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
web page 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.
web page Hiro-o Tokunaga tokunaga
Algebraic geometry 　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.
web page Hirofumi Tsumura tsumura
Analytic number theory, p-adic analysis 　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.
web page 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.
web page Akihiro Kanemitsu kanemitsu
Algebraic geometry 　I work on algebraic geometry, especially on problems related to Fano varieties and vector bundles.
web page Masanori Kobayashi kobayashi-masanori
Algebraic geometry, mirror symmetry, tropical geometry 　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.
web page Takesi 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
web page 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.
web page 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.
web page Manabu Akaho akaho
Symplectic geometry, Floer theory, Gauge 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.
web page 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.
web page Daisuke Kazukawa kazukawa
Geometric analysis, convergence theory, concentration of measure phenomenon I research the convergence theory of Riemannian manifolds and metric measure spaces. More specifically, I study the convergence of such spaces and the geometric analytic properties of convergent sequences of spaces and of their limit spaces. In particular, I am interested in the concentration topology (on the set of all metric measure spaces) based on the concentration of measure phenomenon. This topology is effective to capture the high-dimensional aspects of spaces, and indeed infinite-dimensional spaces appear as the limit spaces.
web page Tomoyuki Hisamoto hisamoto
　We have been studying properties of algebraic varaieties using analytic methods, such as the d-bar equation or the Monge-Ampère equation.
In recent years my research interest is especially around the K-stability and the Kähler-Einstein metric.
3. Analysis
web page 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.
web page Karel Svadlenka karel
Calculus of variations, partial differential equations, applied mathematics 　I work on mathematical analysis and numerical solution of optimization problems in infinite dimensional spaces. In particular, I focus on methods to mathematically understand challenging partial differential equations such as hyperbolic free boundary problems by viewing them as optimization problems. Recently, I am also interested in methods of dealing with vector-valued optimization problems that do not possess any solution. In addition to these theoretical topics, I conduct research aiming at contributing through mathematics to the understanding and solution of real-world problems and tasks, ranging from development of new materials to treatment of diseases, by formulating them as variational problems and analyzing these problems mathematically or performing numerical simulations. Examples of such tasks that I addressed in previous research projects are elucidating the mechanism of cell pattern formation in morphogenesis of sensory organs, investigating the principles of material strengthening through microstructure in magnesium alloys, or clarifying the causes of lung damage in COPD disease.
web page Kazushi Yoshitomi yositomi
Pseudo-differential operators 　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.
web page 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.
web page Yukihiro Seki yseki
Nonlinear partial differential equations, reaction-diffusion systems, asymptotic analysis 　 My research interest is nonlinear partial differential equation.
　 I am interested in "large solutions" in some sense, which play an important role in the mathematical analysis of nonlinear PDEs.
　 Current research topic includes an apparently simple semilinear heat equation, a parabolic equation derived from harmonic map heat flow, and reaction-diffusion systems that describe the motionof slime mold.
　 A common feature in the partial differential equations consists in the occurrence of finite time singularity. I study the asymptotic behavior associated to the singularity formation.
　 The aim of our research is to understand the mechanisms, in terms of analyzing the singularity,that cannot be understood in terms of abstract theory.
　 The knowledge of basic functional analysis, differential equations, and mapping degree are required.
　 The most important thing is, however, to be full of vigor for implementing hard computations without hesitation.
web page Masahiko Simojo shimojo
Reaction diffusion equations, dynamical systems 　 My particular interest lies in analyzing the asymptotic behavior ( spreading & singularity e.t.c.) of solutions to reaction diffusion equations from the point of infinite dimensional dynamical systems.
web page 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
web page 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.
web page Shun Sato shun_sato
Numerical analysis, continuous optimization Differential equations appear as models of various phenomena. Since their exact solutions are often impossible or difficult to obtain, numerical methods are widely used to construct approximate solutions. Among them, I am focusing on structure-preserving numerical methods that inherit important structure of the differential equations to be solved, such as conservation laws and symmetry. I am also focusing on the analogy between numerical analysis of differential equations and continuous optimization.
web page 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.
web page 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.
web page Shun'ichi Yokoyama s-yokoyama
Computational number theory, computer algebra 　My research interest lies in computational (algebraic) number theory. I am interested in elliptic curves, modular forms, and their correspondences from a viewpoint of the modularity conjecture. I am strongly interested in developments in computer algebra research and creating databases, especially using Magma and Sage.
On the other hand, I am also interested in applied mathematics research for pursuing new mathematical frameworks for computer graphics and visual effects (but not ongoing).

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