Graduate School of Science
Hokkaido University

International Course in Graduate School of Science

Mathematics

Mathematics

Our department covers a wide range of research areas—from theory-driven approaches that mainly seek to achieve theoretical sophistication to more empirically oriented approaches that employ computers for calculations on various phenomena. Our research includes diverse fields such as hyperplane arrangements, representation theory, differential geometry, singularity theory, partial differential equations, mathematical physics, chaos, probability theory, and dynamical systems.

Every year, the department sponsors or participates in several research conferences in Sapporo that attract several hundred domestic and foreign researchers. A partial list of the conferences in 2015 gives a sense of the high level of research activity in our department:
-The HU and SNU symposium on Mathematics (held annually)
-Mathematics Conference for Young Researchers (organized annually by graduate students)
-Sapporo Symposium on Partial Differential Equations (held annually)
-Northeastan Symposium on Mathematical Analysis (held annually)

These conferences provide students with first-hand glimpses into current research activities as well as opportunities to present their research results to world-renowned mathematicians. In addition, there are several weekly seminars that students are encouraged to attend. The department has an in-house library containing about 90,000 books and 525 journals, where students may study in a spacious and quiet environment. Graduate students are provided with their own desks in the department building. Our recent major achievements in receiving large research grants include: (1) the 21st Century Center of Excellence (COE) Program “Mathematics of Nonlinear Structure via Singularity ”from 2003 to 2008, and (2)the Japan Society for the Promotion of Science (JSPS) International Training Program “The international sending-elevating project for young mathematicians based on singularity, topology and mathematical analysis: Hokudai model” from 2008 to 2012. The fruitful success of the COE program resulted in our founding the Research Center for Integrative Mathematics in 2008 which was reformed into Research Center of Mathematics for Social Creativity in RIES, later. Our department provides an advanced integrated education program “Ambitious leader's program” for graduate students from 2014.

Research Fields

Algebra

Keywords: Algebraic combinatorics, Algebraic geometry, Arithmetic geometry, Combinatorics, Representation theory, Rings of differential operators, Singularity theory, Special functions, Vertex algebras, Yang-Baxter equations and quantum groups

Geometry

Keywords: Complex geometry, Differential geometry, Differential topology, Dynamical systems, Mathematical physics, Painlevé systems, Real algebraic geometry, Singularity theory, Topology

Analysis

Keywords: Algebraic analysis, Differential equations, Functional analysis, Geometric measure theory, Harmonic analysis, Mathematical fluid dynamics, Mathematical physics, Operator algebras, Partial differential equations, Potential theory, Probability theory, Real analysis
  • Hiroaki Aikawa
    Hiroaki Aikawa

    Professor

    PhD

    Fields of Specialty:Potential theory, real analysis

    I have been studying potential theory. The main theme of potential theory is the investigation of fundamental functions such as (super)harmonic functions, subharmonic functions, plurisubharmonic functions. The importance of the deep analysis of these functions has been recognized as they play crucial role in analysis, geometry, probability, applied mathematics and so on. On the other hand, methods, notions and aims of real and complex analysis, functional analysis and probabilistic analysis have enriched potential theory. Generally speaking, the interior properties of solutions to partial differential equations have been exploited even though the equations are complicated; if the domain is sufficiently smooth, the boundary behavior of the solutions can be studied to some extent. Nevertheless the boundary behavior of harmonic functions has many open problems if the domain is not smooth. My research interest is the boundary behavior of these functions in a wide sense. More specifically, I have been studying the Fatou theorem, minimal fine topology, the integrability of super harmonic functions, the additivity of capacity, the norm estimate of the Green operator, Martin boundary and the boundary Harnack principle.

  • Akihito Hora
    Akihito Hora

    Professor

    PhD

    Fields of Specialty:Functional analysis, probability theory

    I have made random walks around the frontier of probability and group representations. Having the notion of measures as the navigator, my walk often digresses in a random way. I am interested in bridging finite and infinite objects. The latest and biggest concerns of mine are analyzing probabilistic phenomena caused by the actions of huge groups.

  • Naofumi Honda
    Naofumi Honda

    Professor

    PhD

    Fields of Specialty:Algebraic analysis

    I am mainly studying subjects related to “Algebraic Analysis”, in particular, microlocal aspects of asymptotic analysis and exact WKB analysis. The microlocal analysis, which was initiated by Mikio Sato in 70’s, is a quite powerful tool to study solutions of systems of linear partial differential equations, and I am now trying to extend this method to the area of asymptotic analysis.

  • Jun Masamune
    Jun Masamune

    Professor

    PhD

    Fields of Specialty:Global Analysis

    My research interest is to investigate the relationships between the analytical and geometric properties of spaces such as Riemannian manifolds and graphs by analyzing the Laplacian, head kernels, Green functions, and Brownian motion on the spaces. The problems currently I am working on are to determine the selfadjoint extensions of the Laplacians, long-term behaviors of Brownian motion such as conservation ad recurrent property, and the Liouville property of harmonic functions applying the techniques from PDEs and Functional Analysis.

  • Asao Arai
    Asao Arai

    Specially Appointed Professor

    PhD

    Fields of Specialty:Mathematical physics, functional analysis

    Searches and investigations, from all possible aspects, of mathematical ideas and laws which govern quantum phenomena. Subjects that have been considered in recent years include the following:
    Mathematical analysis of quantum systems interacting with quantum fields, mathematical theory of time operator, representation theory of canonical commutation relations, mathematical foundations of quantum electrodynamics, mathematical construction of supersymmetric quantum field theory and a theory of infinite-dimensional Dirac operators.

  • Masaharu Kobayashi
    Masaharu Kobayashi

    Associate Professor

    PhD

    Fields of Specialty:Harmonic Analysis

    My research field is analysis. Especially, I am interested in topics called harmonic analysis and real analysis. Roughly speaking, to study boundedness, continuity, differentiability and integrability of functions, we measure “quantitative properties” of functions. We also study how these “quantitative properties” change when we apply various operators. To measure “quantitative properties”, we use various norms (and we also use function spaces defined by these norms). Recently, I am interested in the theory of modulation spaces which were introduced by H.Feichtinger, and I’m trying to study pseudo-differential operators and partial differential equations in the framework of modulation spaces.

  • Reiji Tomatsu
    Reiji Tomatsu

    Associate Professor

    PhD

    Fields of Specialty:Operator algebras

    I am mainly studying von Neumann algebras and (quantum) group actions on them. My recent interest has been focused on the classification of Rohlin flows on injective factors and the characterization of the Rohlin property for flows.

  • Takahiro Hasebe
    Takahiro Hasebe

    Associate Professor

    PhD

    Fields of Specialty:Probability theory, complex analysis, functional analysis

    Free probability was discovered in the context of operator algebras to study free groups. By regarding the generators of a free group (algebra) as “independent random variables”, a probabilistic structure appears. Thus free probability comes up. Later, applications to the eigenvalue distributions of random matrices were discovered, and then probabilistic research started to increase. Now free probability is a field that involves operator algebras, probability theory, combinatorics, complex analysis and representation theory. I am working in particular on the aspects combinatorics of set partitions and complex analysis.

  • Nao Hamamuki
    Nao Hamamuki

    Associate Professor

    PhD

    Fields of Specialty:Nonlinear partial differential equations?Theory of viscosity solutions

    My major research topic is the study of nonlinear partial differential equations, especially evolution equations such as Hamilton-Jacobi equations and curvature flow equations which appear in materials science and describe a motion of a surface (an interface) separated by two different phases of matter.
    On the basis of the theory of viscosity solutions, which is a notion of weak solutions for differential equations, I aim at introducing a suitable notion of solutions, establishing unique existence of solutions to the initial value problem and tracking the large time behavior of solutions to give mathematical foundations to such nonlinear equations.
    My current interests include a multifaceted understanding of phenomena via the limit process; for instance, finding a connection between a discrete/continuum problem and a microscopic/macroscopic model (homogenization).
    I will be glad if we could make links among various research fields through mathematics and find applications to new fields.

  • Tadahiro Miyao
    Tadahiro Miyao

    Associate Professor

    PhD

    Fields of Specialty:Mathematical physics, functional analysis, condensed matter physics

    I am trying to clarify various mathematical structures behind physical phenomena in the condensed matter physics by applying and cultivating methods of functional analysis.

Applied Mathematics

Keywords: Applied analysis, Biophysical complex systems, Biophysics, Brain theory, Chaotic dynamical systems, Complex systems, Computational neuroscience, Computational topology, Ergodic theory, Free boundary problems, Mathematical modeling, Nonequilibrium statistical mechanics, Numerical analysis, Numerical simulation, Partial differential equations, Probability theory, Reaction-diffusion system, Scattering theory, Time series analysis, Variational methods
  • Shin-ichiro Ei
    Shin-ichiro Ei

    Professor

    PhD

    Fields of Specialty:Nonlinear analysis, nonlinear partial differential equations

    The understanding of various patterns such as snow crystal, combustion, spot patterns of groups of plankton, and other kinds of chemical patterns appearing in nature are one of the most attractive objects of
    study in natural science. My interest is to theoretically study the structure and mechnism of such phenomena. To do it, I use description through model equations, which is one of the most theoretical methods, well known since Newton.
    The model equations which I have studied so far lie in the framework of partial differential equations which describe evolutional processes of certain materials with two mechanisms: (1) the diffusion process in space, and (2) the production and/or extinction of materials. Such model equations are generally called “reaction-diffusion systems” and their use has been well recognized in physics, chemistry, biology and other fields from a mathematical modelling point of view.
    It is important to have interests in patterns appearing in nature, which give a strong motivation and conatus for learning in our laboratory. I welcome such students.

  • Hideo Kubo
    Hideo Kubo

    Professor

    PhD

    Fields of Specialty:Partial Differential Equations associated with Nonlinear Dynamics

    The wave equation is one of the typical partial differential equations and has a long history. Although the wave equation looks like so simple, its mathematical structure is quite rich. In my research the effect from some perturbation such as the nonlinear perturbation, the presence of an obstacle, and so on are analyzed. The main issue is to compare the leading term of a solution to the unperturbed system and that to the perturbed system. For instance, the scattering theory is nothing else but the comparison between the behavior of solutions to these systems as time goes to infinity. We use functional analysis and real analysis for studying the scattering theory. But heavy computations based on calculus are the core of our analysis. It is of special interest to consider the case where the effect from the perturbation is balanced with that from the unperturbed system, because such consideration enables us to see the essential feature of the unperturbed and perturbed systems. Recently, I’m also interested in systems appeared in mathematical physics which are reduced to the wave equation and in the non-commutative structure of some partial differential equations.

  • Shuichi Jimbo
    Shuichi Jimbo

    Professor

    PhD

    Fields of Specialty:Applied analysis, Partial differential equations, Spectral theory

    Singular deformation of domains and spectral analysis: I enjoy walking on mountains in summer and skiing in winter. I walk around the woods, look at trees and feel the atmosphere. It is quite a joyful and comfortable experience. From physical point of view, this can be well simulated and demonstrated by receiving sound waves, light waves or mechanical vibrations induced by natural phenomena. They are the objects which I like to understand very well. These phenomena are written mathematically in terms of PDEs, which are several kinds of wave equations depending on the situations. Elliptic operators appear in these equations. My research interest is to study and analyze the spectra of these operators, centered around their dependence on the geometric properties and several other environment-oriented features.

  • Masaharu Nagayama
    Masaharu Nagayama

    Professor

    PhD

    Fields of Specialty:Reaction-diffusion systems, mathematical modeling, numerical simulation

    The goal of our study is to understand the mechanism of nonlinear phenomena from the mathematical viewpoint, using mathematical modeling, numerical simulation and mathematical analysis. For example, we are studying on the motion of droplets and particles, a chemical reaction and a cell dynamics.

  • Michiko Yuri
    Michiko Yuri

    Professor

    PhD

    Fields of Specialty:Ergodic theory, dynamical systems, complex systems

    The purpose of our project is to present mathematical ideas and methods?which are useful in?predicting asymptotic behavior of complex systems.
    In particular, we are interested in dynamics of complex systems exhibiting “nonhyperbolic” phenomena and in applying our results to a number of the applied sciences, (e.g., in neuroscience, physics, chemistry and economics).
    Our techniques are based on ergodic theory arising from equilibrium statistical physics. We develop a new concept that may be adapted to nonequilibrium steady states exhibiting dissipative phenomena producing non-stationary processes. This allows us to study statistical properties of complex systems admitting both chaotic and fractal structures.

  • Akira Sakai
    Akira Sakai

    Associate Professor

    PhD

    Fields of Specialty:Probability theory, statistical mechanics, mathematical physics

    My major research field is mathematical physics (probability and statistical mechanics). The topics I have been most fascinated with are phase transitions and critical phenomena, and associated scaling limits. For example, the Ising model exhibits a magnetic phase transition; it takes on positive spontaneous magnetization when the temperature of the system is turned down below its critical value. Various other observables also exhibit singular behavior around the critical point, due to cooperation of infinitely many interacting variables. To fully understand such phenomena, it would require development of a theory beyond the standard probability theory. This is a challenging and intriguing problem, towards which I would love to make even a tiny contribution. The mathematical models I have been studying are the Ising model, self-avoiding walk, percolation, the contact process (a model for the spread of an infection) and random walk with reinforcement.

  • Yuzuru Sato
    Yuzuru Sato

    Associate Professor

    PhD

    Fields of Specialty:Complex systems, chaotic dynamical systems

    I study nonlinear dynamical systems and complex systems with interest in complexity of spatio-temporal chaotic dynamics. My current research are focused on random dynamical systems approaches to noise-induced phenomena, time-series analysis of dynamical systems with large degrees of freedom, and information theoretic analysis of spatio-temporal chaos. Applications of nonlinear dynamical systems theory to problems in information theory, computation theory, prediction and control, are also put in perspective.

  • Takao Namiki
    Takao Namiki

    Associate Professor

    PhD

    Fields of Specialty:Ergodic theory, dynamical systems, complex systems

    I study complex system, especially cellular automata and quantum walks. First, we can observe fascinating dynamical behavior of cellular automata, typical complex system like the Game of Life, on their configuration space. Though the definition of cellular automata is various, my interest is the orbit structure of such behavior from the viewpoint of dynamical system and ergodic theory. Second, quantum walk, quantum version of random walk, shows different behavior from normal random walk. In the definition of quantum walk, a unitary operator on given Hilbert space is required and I study the features of quantum walks defined by the pair induced with measure preserving dynamical system. Now, relative to quantum information, the study of quantum walk is so important.

  • Kenji Matsumoto
    Kenji Matsumoto

    Associate Professor

    PhD

    Fields of Specialty:Biophysical complex systems, chaotic dynamical systems

    My main research interest lies in the analysis of time series from complex systems. I focus on analysis of the movement of microorganisms, developments of data acquisition programs for recording the movements of organisms including image processing and numerical simulation of high-dimensional dynamical systems.

  • Masakazu Akiyama
    Masakazu Akiyama

    Assistant Professor

    PhD

    Fields of Specialty:Mathematical modeling, mathematical and theoretical biology, numerical calculation

    A mathematical model of cleavage
    Mathematical analysis of Physarum
    A mathematical model of planar cell polarity
    A mathematical study of cell division model
    Mathematical analysis of gait transitions in quadrupeds.
    A mathematical model for locomotion of the amoeba cell.
    A mathematical model of rivers

  • Marko Jusup
    Marko Jusup

    Assistant Professor

    PhD

    Fields of Specialty:Mathematical Biology, Ecological Modelling, Biophysics

    Research based on an integrative global perspective has the potential to address the problems of practical significance for societies. The specific problems addressed in my research have quite diverse origins, ranging from ecology to biology to social sciences and even economics. Yet, when translated into the universal language of mathematics, the principles and processes underlying all these problems attain a similar form. Recognizing this form by means of the mathematical translations (i.e. models) of complex real-world phenomena and inferring the consequent dynamic properties is at the heart of analyses that I am conducting at Research Institute for Electronic Science (RIES) in collaboration with colleagues from Japan and internationally.
    In summary, the overall aim of my research is to formulate the mathematical models of complex ecological, biological, social, and economic phenomena and analyze their dynamic properties. Special emphasis is put on finding the solutions of practical significance for society and global community.