Graduate School of Human and Environmental Studies Department of Human Coexistence
Job title
Professor
Degree
Doctor of Science
Profile
I am studying so-called ``integrable systems'' in mathematics and mathematical physics by the method of algebraic analysis. The aim is to elucidate algebraic, geometric and combinatorial structures hidden in integrable systems, and to apply such knowledge to mathematical physis. My research interest also ranges over some other areas related to integrable systems, such as enumerative combinatorics, Painleve equations and twistor theory.
Research Assistant, Saitama University (Faculty of Science)
1985
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1991
Research Assistant, Kyoto University (Research Institute for Mathematical Sciences)
1991
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2003
Associate Professor, Kyoto University (College of Liberal Arts; Faculty of Integrated Human Studies; Graduate School of Human and Environmental studies)
2004
Professor, Kyoto University (Graduate School of Human and Environmental Studies)
Analysis and Mathematical Physics. 2 (2012), 171-214 Dec 2011 [Refereed]
A construction of general solutions of the \hbar-dependent Toda hierarchy is presented. The construction is based on a Riemann-Hilbert problem for the pairs (L,M) and (\bar L,\bar M) of Lax and Orlov-Schulman operators. This Riemann-Hilbert problem is translated to the language of the dressing operators W and \bar W. The dressing operators are set in an exponential form as W = e^{X/\hbar} and \...
J. Phys. A: Math. Theor. 45 (2012), 025403 (38pp) Oct 2011 [Refereed]
We study the thermodynamic limit of random partition models for the instanton
sum of 4D and 5D supersymmetric U(1) gauge theories deformed by some physical
observables. The physical observables correspond to external potentials in the
statistical model. The partition function is reformulated in terms of the
density function of Maya diagrams. The thermodynamic limit is governed by a
limit shape ...
Journal of Geometry and Physics 62 (2012), 1135-1156 Dec 2010 [Refereed]
The generating function of double Hurwitz numbers is known to become a tau
function of the Toda hierarchy. The associated Lax and Orlov-Schulman operators
turn out to satisfy a set of generalized string equations. These generalized
string equations resemble those of string theory except that the
Orlov-Schulman operators are contained therein in an exponentiated form. These
equations are...
J. Phys. A: Math. Theor.43:325205, 2010 Mar 2010 [Refereed]
The notion of non-degenerate solutions for the dispersionless Toda hierarchy is generalized to the universal Whitham hierarchy of genus zero with marked points. These solutions are characterized by a Riemann-Hilbert problem (generalized string equations) with respect to two-dimensional canonical transformations, and may be thought of as a kind of general solutions of the hierarchy. T...
The extended Toda hierarchy of Carlet, Dubrovin and Zhang is reconsidered in the light of a 2+1D extension of the 1D Toda hierarchy constructed by Ogawa. These two extensions of the 1D Toda hierarchy turn out to have a very similar structure, and the former may be thought of as a kind of dimensional reduction of the latter. In particular, this explains an origin of the mysterious structure...
Theoretical and Mathematical Physics (to appear) May 2011
This is a summary of a recursive construction of solutions of the hbar-dependent KP hierarchy. We give recursion relations for the coefficients X_n of an hbar-expansion of the operator X = X_0 + \hbar X_1 + \hbar^2 X_2 + ... for which the dressing operator W is expressed in the exponential form W = \exp(X/\hbar). The asymptotic behaviours of (the logarithm of) the wave function and the tau func...
The quantum torus algebra plays an important role in a special class of
solutions of the Toda hierarchy. Typical examples are the solutions related to
the melting crystal model of topological strings and 5D SUSY gauge theories.
The quantum torus algebra is realized by a 2D complex free fermion system that
underlies the Toda hierarchy, and exhibits mysterious "shift symmetries". This
article is ...
The festschrift volume for the 60th anniversary of Tetsuji Miwa Mar 2010
Recent work of Foda and his group on a connection between classical
integrable hierarchies (the KP and 2D Toda hierarchies) and some quantum
integrable systems (the 6-vertex model with DWBC, the finite XXZ chain of spin
1/2, the phase model on a finite chain, etc.) is reviewed. Some additional
information on this issue is also presented.
It has been shown that the dispersionless KP hierarchy (or the Benney hierarchy) is reduced to the chordal Löwner equation. We show that the radial Löwner equation also gives reduction of a dispersionless type integrable system. The resulting system acquires another degree of freedom and becomes the dcmKP hierarchy, which is a ``half'' of the dispersionless Toda hierarchy. The results of th...
"Infinite dimensional algebras and quantum integrable systems", Progress in Mathematics vol. 237, pp. 169--197 (Birkhauser, Basel/Switzerland, 2005) Dec 2003
Recent results on the Grassmannian perspective of soliton equations with an elliptic spectral parameter are presented along with a detailed review of the classical case with a rational spectral parameter. The nonlinear Schrödinger hierarchy is picked out for illustration of the classical case. This system is formulated as a dynamical system on a Lie group of Laurent series with factorization ...
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