Department of MATHEMATICAL SCIENCES

Graduate School of Science Tokyo Metropolitan University

Mathematics is the basis of science.

Our department intends to study systematic theory in
algebra, geometry, analysis, and applied mathematics,
which function as the core in education and research
organically and in a cross-sectoral manner.

Trained in mathematical thinking under our system,
many researchers with high logical ability
and a number of flexible and broadminded people have been produced.

Meanwhile, Graduate School of Science holds an advantage
in cross-fertilizing physics, chemistry, and life science,
which enables us to solve urgent problems of modern society.

1. Algebra
Shigeru Kuroda kuroda
Affine algebraic geometry, polynomial ring theory  My area of research is affine algebraic geometry, or polynomial ring theory. I am developing effective methods for studying multivariate polynomials, and I am investigating various applications of these objects. I have worked on Hilbert's Fourteenth Problem and constructed several new counterexamples. I also have done some research on SAGBI (Subalgebra Analogue of Groebner Bases for Ideals) bases, derivations, and the automorphism groups of polynomial rings.
Hiro-o Tokunaga tokunaga
Algebraic geometry, learning theory  My research on branched Galois covers involves several projects, including (1) the construction of Galois covers with prescribed finite group as the Galois group, (2) the topology of the complements of reduced plane curves, and (3) investigation of the singularities appearing in various Galois covers. Recently I have been interested in relations between machine learning theory and the notion of "finite generation" in algebraic geometry. Graduate students who are interested in working with me should learn computer algebra and plane curves as the first step.
Hirofumi Tsumura tsumura
Analytic theory, zeta functions  My primary research area is number theory. Recently I have been investigating various number theoretic objects derived from the Bernoulli numbers, such as Riemann's zeta-function, Dirichlet series and their multiple analogues, and multiple polylogarithms. In particular I am interested in their relations with Lie algebras and physics.
Masanori Kobayashi kobayashi-masanori
Algebraic geometry, mirror symmetry  The geometry of complex manifolds and their singularities is a central topic in geometry. My research is related to superstring theory, in particular, the mirror symmetry of Calabi-Yau manifolds. This area is developing very rapidly and has already revealed many unexpected connections between mathematics and physics. I also study topics in real algebraic geometry, such as real forms of algebraic varieties and special Lagrangian submanifolds. Interactions between mathematics and other branch of sciences, such as machine learning and molecular biology, have stimulated my research in new directions.
Hokuto Uehara hokuto
Algebraic geometry  I am interested in the classification theory for higher dimensional algebraic varieties, which is a major open problem in algebraic geometry. My current research area is the theory of derived categories of coherent sheaves on algebraic varieties. In particular I am investigating Bridgeland's stability conditions and non-commutative crepant resolutions.
Takeshi Kawasaki kawasaki
Commutative algebra  My research in commutative algebra involves excellent rings and their analogues such as Nagata rings and acceptable rings. In particular I am currently studying Noetherian rings which are universally catenary and are such that (1) all formal fibers of localizations are Cohen-Macaulay and (2) the Cohen-Macaulay locus of each finitely generated algebra is open. These conditions naturally generalize the concept of an excellent ring.
2. Geometry
Teruhiko Soma tsoma
Hyperbolic geometry, dynamical systems  Two-dimensional manifolds (surfaces) are well understood, but the theory of three-dimensional manifolds is much more complicated and it is the subject of important research problems such as the recently proved Poincare Conjecture. I am studying structures on such 3-manifolds from the topological and geometrical points of view. My recent work focuses on the topology of geometric limit hyperbolic 3-manifolds of algebraically convergent sequences of Kleinian groups. Another main theme is the dynamics of Henon maps; this involves studying the existence and non-existence of SRB-measures and the existence of generically unfolding cubic homoclinic tangencies for such maps.
Yoshiyuki Yokota jojo
Knot theory, theory of 3-manifolds  My research concerns the geometric structure of knots and 3-manifolds. Knot theory has a long history as a branch of topology, but recent developments have revealed deep connections with other parts of mathematics and theoretical physics. Understanding these connections is a great challenge to researchers in this field. Currently I am interested in the curious relationship between the limits of quantum invariants and their geometric structure.
Manabu Akaho akaho
Floer theory  My research focuses on Floer homology in symplectic and contact topology. Floer defined the so-called Floer homology, which is a version of Morse homology for infinite-dimensional manifolds. In Lagrangian intersection theory, the infinite-dimensional manifold is a covering space of paths bounded by Lagrangian submanifolds in a symplectic manifold, and the Morse function is the symplectic area functional. Floer's chain complex is generated by the intersection points of Lagrangian submanifolds, and the differential counts pseudo-holomorphic strips. I am investigating Floer homology for properly embedded non-compact Lagrangian submanifolds in non-compact symplectic manifolds.
Tomohiro Fukaya tomohirofukaya
Geometric Group Theory  Geometric Group Theory is an research area devoted to the study of infinite, discrete, and non-commutative groups, via the geometry of the space on which the group acts. Until the mid 20th century, there were already several researches on those groups, like Dehn's work on the word problem for the fundamental group of a surface, and Mostov's work on the regidity of the hypebolic manifolds. Then in the late 20th century, Gromov introduced the notion of ``hyerbolic groups'' and it turn out that it is a rich source of interesting theories. The spaces on which those groups acts are not necessarily smooth spaces, like Riemannian manifolds, but discrete spaces, like infinite graphs. Thus it is important to study the ``Large scale properties'' of the spaces.
Takashi Sakai sakai-t
Differential geometry,submanifold theory Many phenomena in nature can be described from the view point of mathematics in terms of variational problems.A submanifold which attains a critical point of the volume functional is called a minimal submanifold.In the histry of mathematics, many mathematicians have investigated this concept.In this research area, we can use the theory of complex analysis (e.g. Weierstrass representation formula), the theory of harmonic maps and so on.The main topics of my research are related to variational problems of submanifolds in Riemannian symmetric spaces, using Lie theoretic methods.I am especially interested in submanifolds with special properties, such as symmetry or volume minimizing properties.Furthermore, I am investigating some differential geometric properties of Lagrangian submanifolds in Hermitian symmetric spaces.
Asuka Takatsu asuka
 My research is in the area of geometric analysis by using geometry on the space of probability measures. In particular, I use the Wasserstein geometry and the Information geometry. The two geometries are different from each other: On the one hand, the Wasserstein geometry is a distance geometry and relates with the geometry of its underlying space. On the other hand, the Information geometry is a geometry of a metric with a pair of orthogonal connections and does not reflect the geometry of its underlying space. However the two geometries are related to each other, and its relation is useful to investigate functional inequalities, partial differential equations and so on. Among such subjects, I intend to analyze the isoperimetric profile and the concentration function of a probability measure on a metric space.
3. Analysis
Kumiko Hattori khattori
Stochastic processes on fractals  Fractal spaces are examples of inhomogeneous spaces, where the standard tools of analysis on Euclidean spaces break down. While extensive work has been done on Markov processes, much less is known about non-Markov processes such as self-avoiding and self-repelling processes. The study of non-Markov processes on fractals offers a wide variety of interesting yet solvable models.
Kazuhiro Kurata kurata
Partial differential equations, variational problems and eigenvalue problems  The theory of partial differential equations is a central area of mathematics, which pervades the natural sciences. My current research involves studying properties of fundamental solutions to elliptic and parabolic partial differential equations, the structure of solutions to nonlinear elliptic equations, and various optimization problems, including eigenvalue optimization problems. I am also interested in various nonlinear phenomena arising in physics, pattern formations in mathematical biology, nonlinear optimization problems and inverse problems.
Shoichiro Takakuwa takakuwa
Global analysis, partial differential equations  My research involves nonlinear differential equations arising in geometry and physics. Some important recent examples are harmonic maps, the Yamabe problem, Einstein metrics, equations for prescribed curvature and the Yang-Mills equations. Recently I have been investigating the asymptotic behaviour of solutions of nonlinear partial differential equations and renormalization group theory. I am also interested in using computers to simulate and visualize solutions and attractors of differential equations.
Kazushi Yoshitomi yositomi
Differentilal equations, spectral theory  My research field is the spectral theory of linear differential operators. Recently I have been studying spectral gaps of the one-dimensional Schroedinger/Dirac operators with periodic impulses.
Kensuke Ishitani k-ishitani
Mathematical Finance, Probability Theory  My research interests are in the areas of mathematical finance and probability theory. My current research involves application of integration by parts formulae for Winer measures to sensitivity analysis for barrier options, stochastic optimal control problem in finance, and quantitative risk management for banking and insurance.
Yoshihiro Sawano ysawano
Harmonic analysis, reproducing kernel Hilbert spaces  My main two branches are hamonic analysis and reproducing kernel Hilbert spaces. In particular, I am interested in function spaces such as Morrey spaces, Besov spaces, Triebel-Lizorkin spaces; see my textbook in Japanese. As for reproducing kernel Hilbert spaces, I am interested in obtaining the approximation formula as well as the inverse formula. For example, jointly with Professors Saburou Saitoh, Hiroshi Fujiwara and Tsutomu Matsuura, I considered the inverse of the Laplace transform. My major is such a subject but any student will be welcome as long as he /she is interested in analysis. If the student is interested in my laboratory, it is desirable to learn calculus and integration theory.
Masaki Hirata mhirata
Dynamical systems, ergodic theory  The analysis of chaotic phenomena is a modern and important area of mathematical physics. My research involves the study of chaos by measure theoretic methods. Recently, I have been investigating the limit distribution of the return times of chaotic dynamical systems. I am also interested in problems relating to quantum chaos, in particular the problem of distribution of energy levels, which is related to the return time distribution.
4. Applied Mathematics
Chikara Fukunaga fukunaga
Computer architecture, parallel processing  By custom-designing the basic system components it is possible to construct very high performance computation systems. Based on the CSP (Communicating Sequential Process) parallel processing theory, we have designed and constructed a parallel processing chip called TPcore. By combining these chips we are implementing a large network system in which the network topology is reconfigurable. Using this we are investigating optimal designs for the processor element and network topologies, for specific parallel computations. We are also developing application programs for the system (graphic process unit and pattern recognition).
Shigenori Uchiyama uchiyama-shigenori
Information security, algorithmic number theory  I have been engaged in research on public-key cryptography. In particular, I am interested in the proposal and cryptanalysis of public-key cryptosystems based on the intractability of number theoretic or combinatorial problems.
Hiroshi Murakami mrkmhrsh
Numerical methods, symbolic computation, parallel computation  I am working in the following areas: (1) numerical methods for scientific and engineering computations; (2) algorithms for symbolic computations in mathematical problems; (3) parallel computation methods for large scale problems; (4) the efficient use of high performance computing systems including personal computers. Recently I have focused on numerical methods for the solution of eigenproblems for large matrices.
Toshio Suzuki toshio-suzuki
Theory of computing, logical aspects of computational complexity  Logic, randomness and computability are strongly related concepts. For example, a set of natural numbers is computable if and only if it is Turing reducible to a random oracle with probability one (Sacks, 1963). Amongst counterparts of this result in computational complexity, Dowd (1992) revealed relationships between the "NP = coNP ?" question and random oracles. My current research extends Dowd's work, and focuses on: (1) minimum sizes of forcing conditions (with applications to the structure of the complexity class coNP[X]), (2) experimental and theoretical studies on random extraction from bitmap files of hand-drawn curves. For the beginner it is necessary to study P and NP, probabilistic algorithms, first-order logic and C++ programming.
Yukihiro Uchida yuchida
Algorithmic number theory, arithmetic geometry, cryptography  My research area is algorithmic number theory. In particular, I am studying algorithms related to the arithmetic of algebraic curves and Abelian varieties. I am also interested in elliptic and hyperelliptic curve cryptography.

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