Research members & Research introduction
Department of Mathematics, Faculty of Science Division I
Professor Naoyuki Koike (Division Manager)
I research the differential geometry and the geometric analysis (which
is the research field using both the differential geometry and the analysis).
In particular, I research submanifolds in a pseudo-Riemannian manifold,
which is a general notion of a 4-dimensinal Lorentzian manifold treated
as the space-time in the Einstein's general relativity. In this research,
I use a Lie group action on a pseudo-Riemannian manifold and time evolutions
(the mean curvature flow, the inverse mean curvature flow and so on) of
submanifolds in a pseudo-Riemannian manifold. Also, we research the gauge
theory (in the theoretiacl physics) in view-point of the differential geometry.
In this reseach, we use the submanifold theory in an (infinite dimensional)
Hilbert space, and the notions of a Hilbert Lie group action on a Hilbert
space and the regularized mean curvature flow (on the Hilbert space) which
is invariant under the Hilbert Lie group action. In the future, I consider
to construct the theories of the discretization of the above researches
and apply them to the researches of the (wide-sense of) natural science
(the quantum mechanism (the quantum walk etc.), the condensed matter physics
(the higher dimensional geometric symmetry of super-degenerate material,
the method cotrolling interfaces etc.), molecular biology (the mechanism
of DNA・RNA and enzymes), grain boundary and composite materials mechanics
(the method cotrolling the structure and the strength of a grain boundary
and its applications to the composite materials mechanics).
Department of Mathematics, Faculty of Science Division II
Professor Akifumi Sako (Deputy Division Manager)
I aim to understand the noncommutative structure of spacetime with "quantization" as a key concept. Currently, our real world is roughly understood through quantum field theory - often referred
to as the standard
model - and the theory of gravity,
for which a successful quantization has not yet been
achieved. However, there remains a fascinating topic: how the breakdowns of these theories
at high energies, such as in the early universe or within black holes, might
lead to the construction of new theoretical frameworks. Since quantization gave rise to quantum mechanics, it is expected
that a similar "quantization" could also apply to spacetime itself. This necessitates a noncommutative
geometry arising from the quantization of spacetime. However,
the noncommutative geometries studied thus far have not been
sufficient to fulfill this role.
Department of Mathematics, Faculty of Science and Technology
Professor Makiko Tanaka
My primary research interest is geometry related to group actions, particularly
the geometry of homogeneous spaces admitting transitive Lie group actions.
I research the geometry of symmetric spaces, in particular. Symmetric spaces
are endowed with the involutive transformation called the symmetry at each
point, which leads to many nice properties. The ability to explicitly describe
geodesics, curvatures, and so on makes it possible to study geometric structures
and properties in detail. Lie groups can be viewed as symmetric spaces.
In particular, compact Lie groups admit Riemannian metrics invariant under
the symmetries, making them Riemannian symmetric spaces. I have been researching
antipodal sets of compact Lie groups and symmetric spaces these days. An
antipodal set is a discrete subset of which any two points are antipodal
(i.e., the symmetry at one point fixes the other point).
Based on the knowledge obtained
through these studies,
I will work on
collaborations in this research division.
Department of Mathematics, Faculty of Science and Technology
Professor Susumu Hirose
We are invetigating on low-dimensional topology especially on mapping
class groups over surfaces. Mapping class group of surfaces represents
the fiber structure on low-dimensional manifolds, for example hyperbolic
surface bundles over circles or Lefschetz fibrations,which are very important
objects for the recent research on low-dimensional manifolds. Mapping class
groups over surfaces are very interesting objects
from the view point of dymanical system.
On the other hand, braid
group, which is deeply related to the knot theory, is a kind of mapping
class groups. We will also make a progress on the research on knot thery
through the researchs on mapping class groups.
Department of Physics, Faculty of Science Division I
Professor Katsunori Suzuki
We study particle and nuclear physics, and their applications to high-energy
astrophysics, focusing primarily on phenomena related to the strong interaction.
The gauge theory of the strong interaction is known to be quantum chromodynamics
(QCD), in which quarks interact with gluons by exchanging the SU(3) color
charge. Due to its non-Abelian nature, QCD exhibits non-perturbative behavior,
and the vacuum state of our universe, possesses highly complex properties
such as spontaneous symmetry breaking and confinement. In this research
division, we aim to formulate the quantum walk approach in the presence
of gauge fields, or in the curved spacetime, with the goal of describing
non-perturbative phenomena using the quantum walk.
Department of Physics, Faculty of Science Division I
Professor Tetsuro Nikuni
We focus on the theoretical study of quantum many-body systems, with particular
interests in the nonequilibrium dynamics of ultracold atomic gases and
the application of quantum algorithms to optimization and quantum many-body
problems. In this division, we investigate quantum walks on graph structures,
focusing on how the geometric properties of graphs affect their dynamical
behavior.
Department of Chemistry, Faculty of Science Division I
Professor Makoto Tadokoro
WNT (water nanotube cluster), which is confined to quasi-one-dimensional
nanochannel pores in a hydrophilic single crystal, has been structurally
investigated and measured on proton conductivity. WNT confined to the crystalline
nano-pores can form a dynamic WNT (DWNT), due to the influence of some
regularly arranged atoms on the outer wall in narrow spaces. In addition,
DWNT can also form a new pre-melting phase with the mixture between water
and ice. In this research, we aim to develop new water-based materials
such as an artificial gas hydrate and dynamic ion-clathrate hydrate utilizing
DWNTs
Department of Mechanical and Aerospace Engineering, Faculty of Science
and Technology
Professor Shinji Ogihara
We are investigating the mechanical properties in aerospace lightweight materials such as fiber reinforced composites. Our main target is carbon fiber reinforced plastics (CFRP). These materials have unique mechanical properties because of the inhomogeneity
due to the introduction of carbon fibers into resin as reinforcement, considerable
anisotropy and laminate structure. We take approaches based on both experimental and analytical modeling. In experiments, we investigate the deformation behavior
under various loading conditions and evaluate the microscopic damage initiation/propagation behavior. Based on the experimental results, we discuss geometrical approaches to
these material systems.
Department of Mechanical and Aerospace Engineering, Faculty of Science
and Technology
Professor Akiyuki Takahashi
We are conducting research on dislocation behavior in heterogeneous metallic
materials, particularly alloys, using dislocation dynamics simulations
based on dislocation theory. In this work, we develop computational models
that capture the atomistic and continuum mechanics interactions between
dislocations and precipitates. By employing these models, we aim to elucidate
the strengthening mechanisms in alloys and the underlying mechanisms of
size effects in small-scale alloys through numerical analysis.
Department of Mathematics, Faculty of Science and Technology
Associated Professor Hisanori Ohashi
I study groups of several kinds within the scope of algebraic geometry,
mainly from the automorphism (self-symmetry) viewpoint. Historically, the
similarity between the specialness of the discrete objects on which finite
simple groups act and that of the interesting algebro-geometric varieties
is being known for long. Especially, I am interested in the problem of
qualitative descriptions of automorphisms of K3 and Enriques surfaces in
relation with the Golay codes, Mathieu and Conway groups. The main problems
are the combinatorics of discrete invariants of group actions and the construction
of equivariant projective models of relevant varieties. I also study the
related problems such as discrete groups of isometries of hyperbolic spaces,
number theory of Galois extensions and singularity theory.
Department of Mathematics, Faculty of Science and Technology
Associated Professor Kurando Baba
In differential geometry, Riemannian geometry is the field of the study
of manifolds equipped with Riemannian metrics. My research focuses mainly
on symmetric spaces, which are a rich and attractive class of Riemannian
manifolds. The theory of symmetric spaces has led to numerous results across
various areas of mathematics, including geometry, algebra, and analysis.
In particular, symmetric spaces are deeply connected to Lie group theory,
and the structures of Lie groups and their Lie algebras play a central role in understanding geodesics and curvature on these spaces.
The aim of my
project is to further develop the foundational theory established by Élie
Cartan, such as the classification of symmetric spaces and duality, and to
address classification problems arising from Lie group actions on symmetric
spaces. I also seek to uncover new types of dualities within this framework.
Department of Mathematics, Faculty of Science Division I
Associated Professor Daisuke Yamakawa
For a linear ordinary differential equation with rational functions as
coefficients, we can consider what is called monodromy/Stokes datum, which
describes the analytic continuation of the local solutions. It is possible
to deform the differential equation continuously so that the monodromy/Stokes
datum remains essentially unchanged, and the nonlinear differential equations
describing such deformations (called isomonodromic deformations) at the
infinitesimal level appear in the theory of integrable systems, conformal
field theory, gauge theory, and other fields. Currently, I am studying the
symmetry of isomonodromic deformations and their geometric aspects.
Department of Mathematics, Faculty of Science Division I
Associated Professor Natsumi Oyamaguchi
My research lies in low-dimensional topology, particularly knot theory
and spatial graph theory. I study the topological properties of knots and
spatial graphs, which are embeddings of the circle S^1 and finite graphs
into three-dimensional space.
I'm particularly interested in classifying spatial
graphs with 4-valent vertices up to rigid vertex isotopy. These
structures are especially well-suited to modeling dynamic changes in DNA
structure, and I hope to contribute to the interdisciplinary research within this
division from this perspective.
Department of Mathematics, Faculty of Science Division II
Associated Professor Yasufumi Nitta
We study the existence problem of canonical Kähler metrics, focusing on
their relationship with stability in the sense of geometric invariant theory.
Especially, we are deeply interested in the conjecture known as the Yau-Tian-Donaldson
conjecture, which predicts that a polarized manifold is K-stable if and
only if its polarization class contains a constant scalar curvature Kähler
metric. Recently, we have been studying weighted constant scalar curvature
Kähler metrics (wcscK), aiming to approach the existence problems of various
canonical Kähler metrics from a unified perspective. In addition, I am also
working on a cross-sectional study of existence problems of canonical Kähler
metrics of different types using wcscK.
Department of Chemistry, Faculty of Science Division I
Associated Professor Kazuya Otsubo
I am working on the research using transition-metal complexes, which
are composed of metal ions and organic molecules (organic ligands). Unlike
simple inorganic compounds, metal complexes are formed through coordination
bonds, which are of moderate strength compared to covalent or hydrogen
bonds. Furthermore, the combinations of metal ions and organic ligands
are virtually limitless, allowing for highly flexible and versatile material
design. My research focuses on the physical properties and solid-state
chemistry of transition-metal complexes. In particular, I am interested
in the synthesis of novel materials through advanced synthetic and reaction
techniques, as well as in elucidating their physical properties using cutting-edge
characterization methods, including synchrotron X-ray diffraction measurements.
Research Institute for Biomedical Sciences/Graduate School of Biological
Sciences
Associated Professor Masayuki Sakurai
Department of Mathematics, Faculty of Science Division I
Assistant Professor Toru Kajigaya
I am interested in geometric variational problems, particularly in the properties and geometric realizations of maps that arise
as critical points
of classical functionals such as volume
and energy. I investigate such maps by combining algebraic methods, including Lie group theory,
with techniques from geometric
analysis. Recently, I have also conducted fundamental research on the realization of discrete structures such as graphs and have explored their properties
using differential geometric methods. Furthermore, I have a strong interest
in applications that extend beyond the scope of fundamental research. Department of Mathematics, Faculty of Science Division I
Assistant Professor Tomoki Fujii
I am researching solitons for the mean curvature
flow. A soliton for the mean curvature
flow is a
submanifold
whose shape does not change under the mean curvature flow except for an
isometric transformation. Singular points
of the mean curvature flow are classified into Type I and Type II. After rescaling the mean curvature flow
with a sigular point of Type II, a translating soliton which is a kind of soliton for the mean curvature flow is obtained as a model of the
area around the singular point. Therefore, translating solitons
are important to understand the singular points
of the mean curvature flow.
I am mainlyresearching the shape of
translating solitons and other solitons for the mean
curvature flow given
as graphs of functions on Riemannian manifolds. The function is a solution
of a
certain differential equation, so I am analyzing the shape of the graph
of the solution of that differential equation.
Department of Mathematics, Faculty of Science Division I
Visiting Professor Koya Shimokawa
I aim to conduct
research in knot theory and three-dimensional topology
and to apply the results to the study of DNA topology.
Some DNA recombinases alter the topology of DNA during recombination. This
recombination process can be modeled as band surgery on knots and links. By
utilizing topological information before and after recombination, I seek to
elucidate the mechanisms of action of recombinases. Previous studies have
classified how DNA links are simplified by recombinase enzymes from a topological
perspective. In this research, I will incorporate differential geometric methods
in addition to the topological approaches to obtain more detailed information.
Furthermore, I will investigate how the geometric structure of DNA influences
processes such as gene expression. Since the study of DNA recombination
topology can also be applied to the study of vortex knots and defect line
knots, I hope to explore these directions as well.
Department of Mathematics, Faculty of Science Division I
Visiting Associated Professor Kazutoshi Inoue
Dislocations and grain boundaries are defects in the crystal lattice,
and their nanoscale strain distribution significantly impacts the macroscale
mechanical properties. Furthermore, the process of grain growth is an intricate
phenomenon that arises from substantial grain boundary migration during
the high-temperature sintering process. In this study, we investigate a
dislocation and grain boundary model over the lattice using discrete differential
geometry and its continuous limit. A grain boundary theory on the continuum
reflecting the discrete model is also proposed. To determine the conditions
for the optimal grain growth, we investigate the dependence of grain boundary
mobility on crystallographic orientation and develop an effective phase-field
model that is capable of reproducing experimental results.
Department of Mathematics, Faculty of Science Division I
Visiting Lecturer Takamasa Tsukamoto
I focus on "cluster materials," which are ultra-small particulate chemical substances composed of several to several dozen atoms, and I am investigating this class of materials through both experimental and theoretical approaches. In particular, through theoretical research, I recently discovered a peculiar type of cluster—referred to as a "super-degenerate material"—in which the energy levels of molecular orbitals exhibit an extraordinary degree of degeneracy that cannot occur in conventional chemical substances. This phenomenon cannot be explained within the framework of geometric symmetry in three-dimensional space, and it is suggested
to originate from a higher symmetry, known as dynamical symmetry. Currently,
with the aim of gaining a more fundamental understanding of this phenomenon from a perspective different from that of conventional chemistry, I am exploring the construction of a new theoretical
framework based on a mathematical science approach.