Kyushu Integrable Systems Seminar

$B:G=*99?7F|(B: 2017/12/22 15:21

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  • $B9V1i
  • $B%?%$%H%k!'(B Integrable discrete models for one-dimensional soil water infiltration
  • $B35MW(B:

    I will describe some integrable discrete models for one-dimensional soil water infiltration, developed through collaborative research at the IMI Australia Branch. The discrete models are based on the continuum model by Broadbridge and White, which takes the form of nonlinear convection-diffusion equation with a nonlinear flux boundary condition at the surface. It is transformed to the Burgers equation with a time-dependent flux term by the hodograph transformation. We construct discrete models parallel to the continuum model and preserving the underlying integrability. These take the form of self-adaptive moving mesh schemes. The discretizations are based on linearizability of the Burgers equation to the linear diffusion equation, however the naive Euler discretization that is often used in the theory of discrete integrable systems does not necessarily produce a good numerical scheme. Taking desirable properties of a numerical scheme into account, we propose an alternative discrete model with reasonable stability and accuracy.

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  • $B9V1i
  • $B%?%$%H%k!'(B Growth of degrees of lattice equations and its signitures over finite fields
  • $B35MW(B:

    We study growth of degrees of autonomous and non-autonomous lattice equations, some of which are known to be integrable. We present a conjecture that helps us to prove polynomial growth of a certain class of equations including $Q_V$ and its non-autonomous generalization. In addition, we also study growth of degrees of several non-integrable equations. Exponential growth of degrees of these equations is also proved subject to a conjecture. Our technique is to determine the ambient degree growth of the equations and a conjectured growth of their common factors at each vertex, allowing the true degree growth to be found. Moreover, our results can also be used for mappings obtained as periodic reductions of integrable lattice equations. We also study signitures of growth of degrees of lattice equations over finite fields. We propose some growth diagnostics over finite fields that can often distinguish between integrable equations and their non-integrable perturbations.

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  • $B9V1i
  • $B%?%$%H%k!'(B Boundary-aware Hodge decompositions for piecewise constant vector fields
  • $B35MW(B:

    We provide a theoretical framework for discrete Hodge-type decomposition theorems of piecewise constant vector fields on simplicial surfaces with boundary that is structurally consistent with decomposition results for differential forms on smooth manifolds with boundary. In particular, we obtain a discrete Hodge-Morrey-Friedrichs decomposition with subspaces of discrete harmonic Neumann fields \(\cal{H}_{h,N}\) and Dirichlet fields \(\cal{H}_{h,D}\), which are representatives of absolute and relative cohomology and therefore directly linked to the underlying topology of the surface. In addition, we discretize a recent result that provides a further refinement of the spaces \(\cal{H}_{h,N}\) and \(\cal{H}_{h,D}\), and answer the question in which case one can hope for a complete orthogonal decomposition involving both spaces at the same time. As applications, we present a simple strategy based on iterated \(L^2\)-projections to compute refined Hodge-type decompositions of vector fields on surfaces according to our results, which give a more detailed insight than previous decompositions. As a proof of concept, we explicitly compute harmonic basis fields for the various significant subspaces and provide exemplary decompositions for two synthetic vector fields.

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  • $B9V1i
  • $B%?%$%H%k!'(B Raney distribution and random matrix theory
  • $B35MW(B:

    The Raney numbers are a generalisation of the Fuss-Catalan numbers, which occur in ballot type problems. Recently they have been shown to occur in random matrix theory as the moments of eigenvalue probability densities. Some themes resulting from this interpretation will be developed.

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  • $B9V1i
  • $B%?%$%H%k!'(B TThe full Kostant-Toda lattice and the positive flag variety
  • $B35MW(B:

    The full Kostant-Toda lattice hierarchy is given by the Lax equation \[ \frac{\partial L}{\partial t_j}=[(L^j)_{\ge 0}, L],\qquad j=1,...,n-1 \] where $L$ is an $n\times n$ lower Hessenberg matrix with $1$'s in the super-diagonal, and $(L)_{\ge0}$ is the upper triangular part of $L$.

    We study combinatorial aspects of the solution to the hierarchy when the initial matrix $L(0)$ is given by an arbitrary point of the totally non-negative flag variety of $\text{SL}_n(\mathbb{R})$. We define the full Kostant-Toda flows on the weight space through the moment map, and show that the closure of the flows forms a convex polytope inside the permutohedron of the symmetric group $S_n$. This polytope is uniquely determined by a pair of permutations $(v,w)$ which is used to parametrize the component of the Deodhar decomposition of the flag variety. This is a join work with Lauren Williams.

$BBh#6#12s!'(B

$B9V1i(2013-07-19)

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  • $B9V1i
  • $B%?%$%H%k!'(B The Lax pair of symmetric q-Painlevé VI equation of type A3
  • $B35MW(B: The Lax pair of symmetric q-Painlevé VI equation of type A3 (q-P3) is given in the form of 4$B!_(B4 matrices by V.G. Papageorgiou et al. in 1992. In this talk we will show the scalar Lax pair of q-P3. We also show how to construct scalar Lax pair by using the geometric way introduced by Y. Yamada in 2009. This work has been done in collaboration with Joshi Nalini.

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  • $B9V1il(B $BFXIW(B $B!JEl5~Bg3XBg3X1!Am9gJ82=8&5f2J!K(B
  • $B%?%$%H%k!'(B 3D integrability, quantized algebra of functions and PBW bases
  • $B35MW(B: I shall explain how Soibelman's theory of quantized algebra of functions Aq(g) led to a representation theoretical construction of a solution to the Zamolodchikov tetrahedron equation and the 3D analogue of the reflection equation proposed by Isaev and Kulish in 1997. If time allows, I present a theorem relating the intertwiners of representations of Aq(g) with the PBW bases of the positive part of the quantized enveloping algebra Uq(g). (Joint work with Masato Okado and Yasuhiko Yamada.)

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  1. 10:30-12:00
    • $B9V1ipJs(B)
    • $B%?%$%H%k!'(B Construction of the exceptional orthogonal polynomials and its application to the superintegrable system
    • $B35MW(B: To construct systems of polynomial eigenfunctions with jump in degree by means of the theory of Darboux transformation, a class of eigenfunctions of the Sturm-Liouville operator is introduced. Then we show a systematic way to construct the system of polynomial eigenfunctions with jump in degree from the Sturm-Liouville operator of the classical orthogonal polynomial. We classify these systems of the polynomial eigenfunctions with jump in degree according to the contour of integration. Finally, we give a brief review on the superintegrable Hamiltonian derived from the exceptional orthogonal polynomials.
  2. 13:00-14:30
    • $B9V1i
    • $B%?%$%H%k!'(B q = -1 limit of the Askey scheme
    • $B35MW(B: The q=1 limit of the Askey scheme is well known. There is however less trivial q=-1 limit of the same scheme. We give a review of recent results in this area. The Bannai-Ito scheme (i.e. the limiting case of the Askey scheme) is presented . Applications to perfect state transfer in quantum informatics are considered.

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  • $B9V1i
  • $B%?%$%H%k!'(B The second Painlevé hierarchy and mKdV equation
  • $B35MW(B: In this seminar, we will report on the fourth-order autonomous ordinary differential equation (*) which is compatible with the mKdV equation. The pair of (*) and mKdV equation is a partial differential equation in two variables. For this equation, we will present (1) polynomial Hamiltonian structure, (2) symmetry and holomorphy conditions. We will also discuss its phase space.

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  • $B%?%$%H%k!'(B $B%k!<%W72$K$h$k%Q%s%k%t%'(BIII$B7?J}Dx<0$N2r$N9=@.(B
  • $B35MW(B: $B8MED3J;RJ}Dx<07O$r4JLs2=$7$FF@$i$l$k%F%#%D%'%$%+J}Dx<0$O!$(B $B$=$N2r$,Dj5A0h$NJ#AG:BI8$NJP3Q$K0M$i$J$$>l9g$K%Q%s%k%t%'(BIII$B7?J}Dx<0$X$HJQ49$5$l$k!%(B $BK\9V1i$G$O%k!<%W72$rMQ$$$?D4OB

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    • $B%?%$%H%k!'(B Painleve$BJ}Dx<0$NA26aE83+!'(BBoutroux 100
    • $B35MW(B: Painlevé$BJ}Dx<0$NA26aE83+$,!$(B1913$BG/$K(BBoutroux$B$K$h$C$F8&5f$5$l$F$+$i(B100$BG/6a$/$?$C$?!%(B $BA26a2r@O$N8=>u$rGD0.$9$k$N$,Fq$7$/$J$C$F$$$k$3$H$rF'$^$($F!$(B
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    • $B%?%$%H%k!'(B $BNO3X7O$K8=$l$k(BPainlevé$BJ}Dx<0(B
    • $B35MW(B: $B%Y%/%H%k>l$NHsAP6J7?ITF0E@$NHsAP6J@-$O!$(B $B%V%m!<%"%C%W$K$h$j2r>C$9$k$3$H$,$G$-$k!%(B $B%V%m!<%"%C%W6u4V$G$O(BPainlev´$B@-$r;}$DJ}Dx<0$,8=$l$k$3$H$,B?$$!%(B $B$3$3$G$O$"$k%/%i%9$NNO3X7O$NFC0[@]F0LdBj$KBP$7!$(BBoutroux$B$K$h$k(BPainlevé$BJ}Dx<0$NA26aE83+$,!$(B $BNO3X7O$N5sF0$r7hDj$9$k$N$K=EMW$JLr3d$r2L$?$9$3$H$r<($7$?$$!%(B

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  • $B%?%$%H%k!'(B $B%(%k%_!<%HBP>N6u4V(B (S_N^C,S_N) $B$KBP$9$k(B Kazhdan--Lusztig $BB?9`<0$K$D$$$F(B
  • $B35MW(B: $B%(%k%_!<%HBP>N6u4V!J(BS_N^C.S_N$B!K$KBP1~$9$k(B parabolic Kazhdan-Lusztig $BB?9`<0$K4X$7$F!$(B $B%X%C%14D$,0[$J$k%Q%i%a!<%?$r;}$D$H$-$K!$$=$l$i$r7W;;$9$kAH$_9g$o$;O@E*$J5,B'$rM?$($k!%(B $B$3$l$i$O!$OD(B Ferres $B?^$r4JC1$J5,B'$G(B"Ballot $BBS(B"$B$K$h$jI_$-5M$a$k$3$H$GF@$i$kJ,G[4X?t$HF1CM$G$"$k!%(B $B$^$?$b$&0l$D$N7W;;K!$H$7$F!$(BLascoux-Schutzenberger $B$K$h$C$FF3F~$5$l$?FsJ,LZ;;K!$r3HD%$9$k!%(B

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  • $B%?%$%H%k!'(B $BD6N%;6(BAllen-Cahn$BJ}Dx<0(B
  • $B35MW(B: $BD6N%;62=$OM?$($i$l$?:9J,J}Dx<0$r%;%k!&%*!<%H%^%H%s$KJQ49$9$k6K8BA`:n$G$"$k!%(B $B$^$?!$$3$No2r$d?J9TGH2r$*$h$SBg0h2r$rM?$($k!%(B $B$3$l$i$N2r$O85$NJ}Dx<0$N2r$HN`;w$7$F$$$k$3$H$,J,$+$k!%(B

$BBh#5#22s!'(B

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  1. 15:00-16:30
    • $B9V1ipJs(B)
    • $B%?%$%H%k!'(B A unified approach to q-difference equations of the Laplace type
    • $B35MW(B: We propose a unified approach to q-difference equations, which are degenerations of basic hypergeometric functions 2φ1. We obtain a list of seven different class of q-special functions, including two types of the q-Bessel functions, the q-Hermite-Weber functions, two different types of the q-Airy functions. We discuss a relation between this unified approach and particular solutions of the q-Painlevé equations.
  2. 16:45-18:15
    • $B9V1ipJs(B)
    • $B%?%$%H%k!'(B Connection formulae of second order linear q-difference equations
    • $B35MW(B: We show connection formulae of some q-special functions: two types of the q-Airy functions, the Hahn-Exton q-Bessel function,... . These connection formulae are obtained by using the q-Borel transformation and the q-Laplace transformation which are introduced by C. Zhang. They are useful to consider connection problems between the origin and the infinity. We also review recent development on connection problems of linear q-difference equations.

$BBh#5#12s!'(B

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  • $B9V1i
  • $B%?%$%H%k!'(B Rigidity index and middle convolution of q-difference equations
  • $B35MW(B: We consider q-difference system ER: Y(qx)=R(x)Y(x) with rational function in element matrix coefficient. At first we define spectral type and rigidity index of ER. Next we obtain classification of 2nd order irreducible rigid equations. Moreover we define q-middle convolution algebraically. We can recompose that as analytical transformation using Euler transformation. Finally we consider about the important properties that q-middle convolution satisfies.

$BBh#5#02s!'(B

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  • $B9V1i>2H(B $B7I2p(B($BElBg?tM}(B)
  • $B%?%$%H%k!'(B $BGzH/$9$k2r$r$b$DHs@~7AGHF0J}Dx<0$NN%;62=(B
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  1. 13:00-14:30
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  2. 15:00-16:30
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    • $B%?%$%H%k!'(B Casorati solutions to non-autonomous DSKP and DSKdV equations
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  2. 15:00-16:30
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  • $B9V1i
  • $B%?%$%H%k!'(B Totally nonnegative Grassmann cells and soliton solutions of the KP equation
  • $B35MW(B: Let Gr(N,M) be the real Grassmannian defined by the set of all N-dimenaional subspaces of RM. Each point on Gr(N,M) can be represented by an NxM matrix A of rank N. If all the NxN minors of A are nonnegative, the set of all points associated with those matrices forms the totally nonnegative part of the Grassmannian, denoted by Gr+(N,M). In this talk, I will give a realization of Gr+(N,M) in terms of the soliton solutions of the KP equation, and construct a cellular decomposition with the permutation group SM. Each element of SM can be uniquely expressed by a chord diagram. I will then discuss a classification problem of the soliton solutions of the KP equation based on the cellular decomposition of Gr+(N,M), and show that the chord diagrams can be used to analyze the asymptotic behavior of certain initial value problems of the KP equation. I will also present some movies of real experiments of shallow water waves which represent some of new solutions obtained from the classification problem.

$BBh#3#82s!'(B

$B6e=#Bg3XH":j%-%c%s%Q%9M}3XIt(B1$B9f4[(B4F$B!&(B1401$B9f< $B:#2s$N%;%_%J!<$O4v2?3X%;%_%J!<$H9gF1$G9T$o$l$^$9!%(B

  • $B9V1i
  • $B%?%$%H%k!'(B Discrete Hilbert-Einstein functional: history and applications
  • $B35MW(B: The Hilbert-Einstein functional S_M for a manifold M maps a Riemannian metric g on M to the integral of its scalar curvature. Remarkably, the critical points of S_M correspond to Einstein metrics on M.
    There is a discrete Hilbert-Einstein functional, also known as the Regge functional. For 3-dimensional manifolds, its critical points are metrics of constant sectional curvature.
    In this talk, we review the history of the discrete Hilbert-Einstein functional and describe some recent applications of it.

$BBh#3#72s!'(B

$B6e=#Bg3XH":j%-%c%s%Q%9M}3XIt(B3$B9f4[(B3F$B!&(B3311$B9f<

  • $B9V1i
  • $B%?%$%H%k!'(B Soliton solutions and asymptotic behaviors of initial value problems in Kadomtsev-Petviashvili II equation
  • $B35MW(B: Recently a large variety of new soliton solutions of Kadomtsev-Petviashvili II (KPII) equation have been found and studied extensively. These solutions enable us to describe more general interactions of line solitons. In order to investigate the importance of this solutions, we consider initial value problems for V-shape initial waves consisting of two semi-infinite line solitons. By the numerical simulations it is shown that the time developments of the interactions settle to almost steady states (asymptotic patterns) which is described by the corresponding soliton solutions. This relationship is explaned using sub-chord diagram derived from the initial condition.
    This is a joint work with Y. Kodama(Ohio State Univ.) and M. Oikawa(Kyushu Univ.).

$BBh#3#62s!'(B $B>.JY6/2q!V4X?tJ}Dx<0$NJQ49M}O@$H2D@QJ,7O!W(B

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3 $B7n(B 16 $BF|(B ($B7n(B) $B!!(B 

13:00$B!<(B15:00 $BDEED>H5W(B ($B6e=#Bg(B) 
UC $B3,AX$H%b%N%I%m%_!eBs;V(B ($BEl5~Bg(B) 
$B0lHLBg5WJ]7?J}Dx<0$H(B middle convolution $B$N3HD%$K$D$$$F!$(BI 

3 $B7n(B 17 $BF|(B ($B2P(B) $B!!(B 

10:00$B!<(B12:00 $B@n>eBs;V(B ($BEl5~Bg(B) 
$B0lHLBg5WJ]7?J}Dx<0$H(B middle convolution $B$N3HD%$K$D$$$F!$(BII 

14:00$B!<(B16:00 $B3a867r;J(B ($B6e=#Bg(B) 
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$BBh#3#52s(B

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  1. 14:30-16:00
    • $B9V1i
    • $B%?%$%H%k!'(B $BB?@.J,#A#S#E#P$N8GM-CM$K$D$$$F(B
    • $B35MW(B: $B#1e$NB?N3;R7O$N3NN(2aDx$GHsBP>NC1=cGSB>2aDx(B $B!J(BAsymmetric Simple Exclusion Process, ASEP$B!K$H8F$P$l$kHsJ?9U%b%G%k$,$"$k!#(B $B9)3XE*$K$O8rDLN.$N4pK\%b%G%k$H$7$F0LCV$E$1$i$l!$?t3XE*$K$OD>8rB?9`<0!$%i%s%@%`9TNs$J$I$H4XO"$,<($5$l$k$J$I6=L#?<$$!#(B
      $B:#2s$OB?@.J,#A#S#E#P!JN3;R$N@_$K$h$C$F2D2r$G$"$k!%(B $B0[$J$k%;%/%?!<4V$G$N8GM-CM$N4X78$K$D$$$F$o$+$C$?$3$H$r=R$Y$k!%(B $BK\8&5f$OElBg$N9q>lFXIW;a!$:fOB8w;a!$?tM}%7%9%F%`$NBtJU9d;a$H$N6&F18&5f$G$"$k!%(B
  2. 16:15-17:45
    • $B9V1i
    • $B%?%$%H%k!'(B A completeness study on certain 2X2 Lax pairs
    • $B35MW!'(BA Lax pair is a pair of linear equations that are associated with an integrable nonlinear equation through a compatibility condition. This talk will explain a method to identify all possible Lax pairs of a certain type (described below) without making assumptions about the form of the spectral parameter. We will focus on Lax pairs that are in the 2X2 matrix form, where each entry of the Lax matrices contains only one quantity. These quantities can be separated into a product of two parts, one that depends on the spectral variable and one that does not. The Lax pairs are otherwise free. In fact, of all the potential Lax pair candidates identified by this method, only two lead to interesting evolution equations. These are higher order varieties of the lattice sine-Gordon and the lattice modified KdV equations.

$B%o!<%/%7%g%C%W!V%F!<%?4X?t$H2D@QJ,7O!W(B

12 $B7n(B 20 $BF|(B ($BEZ(B)

10:00-11:00
Yasuhiko Yamada (Kobe University):
"$BBJ1_(BPainleve$BJ}Dx<0$N4v2?3XE*(BLax$B7A<0(B"

11:10-12:10
Yoshihiro Onishi (Iwate University):
"Two topics in concrete theory of Abelian functions"

14:00-15:00
Kanehisa Takasaki (Kyoto University):
"Hurwitz numbers and tau functions"

15:30-16:30
Takahiro Shiota (Kyoto University):
"On recent progress on Schottky problem (1)"

12 $B7n(B 21 $BF|(B ($BF|(B)

10:00-11:00
Keiji Matsumoto (Hokkaido University):
"Formulas of Thomae type and mean iteration"

11:10-12:10
Tomohide Terasoma (University of Tokyo):
"Thomae$B$N8x<0$K$"$i$o$l$kDj?t$H(Bbinary graph"

14:00-15:00
Atsushi Nakayashiki (Kyushu University):
"Modular invariant tau functions of the KP hierarchy"

15:30-16:30
Takahiro Shiota (Kyoto University):
"On recent progress on Schottky problem (2)"

12 $B7n(B 22 $BF|(B ($B7n(B)

10:00-11:00
Ryohei Hattori (Hokkaido University):
"On periods of cyclic triple covering of the complex projective line"

11:10-12:10
Andrey Mironov (Sobolev Institute, Novosibirsk):
"Baker-Akhiezer modules on rational varieties"

$BBh#3#42s(B

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    • $B9V1ieLn(B $BM40l(B $B;a!J?@8MBgM}!K(B
    • $B%?%$%H%k!'(B Polynomial Hamiltonians for Quantum Painlevé Equations
    • $B35MW(B: $BNL;R(BPainlevé$BJ}Dx<0$N$"$k@5B'@-$K$h$kFCD'IU$1$r9T$&!%$9$J$o$A!$(B $BNL;R(BPainlevé$BJ}Dx<0$N%O%_%k%H%K%"%s7O$,$^$?!$B?9`<0%O%_%k%H%K%"%s7O$KJQ49$5$l$k$h$&$J@5=`JQ49$rF3F~$7!$(B $B5U$K!$$3$N@5B'@-$K$h$j!$%O%_%k%H%K%"%s$,$?$@#1$D$KFCD'IU$1$i$l$k$3$H$r<($9!%(B $B$^$?!$(BH.Nagoya$B$K$h$jM?$($i$l$?%O%_%k%H%K%"%s$H$N4X78$b=R$Y$k!%(B
  2. 16:30-17:50
    • $B9V1i
    • $B%?%$%H%k!'(B A$B7?%I%j%s%U%'%k%H!&%=%3%m%U3,AX$KM3Mh$9$k7k9g7?%Q%s%k%t%'(BVI$B7O(B
    • $B35MW!'%Q%s%k%t%'(BVI$BJ}Dx<0$O$"$k5(1) $B7?Ln3$!&;3ED7O$K5"Ce$9$k$h$&$J7k9g7?%Q%s%k%t%'(BVI$BJ}Dx<0$r?7$?$KF3F~$9$k!%(B $B$^$?$=$l$HJ;$;$F!$(BA4(1), A5(1) $B7?Ln3$!&;3ED7O$N?7$7$$%i%C%/%97A<0$bF3F~$9$k(B.

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  • $B%?%$%H%k!'(B Mean iterations and hypergeometric functions
  • $B35MW!'(B $B@5?t(B a,b $B$N;;=Q4v2?J?6Q$,D64v2?4X?t$rMQ$$$FI=<($G$-$k$3$H$,(B Gauss $B$K$h$C$F(B 1799$BG/$K<($5$l$?!%$3$N9V1i$G$O!$D64v2?4X?t$N$_$?$94X?tEy(B $B<0$KCeL\$7!$;;=QJ?6Q$d4v2?J?6Q0J30$K$b$$$/$D$+$NJ?6Q$rDj$a!$$=$l$i$NJ?(B $B6QA`:n$N7+JV$7$K$h$k6K8B$rD64v2?4X?t$NFC

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  • $B9V1i
  • $B%?%$%H%k!'(B On Discrete Differential Geometry and its Applications
  • $B35MW!'(BThe first soliton equations were found mid 19th century in the context of physics (the KdV equation) as well as in geometry (SineGordon Eq.). Many of the properties we now associate with integrability where already known then. With the example of arc-length preserving flows on plane curves one can see how soliton equations and their transformations can arise from simple geometric assumptions. This construction can be discretized in a straight forward way and results in some discrete geometry as well as integrable difference equations. Moreover it can be applied to surfaces as well giving rise to many classes of discrete surfaces as well as discretized notions that have applications in other fields.

$BBh#2#82s(B

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  • $B%?%$%H%k!'(B $B%H%m%T%+%kBJ1_6J@~$HD6N%;6(BQRT$B7O(B
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  • $B9V1i8E20(B $BAO(B $B;a!!!J7DXfBgM}9)!K(B
  • $B%?%$%H%k!'(B Quantum Painlevé equations and quantum isomonodromic deformation
  • $B35MW(B: $BK\9V1i$G$O%Q%s%k%t%'J}Dx<0$NBP>N@-$r0z$-7Q$0NL;R2=$NDs<($*$h$S(B $BBP>N@-$,(B Al(1) $B7?$N>l9g$K(B Lax $B7A<0$rMQ$$$?0lHL2=$N9=@.$r9T$&!#$^$?(B $B6&7A>lM}O@$K$*$1$kAj4X4X?t$NK~$?$9HyJ,J}Dx<0$G$"$k(B Knizhnik-Zamolodchikov $BJ}Dx<0$H%b%N%I%m%_!e5-$NBP>N@-$r;}$DNL;R%Q%s%k%t%'(B $BJ}Dx<0$,(B QPV $B$+$i(B QPI $B$^$G$K$*$$$FF@$i$l$k$3$H$r<($9!#(B

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  • $B%?%$%H%k!'(B ODE on Rational Surfaces
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    • $B%?%$%H%k!'(B KP$B3,AX$+$i$N%Q%s%k%t%'(BVI$B!$$*$h$S$=$NFC
    • $B35MW!'(BTBA

$B%W%m%0%i%`(B:
9:30-11:30 $Bd*(B
13:00-14:30 $B0f>e(B($BA0H>(B)
14:30-15:00 coffee break
15:00-16:30 $B0f>e(B($B8eH>(B)


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  1. 15:00-16:00
    • $B9V1i
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    • $B35MW(B: $B8rDLN.8&5f$NBP>]$O!$0lDjJ}8~$K1?F0$9$kB?N3;R7O$K$*$$$F!$1?F0$NHsBP>N@-!$N3;R(B $B$NBN@Q$K$h$kGSB>@-!$H?1~$NCY$l$J$I$K$h$C$F@8$8$k=8CD8=>]$G$"$k!%(B $B=BBZ$O$=$NE57?E*$JNc$G$"$j!$$=$NH/@8$H2r>C$N2aDx$O8rDLN.$KFHFC$N%@%$%J%_%/%9(B $B$K$h$k$b$N$H9M$($i$l$k!%(B $B8rDLN.%@%$%J%_%/%9$N%b%G%k2=$K$O!$7hDjO@$^$?$O3NN(O@!$O"B3$^$?$ON%;6$N(B $BAH$_9g$o$;$,$"$j!$3NN(O@$K4p$E$/%b%G%k$OHsJ?9UE}7WNO3X7O$NE57?E*$JNc$H(B $B$7$F@9$s$K8&5f$5$l$F$$$k!%0lJ}$G!$7hDjO@$K4p$E$/%b%G%k$K$D$$$F$O%=%j%H(B $B%sM}O@$K4p$E$$$?87L)2r$N8&5f$b9T$o$l$F$$$k!%(B $B:#2s$N%;%_%J!<$G$O!$$$$/$D$+$N4pK\E*$J%b%G%k$H$3$l$i$N<($98=>]$K$D$$$F(B $B@bL@$9$k!%$^$?!$:G6a$N8&5f$+$i!$$"$k3NN(%b%G%k$,FCJL$J>l9g$KD64v2?4X?t(B $B$r;H$C$F87L)$K2r$1$k$3$H!$$*$h$S;~4VCY$lHy:9J,J}Dx<0$GDj5A$5$l$k7hDjO@(B $BE*%b%G%k$KBP$7$F9-ED$NJ}K!$K$h$j?7$7$$2r$,F@$i$l$k$3$H$J$I$r>R2p$9$k!%(B
  2. 16:15-17:15
    • $B9V1i
    • $B%?%$%H%k!'(B $BD9GHC;GH6&LDAj8_:nMQJ}Dx<0$K$D$$$F(B
    • $B35MW!'(BTBA

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  • $B9V;U!'@>2,(B $B@F<#(B $B;a!JElBg?tM}!K(B
  • $B%?%$%H%k!'(B A7(1)$B7?(Bq-Painlevé$BJ}Dx<0$N2r$ND61[@-(B
  • $B35MW(B: A7(1)$B7?(Bq-Painlevé$BJ}Dx<0$H$O!$(B
    f(qt) f(t)2 f(t/q) = t (1-f(t))
    $B$H$$$&(Bq-$B:9J,J}Dx<0$G$"$k!#$b$7(B0$B$G$J$$J#AG?t(B q $B$,(B 1 $B$N6R:,$G$J$$$J$i!"$3$NJ}Dx<0$N2r(B $B$OJ#AG78?tM-M}4X?tBN>eD61[E*$G$"$k!#(B

$BBh#1#82s(B

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Speakers:
Chris Ormerod University of Sydney (Australia)
Toshiyuki Mano$B!!(B($BbCLn(B $BCR9T(B) Kyoto University

Program:
15:00-16:00 Ormerod
16:15-17:15 Mano

Title and abstract
  • C. Ormerod
    • Title: Connection matrices for ultradiscrete linear problems
    • Abstract: One may consider the appropriate domain for ultradiscrete equations to be the max-plus semiring. We develop the theory of Birkhoff and his school for systems of linear difference equations over the max-plus semiring. We use such theory to provide evidence for the integrability of some ultradiscrete difference equations. We show that by considering the Birkhoffs fundemental solutions, we may recover some a select few hypergeometric solutions to the ultradiscrete equations not obtainable via the ultradiscretization method.

  • T. Mano
    • Title: Monodromy preserving deformations of the linear differential equations on elliptic curves and the sixth Painlevé equation
    • Abstract: K. Okamoto derived Hamiltonian systems which govern isomonodromic deformation of the linear differential equations on elliptic curves. We wish to investigate properties of their solutions. This talk is an attempt for this object. We degenerate elliptic curves to a rational curve with one ordinary double point. Then we consider isomonodromic deformations on this singular rational curve. We can relate the solutions of the isomonodromic deformation problem of this singular rational curve to the solutions of PVI (they govern isomonodromic deformation on a non-singular rational curve) via resolution of singularity. And inversely, we can recover the solutions of the isomonodromic deformation problem on singular curve from the solutions of PVI and additional data. This result informs us about conjectural properties of solutions of isomonodromic deformation problem on elliptic curves.


$BBh#1#72s(B

$B6e=#Bg3XH":j%-%c%s%Q%9M}3XIt#39f4[(B 3303 $B9f<

$B:#2s$N%;%_%J!<$OBe?t3X%;%_%J!<$H$N6&:E$G$9!%(B

$B9V1i Prof. Bruce C. Berndt University of Illinois at Urbana-Champaign $B3a86(B $B9/;K(B $B:eBg!&M}(B

$B%W%m%0%i%`(B:
14:00-15:00 Berndt
15:00-15:20 tea
15:20-16:20 $B3a86(B

$B9V1iBjL\$H35MW(B
  • B. Berndt
    • Title: Ramanujan's Lost Notebook with particular attention to the Rogers--Ramanujan and Enigmatic Continued Fractions
    • Abstract: In the spring of 1976, George Andrews visited the library at Trinity College, Cambridge, and found a sheaf of 138 pages containing approximately 650 unproved claims of Ramanujan. In view of the fame of Ramanujan's notebooks, Andrews called his finding "Ramanujan's Lost Notebook." I will provide a history and description of the lost notebook. I will then give a survey on several entries of the lost notebook with emphasis on the Rogers-Ramanujan and "enigmatic" continued fractions.

  • $B3a86(B $B9/;K(B
    • Title: Multiple hypergeometric transformation formulae with different dimensions
    • Abstract: About 30 years ago, hypergeometric series in SU(n + 1) (or hypergeometric series of type An) have been introduced by Holman, Biedenharn and Louck in need of the explicit expressions of the Clebsch-Gordan coefficients for irreducible representation of the unitary group SU(n+1). Including its basic (often called as Milne$B!G(Bs class), elliptic and root system analogues, multiple hypergeometric series has been investigated in several points of view. Also some applications have been found, for example, infinite series of the sum of squares formulae by S.C.Milne and multivariate orthogonal polynomials of Heckman-Opdam type by several authors. In this talk, I will present some transformation formulae for multiple hypergeometric series of type A with different dimensions by starting from the Cauchy's reproducing kernel. In the course of derivations, symmetries of the Cauchy kernel and a certain divided difffence operator (a special case of Macdonald$B!G(Bs q-difference operators) will be used. By combining the transformation formulae with different dimensions, a number of hypergeometric transformations of type An can be obtained including known ones. If my time will remain, the symmetries of several class of hypergeometric series of type An will be discussed. There, a Coxeter group which has not been in literature arises as a group descibing the symmetries of a class of An hypergeometric series.

      [1] Yasushi Kajihara: $B!H(BEuler transformation formula for multiple basic hypergeometric series of type A and some applications.$B!I(B, Advances in Mathematics 187 (2004), pp53-97.
      [2] Yasushi Kajihara: $B!H(BOn multiple hypergeometric transformation formulae arising from the balanced duality transformation.$B!I(B preprint.


$BBh#1#62s(B

$B6e=#Bg3XH":j%-%c%s%Q%9M}3XIt#39f4[#3#3#0#39f<

$B:#2s$N%;%_%J!<$O(B21$B@$5*(BCOE$B%W%m%0%i%`(B $B!V5!G=?tM}3X$N9=C[$HE83+!W(B$B$N1g=u$r

  • $B9V;U!'(B$B;y6L(B $BM5<#(B $B;a!!!J(BOhio State University$B!K(B
  • $B%?%$%H%k!'(B Positive Grassmannian cells and soliton solutions of the KP equation
  • $B35MW(B: Starting with an elementary introduction of the finite dimensional Grassmannians Gr(N,M), the set of N-dimensional subspaces in M-dimensional Euclidian space (M>N), we describe soliton solutions of the KP equation based on the Schubert decomposition of Gr(N,M). Each soliton solution can be identified as a point on a positive Grassmann cell which is given by a further decomposition of the Schubert cells. We also show that the classification of N-soliton solutions can be obtained by the chord diagrams having overlapping and crossings. This is a joint work with S. Chakravarty.

$BBh#1#52s(B

$B6e=#Bg3XH":j%-%c%s%Q%9M}3XIt#39f4[(B 3311 $B9f<

$B9V1i Ramajayam Sahadevan University of Madras, India $B5FCO(B $BE/Li(B $BElBg!&?tM}(B Gino Biondini State University of New York at Buffalo, USA

$B2>%W%m%0%i%`(B:
13:30-14:30 Sahadevan
14:45-15:45 $B5FCO(B
16:00-17:00 Biondini

$B9V1iBjL\$H35MW(B
  • R. Sahadevan
    • Title: Higher dimensional autonomous and nonautonomous integrable mappings
    • Abstract: Recent work on the investigation of completely integrable fourth order autonomous mappings
      w(n+4) = F(w(n),w(n+1),w(n+2),w(n+3))
      is presented. After reviewing the various working definitions of complete integrability of differential, differential - difference and pure difference equations, we derive the conditions on the function F for which the fourth order mapping is symplectic and admitts two independent integrals of motion ensuring complete integrability in the sense of Liouville. A systematic method is presented to construct explicitly two or three independent integrals of the above mapping. Also, the question of deautonomizing the obtained autonomous integrable equations is briefly discussed.

  • $B5FCO(B $BE/Li(B
    • Title: q-Painlevé equations arising from q-KP hierarchy
    • Abstract:A q-analogue of An(1) generalized Drinfeld-Sokolov hierarchies is proposed as a reduction of multi-component q-KP hierarchy. Applying similarity reduction to the hierarchy, one can obtain q-Painleve equations. Especially, in the three-component case, the q-Painleve VI equation is obtained.

  • G. Biondini
    • Title: Introduction to the mathematical foundations of optical fiber communication systems
    • Abstract:

      The development of high-data-rate optical fiber communications is one of the great technological achievements of the late 20th century. Data rates continue to grow according to Moore's law, with doubling times of a year or less -- even faster than for computers. The mathematical modeling of these systems is highly nontrivial, however, due to the many nonlinear and stochastic effects that contribute to determine the system behavior. In addition, because of the extremely small error rates required of these systems, quantifying the system performance presents an analytical and computational challenge.

      This talk aims at providing an overview of the fundamental effects governing optical fiber communication systems and the mathematical techniques that are used to study them. After briefly mentioning the various components of optical fiber communication systems, we will see how the nonlinear Schroedinger equation is obtained as a model to describe light propagation in optical fibers. We will then review the most important perturbations affecting system behavior, together with the mathematical tools appropriate to describe each of them. We will end the talk by discussing some of the most important open research problems.

Faculty of Science No.3 building 3F Room 3311, Hakozaki Campus of Kyushu University, June 8 2006(Thurs.) 13:30-17:00

Speakers:
Ramajayam Sahadevan (University of Madras, India)
Tetsuya Kikuchi (University of Tokyo)
Gino Biondini (State University of New York at Buffalo, USA)

Tentative program:
13:30-14:30 Sahadevan
14:45-15:45 Kikuchi
16:00-17:00 Biondini


$BBh#1#42s(B

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  • $B9V;U!'>>K\(B $B>[(B $B;a!!!J6eBg?tM}!K(B
  • $B%?%$%H%k!'(B $B%O%$%Q!<9TNs<0$r;H$C$?%7%e!<%"4X?t$NOB8x<0(B
  • $B35MW(B:$B%O%$%Q!<9TNs<0$O9TNs<0$NC1=c$J3HD%$G!"(B19$B@$5*$KDj5A$5$l$?$b$N$G(B $B$"$k!%:#2s$O$3$N%O%$%Q!<9TNs<0$r;H$C$?%7%e!<%"4X?t$NOB8x<0$N>R2p$rCf?4(B $B$KOC$r$9$k!%$^$?!"%F%W%j%C%D9TNs<0$N%O%$%Q!


$BBh#1#32s(B

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$B9V1i $B9b66(B $BBgJe(B ($BAa0pEDBg!&M}9)(B) $B>>1:(B $BK>(B ($BJ!2,Bg!&M}(B) Dr. Mariusz Bialecki ($BEl5~Bg!&?tM}(B)

$B2>%W%m%0%i%`(B:
13:30-14:30 $B9b66(B
14:45-15:45 $B>>1:(B
16:00-17:00 Bialecki

$B9V1iBjL\$H35MW(B
  • $B9b66(B $BBgJe(B
    • Title: $BL@<(E*$J%j%c%W%N%U4X?t$r;}$DHyJ,!&:9J,!&D6N%;6J}Dx<0(B
    • Abstract: $BL@<(E*$J%j%c%W%N%U4X?t$r;}$A!$AjJ?LLFb$K$*$$$FG$0U$N=i4|CM$+$i$N(B $B2r$,JD6J@~$J$I$NC1=c$J%"%H%i%/%?$KMn$A9~$`$h$&$J!$$"$k%/%i%9$N#23,(B $BHs@~7A:9J,J}Dx<0$K$D$$$F2r@b$9$k!%$3$l$iJ}Dx<0$N0lIt$K$OO"B36K8B!&(B $BD6N%;66K8B$,B8:_$7!$F@$i$l$?HyJ,!&D6N%;6J}Dx<0$bF1MM$N@-/$7$G$b3H$,$l$P$H$$$&$N$,4j$$$G$"$k!%(B

  • $B>>1:(B $BK>(B
    • Title: TBA
    • Abstract:

  • Dr. Mariusz Bialecki
    • Title: Algebro-Geometric Solution of the Discrete KP Equation over a Finite Field out of a Hyperelliptic Curve
    • Abstract: The algebro-geometric method of construction of solutions of the discrete KP equation in the finite field case will be presented. We point out the role of the Jacobian of the underlying algebraic curve in construction of the solutions. Using the description of the Jacobian of a hyperelliptic curve we will show in details how this method works.

Faculty of Science No.3 building (room to be determined), Hakozaki Campus of Kyushu University, March. 14 2006(Tue) 13:00-17:00

Mini-workshop "Recent Topics in the Discrete and Ultradiscrete Integrable Systems"

Speakers:
Prof. Daisuke Takahashi (Waseda University)
Dr. Nozomu Matsuura (Fukuoka University)
Dr. Mariusz Bialecki (University of Tokyo)

Tentative program:
13:30-14:30 Takahashi
14:45-15:45 Matsuura
16:00-17:00 Bialecki

$BBh#1#22s(B

$B6e=#Bg3XH":j%-%c%s%Q%9M}3XIt#39f4[#3#3#0#49f<

Faculty of Science No.3 building (room to be determined), Hakozaki Campus of Kyushu University, Jan. 24 2006(Tue) 14:00-17:00

$B%_%K%o!<%/%7%g%C%W!V(BSolitons in Mathematics and Physics$B!W(B
(21$B@$5*(BCOE$B%W%m%0%i%`(B $B!V5!G=?tM}3X$N9=C[$HE83+!W(B$B$N1g=u$r

Mini-workshop "Solitons in Mathematics and Physics "
(Supported by 21st century COE program "Development of Dynamic Mathematics with High Functionality")

$B:#2s$O(B$B8=>]?tM}%;%_%J!<(B $B$H$N6&:E$G$9!%(B

This is a joint seminar with Seminar on Nonlinear Phenomena and Analysis.

Speakers:
Prof. Jon Nimmo (University of Glasgow and University of Tokyo)
Dr. Pearl Louis (Osaka City University)
Dr. Ken-ichi Maruno (Kyushu University)

Tentative program:
14:00-15:00 Louis
15:15-15:45 Maruno
16:00-17:00 Nimmo

Titles and abstracts:
  • Dr. Pearl Louis
    • Title: Matter-wave solitons in optical lattices and superlattices
    • Abstract: In recent years there has been increasing interest in using Bose-Einstein condensates (BECs) or macroscopic coherent matter-waves to study a variety of different quantum and nonlinear phenomena including solitons. Until recently, bright solitons could only be formed in BECs with attractive interatomic interactions [1,2]. However, working with bright solitons in BECs with repulsive interatomic interactions is more desirable as attractive BECs collapse if the atomic density reaches a critical level. Recently in an important experiment, a bright soliton was created in a BEC with repulsive interatomic interactions by using a periodic potential formed by standing light waves known as an optical lattice [3]. In this work we look at the properties of bright gap solitons and dark in-band solitons in repulsive BECs in optical lattices. One of the principle advantages of optical trapping potentials is that their properties can be easily and precisely controlled. We show examples of how we can use this feature of optical potentials to manipulate the properties of the solitons formed and also how they interact with each other, focusing on a special type of optical lattice known as an optical superlattice. We also discuss different ways of creating gap solitons in BECs loaded into optical lattices.

      [1] L. Khaykovich et al., Science 296, 1290 (2002).
      [2] K. E. Strecker et al., Nature 417, 150 (2002).
      [3] B. Eiermann et al., Phys. Rev. Lett. 92, 230401 (2004).

  • Dr. Ken-ichi Maruno
    • Title: Study of higher-dimensional integrable mappings
    • Abstract: We study a class of integrable mappings which have bilinear forms. We propose a method to construct integrals of higher-order mappings using bilnear forms. The key of construction is conservation law of discrete bilinear forms of AKP and BKP equations. This is a joint work with Prof. Quispel.

  • Prof. Jon Nimmo
    • Title: On a nonabelian generalisation of the Hirota-Miwa equation
    • Abstract: Perhaps the most well known three dimensional, fully discrete integrable system is the Hirota-Miwa or discrete KP equation. This talk will begin with a review some properties of this system paying particular attention to the construction of a family of exact solutions in the form of casoratian determinants by means of Darboux transformations. There are a number of features of the Hirota-Miwa equation that are essential for the construction but others aspects are inessential. By relaxing the inessential ones as much as possible while keeping the essential ones, we will obtain a integrable generalisation of the system with solutions in an associative algebra $\mathcal A$, which is in general nonabelian. It will be seen that the action of Darboux transformations for the generalised system is very natural, corresponding merely to movement on a higher dimensional lattice. In the most abstract form the solutions obtained by Darboux transformations are expressed as entries in the inverse of a matrix over $\mathcal A$. When we take $\mathcal A$ to be a matrix algebra these expression lead to more familiar expressions for the solutions as ratios of multicomponents casoratian determinants.

$BBh#1#12s(B

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    • $B9V;U!'NkLZ(B $B5.M:(B $B;a!J?@8MBg!&M}!K(B
    • $B%?%$%H%k!'(B D4(1) $B7?%I%j%s%U%'%k%H!&%=%3%m%U3,AX$H%Q%s%k%t%'Bh#6J}Dx<0(B
    • $B35MW(B: D4(1)$B7?%"%U%#%s!&%j!N@-$NM3Mh$K$D$$$F$b(B $B9M;!$9$k!%(B
    • $B9V;U!'F#(B $B7rB@(B $B;a!J?@8MBg!&<+A3!K(B
    • $B%?%$%H%k!'(B D$B7?(Bp$B$N(BLax Pair$B$KM3Mh$9$k9b3,(BPainlevé$BJ}Dx<0(B
    • $B35MW(B: Painlevé VI $B$N(BLax Pair$B$H$7$F(Bso(8)$B$N$b$N$,$"$j$^$9!%(B $B$=$l$r(Bso(2n)$B$N$b$N$X$H3HD%$9$k$3$H$G!$(B n$B$,6v?t$N>l9g$O(Bcoupled Painlevé VI $B$,(B, n$B$,4q?t$N>l9g$O(Bcoupled Painlevé V $B$,F@$i$l$^$7$?!%(B

$BBh#1#02s(B

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  • $B9V;U!'2
  • $B%?%$%H%k!'(B Garnier $B7O$NFC0[E@$K$*$1$k2r$NB2$K$D$$$F(B
  • $B35MW(B: $B#2


$BBh#92s(B

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$B6e=#Bg3XH":j%-%c%s%Q%9M}3XIt#39f4[#3#3#0#49f<

  • $B9V;U!'(BDr. Raimundas Vidunas $B!J6eBg?tM}!K(B
  • $B%?%$%H%k!'(B Quadratic Transformations of the Sixth Painlevé Equation
  • $B35MW(B:

    In 1991, Kitaev showed existence of quadratic transformations between Painleve VI equations with the local monodromy differences $(1/2,a,b,1/2)$ and $(a,a,b,b)$. The aim of this talk is to present compact formulas for this transformation, developed in colloboration with Kitaev.

    Quite recently, Manin and Ramani-Grammaticos-Tamizhmani found simpler quadratic transformations between Painleve VI equations with the local monodromy differences $(0,A,B,1)$ and $(A/2,B/2,B/2,A/2+1)$. They are related to Kitaev's transformation via a pair of Okamoto transformations.


$BBh#82s(B

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  • $B9V;U!'>>K\(B $B7=;J;a!JKLBgM}!K(B
  • $B%?%$%H%k!'(B A Heun differential equation derived from the Gauss hypergeometric differential equation
  • $B35MW(B: We study a Heun differential equation derived from the Gauss hypergeometric differential equation. We show that the periods for the family of cubic curves of the Hesse normal form satisfy this differential equation for some parameters.We give a monodromy representation of this differential equation; we find parameters such that the monodromy group is isomorphic to the fundamental group of the complement of the Borromean rings.

$BBh#72s(B

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  • $B9V;U!';y6L!!M5Lj(B $B;a!J(BOhio State University$B!K(B
  • $B%?%$%H%k!'(BToda lattice, cohomology of compact Lie groups and finite Chevalley groups.

  • $B35MW!'(BI will describe a connection that exists among

    1. the number of singular points along the trajectory of Toda flow,
    2. the cohomology of a compact subgroup $K$, and
    3. the number of points of a Chevalley group $K({\mathbb F}_q)$

    related to K over a finite field Fq. The Toda lattice is defined for a real split semisimple Lie algebra $\mathfrak g$, and K is a maximal compact Lie subgroup of G associated to $\mathfrak g$. Relations are also obtained between the singularities of the Toda flow and the integral cohomology of the real flag manifold G/B with B the Borel subgroup of G (here we have G/B=K/T with a finite group T). We also compute the maximal number of singularities of the Toda flow for any real split semisimple algebra, and find that this number gives the multiplicity of the singularity at the intersection of the varieties defined by the zero set of Schur polynomials. This is a joint work with Luis Casian.

  • $B#2#1@$5*(BCOE$B!V5!G=?tM}3X$N9=C[$HE83+!W$N;Y1g$r

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    1. C.T. McMullen, Dynamics on K3 surfaces: Salem numbers and Siegel disks, J. reine angew. Math. 545 (2002),201--233.
    2. S. Cantat, Dynamique des automorphismes des surfaces K3, Acta Math. 187 (2001), 1--57.

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