Powering the Future - 立命館大学

Ritsumeikan University College of Science and Engineering
Department of Mathematical Sciences
Research/Development Areas
Stochastic process and its application,
financial mathematics and its application
Research/Development Areas
Structure analysis of von Neumann
algebras
Professor / Jiro AKAHORI, Assistant Professor / Takahiro TSUCHIYA
My interest lies in probability theory, financial mathematics and
the various fields that are related to them. This involves various
subjects that include abstract mathematics as well as its applications;
stochastic differential equation on topological groups,
quadratic Wiener functionals and infinite dimensional Lie algebras,
the pricing of financial derivatives, sustainable economic
growth problems, and so on. These research projects also involve
a number of postgraduate students. International exchanges
take active place with guests frequently visiting my
laboratory from all over the world, from whom we can all learn a
lot. The number of foreign students at our laboratory is also increasing
while some of our students occasionally get sent to
foreign universities. We also frequently have the opportunity to
travel overseas to attend academic society meetings. Graduates
from our laboratory
often enter professions
in the banking
industry.
Associate Professor / Hisashi AOI
As we live in a three-dimensional world the idea of “fourdimensions”
can be quite challenging but it is considered quite
routine in mathematics, with well developed arguments for it in
place. However, contrarily enough five-dimension or sixdimension
worlds appear to have been taken for granted.
My interest is in the “infinite dimensional” world that could be
considered to exist at the beyond of “finite-dimensional” worlds
where phenomena considered impossible in a finite-dimensional
world could occur. The subject of “operator algebras” can be
considered something that “acts” on this marvelous world. The
study of this is classified as “analysis”; however, it is also closely
related to algebra and geometry. In the real world quantum
mechanics and knot theory etc are also related to it.
This field is comparatively new in mathematics and has a lot of
unknown problems, thus making it a challenging research
subject.
College of S cience and Engineering
■Commemorative laboratory photo
Research/Development Areas
Semi-classical Analysis of Schrödinger
Equations
Professor / Setsuro FUJIIE
Semi-classical analysis is an asymptotic analysis where the
Planck constant appearing in the Schrödinger equation is regarded
as a small parameter. Under certain conditions, quantum
mechanics is expected to approach classical mechanics in
the semi-classical limit (Bohr’s correspondence principle). The
asymptotic distribution of eigenvalues or resonances created by
a bound or semi-bound state, respectively, is closely related to
the existence and the geometry of “trapped” trajectories of the
corresponding classical dynamics.
This problem is an extension of the famous question “Can one
hear the shape of the drum?” (M. Kac), which examines the relationship
between the geometry of a bounded domain and the
asymptotic distribution of eigenvalues of its Dirichlet Laplacian.
The useful WKB method consists of constructing an asymptotic
power series solution globally with respect to the Planck constant.
This power series diverges and the asymptotic form
changes discontinuously when passing through turning points
or caustics. This so-called Stokes phenomenon is a key to
solve the above problem.
Research/Development Areas
Application of gauge theory to V-manifolds
and its three-dimensional manifold in the
same boundary
Associate Professor / Yoshihiro FUKUMOTO
Homologically the same boundary groups configured with
whole three-dimensional homological spheres are an important
subject of research related to the unsolved expectation of triangles
being divisible by high-dimensional manifolds, however,
very little is known about the structure except the fact that it is
a finitely generated Abelian group. My research involves homologically
the same boundary invariants in seeking structures that
particularly include the integer lifting of classic Rochlin invariants
by applying gauge theory to V-manifolds. Gauge theory can be
used to extract topology information from nonlinear partial differential
equations describing the field (particle) on the manifold.
I focus on the contribution made by the singular point of a V-
manifold and configure the integer lift of an Ochanine invariant
based on elliptic genus and unbound algebra related to the
same boundary of the
three-dimensional manifold
and the functor in a
certain type of zone
with a commutative ring
in order to consider the
relationship between
basic group, homological
algebra and gauge
theory more.
■Application of gauge theory to topology
Ritsumeikan University
Powering the Future
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