator? Can we find a perfect pseudo-random generator? People have believed it to be impossible for a long time. But in 1980s, a new notion of "computationally secure pseudo-random generator" lets people believe that an imperfect pseudo-random generator has some possibility to be useful for practical purposes. A few years ago, I constructed and implemented a perfect pseudo-random generator for Monte-Carlo integration, i.e., one of Monte-Carlo methods which computes the mean values of random variables by utilizing the law of large numbers. In these days, I am considering about probabilistic approach to computationally secure pseudo-random generators.

### Hiroki SUMI

**Complex Dynamical Systems **

In mathematics, we have several fields in which we try to describe how the things vary as time goes by, for example, differential equations, discrete dynamical systems (systems constructed by iteration of a map in a space), stochastic processes, etc. Related to those theories, there are various mathematical models in biology, economics, etc. For example, regarding the discrete dynamical systems theory, for a kind of insect, we have a model stating that the number of the insect in the (n+1)-th year is the image of the number in the n-th year under a polynomial map f of degree two, where the map f does not depend on n. In this model, to get a wider field of view, we sometimes consider initial values of complex number. Then, we have a discrete dynamical system in the complex plane, using the polynomial map f. As the above, a system constructed by iteration of a polynomial (or holomorphic) map in the complex plane (or in a complex manifold) is called a "complex dynamical system". To analyse such systems mathematically, we use complex analysis deeply. Regarding the iteration of a polynomial map f of degree two or greater in the complex plane, the set of initial values in the complex plane which has initial sensitivity (or chaotic behavior), which is called the "Julia set" for f, is always not empty, and it has "self-similarity". That is, if we magnify the detail of the Julia set, then it is similar to the whole Julia set. Recall that, we are very familiar with various sets having self-similarity, for example, clouds, trees, leafs, surfaces of mountains, etc. Those kinds of complicated (but beautiful) sets are called "fractal sets", of which research was initiated by Mandelbrot in 1975. So, the Julia set of a polynomial map is one of typical examples of fractal sets. Regarding the complex dynamical systems in the complex plane, if we want to go further, we can use the theory of deformation of surfaces. By using it, we can know some information of the orbit of an initial value that does not have initial sensitivity. Going along this direction, you would notice that the theory of complex dynamical systems (of one variable) is very similar to that of discrete groups of linear fractional functions in the field of 2- or 3-dimensional geometry, which gives us much interest. I am a researcher of complex dynamical systems, and especially, I am one of pioneers of theory of dynamics of semigroups generated by polynomials (or rational functions). In this topic, we consider several polynomial maps f,g,h... etc. and consider any images of initial values under the all maps f,g,h... , and consider the images again, and we continue this procedure. In this setting, we can consider the following: (1) an estimate of "fractal dimension", of which value may not be an integer, of the Julia set for a system, (2)random complex dynamical systems, in this direction we can get functions in the complex plane which are similar to the "devil's staircase" (3) the investigation on how small copies of the whole Julia sets inside the Julia sets overlap; we use a kind of "cohomology theory" (a kind of algebra). Studying the complex dynamical systems, we can deal with both the real world in the nature and one of the deepest theory of mathematics, which is the fascination of this field.

### Hideaki SUNAGAWA

**Partial Differential Equations **

My research field is Partial Differential Equations of hyperbolic and dispersive type. They arise in mathematical physics as equations describing wave propagation, so there are a wealth of applications and plenty of problems to be studied. Of my special interest is the nonlinear interactions of hyperbolic waves. Since the analysis of nonlinear PDE is still a developing subject, there are few general conclusions about that. To put it another way, it means that there are possibilities for coming across wonderful phenomena which no one has ever seen before.

### Joe SUZUKI

**Information Mathematics **

My expertise includes Cryptography, error-correcting codes, data compression of digital information. For cryptography, I consider the discrete logarithm problem (DLP), i.e. given a generator α and another element β of a finite cyclic group, what is the l (discrete log) such that α = β ? The larger the discrete log, the harder the DLP. This property has been applied to cryptography of digital data. If G is the Jacobian group of a curve over a finite field, we call it algebraic curve cryptography. For example, in elliptic curves, i.e., curves with genus one, the set of points with the x- and y- coordinates being in the finite field and the point at infinity make a group under a specified arithmetic. Fast arithmetic (encryption) and fast order-counting of Jacobian groups, efficient solutions of DLP are included in the main topics. For error-correcting codes (recovering received data which was sent through a noisy channel), I consider algebraic geometry codes and error-correcting codes based on Bayesian networks with an application to Turbo codes for CDMA. For data compression, assuming a sequence has been emitted from a stationary ergodic process, we predict the future data based on the law of large numbers. Then, the higher the probability of a sequence is, the shorter we assign it to so that the average length can be the shortest. The current algorithm such as gzip, zip, uuencode, LHA etc. are based on this property. Recently, with Russian researchers, I derived several results on evaluation using Kolmogorov complexity and Hausdorff dimension.

### Atsushi TAKAHASHI

**Algebraic Geometry, Mathematical Physics **

My current interests are mathematical aspects of the superstring theory, in particular, algebraic geometry related to the mirror symmetry.

More precisely, I am studying homological algebras and moduli problems for categories of "D-branes" that extend derived categories of coherent sheaves on algebraic varieties.

Indeed, I am trying to construct Kyoji Saito's primitive forms and their associated Frobenius structures from triangulated categories defined via matrix factorizations attached to weighted homogeneous polynomials.

For example, I proved that the triangulated category for a polynomial of type ADE is equivalent to the derived category of finitely generated modules over the path algebra of the Dynkin quiver of the same type.

Now, I extend this result to the case when the polynomial corresponds to one of Arnold's 14 exceptional singularities and then showed the "mirror symmetry" between weighted homogeneous singularities and finite dimensional algebras, where a natural interpretation of the "Arnold's strange duality" is given.

### Naohito TOMITA

**Real Analysis **

My research field is Fourier analysis, and I am particularly interested in the theory of function spaces. Fourier series were introduced by J. Fourier(1768-1830) for the purpose of solving the heat equation. Fourier considered as follows:"Trigonometric series can represent arbitrary periodic functions". However, in general, this is not true. Then, we have the following problem: "When can we write a periodic function as an infinite (or finite) sum of sine and cosine functions?". Lebesgue space which is one of function spaces plays an important role in this classical problem. Here Lebesgue space consists of functions whose p-th powers are integrable. In this way, function spaces are useful for various mathematical problems. As another example, modulation spaces were recently applied to pseudodifferential operators which are important tool for partial differential equations, and my purpose is to clarify their relation.

### Motoo UCHIDA

**Algebraic Analysis and Partial Differential Equations **

My research field is Algebraic Analysis and Microlocal Analysis of partial differential equations. The viewpoint of microlocal analysis (with cohomology) is a new important point of view in analysis introduced by Mikio Sato in the 1970´s. Thinking from a microlocal point of view helps us to well understand a number of mathematical phenomena (at least for PDE) and to find a simple hidden principle behind them. Even for some classical facts (scattered as well-known results), sometimes we can find a new unified way of understanding from an algebro-analytic viewpoint. It is a welcome thing to have a unified way of understanding, and it also brings such a small feeling of bliss as we have at first knowing the simple mechanism of the blue sky and the sunset glow.

### Kazushi UEDA

**Geometry **

My research interest lies in the field where theoretical physics intersects with mathematics.

I am particularly interested in string theory and its relation with algebraic geometry.

String theory has various dualities, and efforts to formulate these dualities rigorously have resulted in a number of astonishing discoveries.

Mirror symmetry is one of those dualities where the mathematical structure is best understood and many non-trivial predictions by physicists has been checked by mathematicians, although a large part of the picture still remains conjectural.

Through the study of mirror symmetry and its ramifications, I hope to shed some light on the mystery surrounding string theory and its dualities.

### Sampei USUI

**Algebraic Geometry**

I research the relationship between algebraic geometry and Hodge theory. To watch this world from the infinity, I investigate this relationship by using log geometry. More precisely, it is called Torelli problem to ask how algebraic varieties are determined by their periods of integrals. Five approaches are known to this problem. Among these five, the inductive approach by using degenerations seems to be effective for surfaces of general type, Calabi-Yau threefolds etc. By the joint work with Kazuya Kato, we succeeded to establish log geometric generalizations of Griffith' theory of period maps and toroidal compactifications by Mumford et.al. Recently, the joint work with Kato and Chikara Nakayama for mixed version of log Hodge theory is developing. This project has been already advanced more than half, and log intermediate Jacobians are constructed. The relationship of these results with geometry and physics, such as Hodge conjecture, mirror symmetry, minimal model program etc., are gradually watched. This area is widely open. Lots of interesting problems are waiting for young energetic students.

### Takao WATANABE

**Number Theory**

My current interest is the Geometry of Numbers. The Geometry of Numbers was founded by Hermann Minkowski in the beginning of the 20th century. Minkowski proved a famous theorem known as "Minkowski's convex body theorem", which asserts that "there exists a non-zero integer point in V if V is an o-symmetric convex body in the n-dimensional Euclidean space whose volume is greater than 2^n". When V is an ellipsoid, this theorem is refined as follows. Let A be a non-singular 3 by 3 real matrix and K(c) the ellipsoid consisting of points x such that the inner product (Ax, Ax) is less than or equal to c > 0. For i = 1,2,3, we define the constant c_i as the minimum of c > 0 such that K(c) contains i linearly independent integer points. Then c_1, c_2, c_3 satisfies the inequality c_1c_2c_3 <= 2|det A|^2. This is called "Minkowski's second theorem". A similar inequality holds for any n-dimensional ellipsoid. Namely, if A is a non-singular n by n real matrix and K(c) is the n-dimensional ellipsoid defined by (Ax, Ax) <= c, we can define c_i for i = 1,2, ..., n as the minimum of c > 0 such that K(c) contains i linearly independent integer points. Then the inequality c_1c_2...c_n <= h(n)|det A|^2 holds for any A. The optimal upper bound h(n) does not depend on A, and is called Hermite's constant. We know h(2) = 4/3, h(3) = 2, h(4) = 4, ..., h(8) = 256, but h(n) for a general n is not known. A recent major topic of this research area is the determination of h(24). In 2003, Henry Cohn and Abhinav Kumar proved that h(24) = 4^24. (Incidentally, h(3) was essentially determined by Gauss in 1831, and h(8) was determined by Blichfeldt in 1953. If you would determine h(9), then your name would be recorded in treatises on the Geometry of Numbers.) Now I study (an analogue of) the Geometry of Numbers on algebraic homogeneous spaces. One of my results is a generalization of Minkowski's second theorem to a Severi-Berauer variety. In addition, I am interested in the reduction theory of arithmetic subgroups, automorphic forms, the algebraic theory of quadratic forms and Diophantine approximation.

### Yohei YAMASAKI

**Discrete Mathematics**

It is said that "Mathematics is a science of the infinity". This seminar, however, treats finite discrete structures, without concerning strong symmetry. The graphs form a typical class among them. A graph is an abstract structure which expresses the incidence between "vertices" and "edges". We do not mind whether or not edges cross when they are drawn on a plain. Any graph can be embedded into the eucledean space of dimension 3, avoiding any cross of edges. So we utilize continuity terms as "connectivity", but have no idea of "convergence". We can not expect too much on the ideas introduced from an infinity model. For example, we have two different ideas of the "2-connectivity" as generalizations of the "connectivity". One of them is the "2-vertex-connectivity", which means the connectivity after removing any vertex. The other the "2-edge-connectivity" is defined similarly. Discrete structures often make us reconsider the concepts. "Matroids" and "hypergraphs" are such concepts emerging from "graphs". This field treats many objects of various structures. An effective treatment of such a structure is not virtual any more, with the progress of computers. You will become skillful to deal with various matters which occur around you, after mastering something in this field.

### Kenji YAMATO

**Differential Topology**

Differential forms which have local normal forms define geometric structures on manifolds. Foliations, contact structures and symplectic structures are the examples of the geometric structures. I am interested in the topology of the geometric structures defined by the differential forms. The basic problem is to decide whether the two geometric structures having the same local normal forms are isomorphic. These structures have the same local normal forms, so that the geometric structures are locally isomorphic and there are no local invariants. Therefore, it is a problem of topology and the purpose is to find the global invariants which distinguish the two geometric structures. The first example of the global invariant was found by Godbillon and Vey for foliations of codimension 1, which was called Godbillon-Vey characteristic class. I studied the characteristic class defined by Lazarov and Pasternack for riemannian foliations of codimension 2. I want to study the topology of the geometric structures which different from foliations, contact structures and symplectic structures.

### Seidai YASUDA

**Number Theory**

Sytems of polynomials with integral coefficients are studied in number theory. It is often very difficult to find the integral solutions of such a system. Instead, we simultaneouly deal with the solutions in various commutative rings. The solutions in various rings forms a scheme, which provides geometric methods for studying the system of polynomials.

Hasse-Weil L-functions of an arithmetic scheme are defined using geometric cohomology. I am interested in the special values of these L-functions. The special values are believed to be related to motivic cohomologies, which are defined by using algebraic cycles or algebraic K-theory and are usually they hard to know explicitly. It is a very deep prediction to expect that such abstract objects should be related to more concrete L-functions.

It is expected that motives are related to automorphic representations. The expectation is important since we have various methods for studying automorhic L-functions. Some relations between motives and automorphic representations are realized by using Shimura varieties. In a joint work with Satishi Kondo, I have proved a equality relating motivic cohomologies and special values of Hasse-Weil L-functions for some function field analogues of Shimura varieties.

Hasse-Weil L-functions are defined via some Galois representations. We need to study such Galois representations. For some technical reasons it is important to study Galois representations of p-adic fields with p-adic coefficients, and p-adic Hodge theory provides some tools for studying such representations. For recent years there have been much development in p-adic Hodge theory, and a lot of beautiful theories have been constructed. However the theory is not fully established and many aspects of the theory remains mysterious. I am now trying to make the integral p-adic Hodge theory more convenient for practical study.

### Takehiko YASUDA

**Algebraic Geometry, Singularity Theory**

My main research object is singularities of algebraic varieties. An algebraic variety is a "figure" formed by solutions of algebraic equations. Such a figure often has points where the figure is sharp-pointed or intersects itself. Singularities make the study of an algebraic variety. However since they often appear under various constructions, it is important to study them. Also singularities are interesting research object themselves.

More specifically, I am interested in resolution of singularities, the birational-geometric aspect of singularities, the McKay correspondence. Although these are classical research areas, changing a viewpoint or the setting of a problem, one can sometimes find a new phenomenon. Such a discovery is the greatest pleasure in my mathematical research. To pursue research, I use various tools like motivic integration, Frobenius maps, moduli-theoretic blowups, non-commutative rings, and sometimes make ones by myself.

Recently I am fascinated by mysterious behaviors of singularities in positive characteristic (a world where summing up several 1's gives 0.)

### Number Theory Seminar

Number theory seminar at Osaka University is a seminar for faculty members and graduate students of Osaka University or researchers studying nearby Osaka University. The seminar is usually held on Fridays, once every two weeks. The subject of the seminar covers wide topics concerning Number theory, especially, algebraic number theory, analytic number theory, modular forms, arithmetic geometry, representation theory and algebraic combinatorics. In this seminar, we have reports of new results on these topics and we exchange ideas and technics of our research.

### Algebraic Geometry and Complex Geometry Seminar

The seminar is held every Friday, from 14:30 to 18:00, and usually has one speaker for each. The purpose is to learn important results by active researchers in Algebraic Geometry and Complex Geometry, providing new perspectives on the areas through lectures and discussions. We also have survey lectures by experts for graduate students and young researchers. We have guest speakers not only from domestic universities but also from foreign countries, reflecting various aspects of the research area.

### Geometry Seminar

This seminar on Mondays is intended for talks that will be of interest to a wide range of geometers. Topics discussed include Riemannian, complex, and symplectic geometry; PDEs on manifolds; mathematical physics.geometry

### Topology Seminar

In our research group of topology, we hold four kinds of specialised seminars regularly: the low-dimensional topology seminar focusing on the knot theory, three-manifolds, and hyperbolic geometry (Ken'ichi Ohshika, Hideki Miyachi); the differential topology seminar focused on contact structures, symplectic structures, foliations, and dynamical systems (Kenji Yamato); the seminar on transformation groups (Yasuhiro Hara); and the seminar on four-manifolds focused on 4-manifolds and complex surfaces (Kazunori Kikuchi). We also hold a monthly topology seminar encompassing all fields of topology, where all of us meet together.

### Seminar of Differential Equations

Our seminar is held every Friday from 15:30 to 17:00. One of the features of the seminar is to cover a wide variety of topics on Qualitative Analysis of Differential Equations. In fact, we are interested in ordinary differential equations, partial differential equations, linear differential equations, nonlinear differential equations and so on. Lecturers are invited from not only domestic universities, but also foreign countries and present us their original results or survey of recent development of their fields. Furthermore, this seminar provides opportunities to give a talk for our colleagues and Ph.D. students majoring in differential equations. Moreover, we should mention that we are pleased to have participants from other universities located close to ours. In this way we communicate with each other and try to contribute to the progress of the theory of differential equations.

### Seminar on Probability

"Seminar on Probability" is held Tuesday evenings by probabilists belonging to the graduate school of science and the graduate school of engineering science. The range of the seminar topics are the following:

**(1) Probability theory;**

Stochastic analysis and infinite dimensional analysis (problems arising from other areas of mathematics such as real analysis, differential equations and differential geometry);

**(2) Problems related to the probability theory**

arising from statistical physics, stochastic control theory and mathematical finance.

We enjoy visits and talks by many researchers from other universities, domestic and abroad, and wish this seminar to continue stimulating research interaction.