$B:G=*99?7F|(B: 2013/07/22 11:06 $B6e=#2D@QJ,7O%;%_%J!<$N%Z!<%8$G$9!%(B $B9V1io$KJg=8$7$F$$^9!%2D@QJ,7OK!J[sNAgCHGb!'Hs!V2D@QJ,7O!WGb2D!K4XO"9kJi!$$I$N$h$&$JOCBj$G$b4?7^$7$^$9!%(B $B9V1i$7$?$$J}!$$^$?$O9V1iR2p$7$F$$?@1kJ}O@'Hs0J2<^G4O"Mm2<5$$!%(B $B$,$"$l$P9V1i $BO"Mm@h(B  $B3a86(B $B7r;J(B ($B6eBg(BIMI) kaji_AT_imi.kyushu-u.ac.jp

• $B@$OC?M$N0l?M$@$C$?(B$B4]Ln7r0l$5$s(B$B$O(B 2006$BG/(B8$B7n$K(B Department of Mathematics, University of Texas-Pan American $B$K0\$j$^$7$?!%(B
• $B@$OC?M$N0l?M$@$C$?4d:j9nB'$5$s$O(B2010$BG/(B4$B7n$KKL3$F;Bg3XBg3X1!M}3X8&5f1!?t3XItLg$KE>=P$5$l$^$7$?!%(B • $B@$OC?M$N0l?M$@$C$?DEED>H5W$5$s$O(B2011$BG/(B4$B7n$K0l66Bg3X$KE>=P$5$l$^$7$?!%(B • $B@$OC?M$N0l?M$@$C$?Cf1?.9'$5$s$O(B2013$BG/(B4$B7n$K%7%I%K!=P$5$l$^$7$?!%(B

### $BBh#6#22s!'(B $B6e=#Bg3X0KET%-%c%s%Q%9?tM}3X8&5f650iEo(B($B0KET?^=q4[(B) $B#33,(B $B>.9V5A<<#1!#0#1#3G/#87n#1#2F|!J7n!K#1#5!'#0#0!A#1#6!'#3#0(B • $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

$B9V1i(2013-07-19) $B6e=#Bg3X0KET%-%c%s%Q%9?tM}3X8&5f650iEo(B($B0KET?^=q4[(B) $B#33,(B $BCf%;%_%J!<<<#7!#0#1#3G/#77n#1#9F|!J6b!K#1#5!'#3#0!A#1#7!'#0#0(B • $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.

### $BBh#6#02s!'(B $B6e=#Bg3X0KET%-%c%s%Q%9?tM}3X8&5f650iEo(B($B0KET?^=q4[(B) $B#33,(B $BCf%;%_%J!<<<#7!#0#1#3G/#67n#5F|!J?e!K#1#5!'#3#0!A#1#7!'#0#0(B • $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
• $B35MW(B: I shall explain how Soibelman's theory of quantized algebra of functions Aq(g) led to a representation theoretical construction of a solution to the Zamolodchikov tetrahedron equation and the 3D analogue of the reflection equation proposed by Isaev and Kulish in 1997. If time allows, I present a theorem relating the intertwiners of representations of Aq(g) with the PBW bases of the positive part of the quantized enveloping algebra Uq(g). (Joint work with Masato Okado and Yasuhiko Yamada.) ### $BBh#5#92s!'(B

$B6e=#Bg3X0KET%-%c%s%Q%9?tM}3X8&5f650iEo(B($B0KET?^=q4[(B) $B#33,(B $BCf%;%_%J!<<<#4!#0#1#3G/#57n#2#1F|!J2P!K#1#0!'#3#0!A#1#4!'#3#0(B

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.

### $BBh#5#82s!'(B $B6e=#Bg3X0KET%-%c%s%Q%9?tM}3X8&5f650iEo(B($B0KET?^=q4[(B) $B#33,(B $BCf%;%_%J!<<<#7!#0#1#2G/#1#07n#5F|!J6b!K#1#5!'#3#0!A#1#8!'#0#0(B • $B9V1i
• $B%?%$%H%k!'(B The second Painlevé hierarchy and mKdV equation
• $B35MW(B: In this seminar, we will report on the fourth-order autonomous ordinary differential equation (*) which is compatible with the mKdV equation. The pair of (*) and mKdV equation is a partial differential equation in two variables. For this equation, we will present (1) polynomial Hamiltonian structure, (2) symmetry and holomorphy conditions. We will also discuss its phase space. ### $BBh#5#72s!'(B

$B6e=#Bg3X0KET%-%c%s%Q%9?tM}3X8&5f650iEo(B($B0KET?^=q4[(B) $B#33,(B $BCf%;%_%J!<<<#7!#0#1#2G/#57n#1#5F|!J2P!K#1#5!'#0#0!A#1#6!'#3#0(B

• $B9V1i • $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$$>l9gK%Q%s%k%t%'(BIIIB7?J}Dx<0XHJQ495lk!%(B BK\9V1iGO%k!<%W72rMQ$$$?D4OB ### $BBh#5#62s!'(B

$B6e=#Bg3X0KET%-%c%s%Q%9?tM}3X8&5f650iEo(B($B0KET?^=q4[(B) $B>.9V5A<<(B2$B!#0#1#2G/#27n#2F|!J?e!K#1#4!'#0#0!A(B

• $B9V1ipJs(B) • $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$$k3HrF'^(F!(B • B@lLg2H0J30NJ}XNF~Lg(B • BBJ1_A26a2r@OHY-5i?tA26a2r@O(B • B7A<0E*Y-5i?tN<}B+@-(B BN(B3BE@KD$$$F2r@b$9$k!%(B
• $B9V1i • $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?$$!%(B B33GO"k%/%i%9NNO3X7ONFC0[@]F0LdBjKBP7!(BBoutrouxBKhk(BPainlevéBJ}Dx<0NA26aE83+,!(B BNO3X7ON5sF0r7hDj9kNK=EMWJLr3dr2L?93Hr<(7?$$!%(B

### $BBh#5#52s!'(B $B6e=#Bg3X0KET%-%c%s%Q%9?tM}3X8&5f1!!?(BIMI $B#33,(B $BCf%;%_%J!<<<#1!#0#1#1G/#1#27n#6F|!J7n!K#1#5!'#0#0!A#1#6!'#3#0(B

• $B9V1i • $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$$F(B • B35MW(B: B%(%k%_!<%HBP>N6u4V!J(BS_N^C.S_NB!KKBP1~9k(B parabolic Kazhdan-Lusztig BB?9<0K4X7F!(B B%X%C%14D,0[Jk%Q%i%a!<%?r;}DH-K!$$=$l$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

### $BBh#5#42s!'(B $B6e=#Bg3X0KET%-%c%s%Q%9?tM}3X8&5f1!!?(BIMI$B#33,(B $BCf%;%_%J!<<<#7!#0#1#1G/#1#27n#5F|!J7n!K#1#5!'#3#0!A#1#7!'#0#0(B

• $B9V1i • $B%?%$%H%k!'(B A$B7?%I%j%s%U%'%k%H!&%=%3%m%U3,AX$N(Bq$BN%;62=$H(Bq$B%Q%s%k%t%'(BVI$BJ}Dx<0$N9b3,2=(B
• $B35MW(B: $B%I%j%s%U%'%k%H!&%=%3%m%U3,AX$O(BKP$B3,AX!J$^$?$O(BmKP$B3,AX!K$N%"%U%#%s!&%j!e5-$N7k2L$r(Bq$BN%;62=$9$k$3$H$G$"$k!#(B $B6qBNE*$K$O!"(BDS$B3,AX$N$&$A(B2$B@.J,(BmKP$B3,AX$KAjEv$9$k%/%i%9$N(Bq$BN%;62=!"(B $B5Z$S$=$NAj;w4JLs$rDj<02=$9$k!#$3$N$h$&$K$7$FF@$i$l$?N%;6J}Dx<07O$O!"(B $B?@J]!&:d0f$K$h$k(Bq$B%Q%s%k%t%'(BVI$BJ}Dx<0$N9b3,2=$H$J$C$F$*$j!"$=$N;v

### $BBh#5#32s!'(B $B6e=#Bg3X0KET%-%c%s%Q%9?tM}3X8&5f1!!?(BIMI$B#33,(B $BCf%;%_%J!<<<#3!#0#1#1G/#1#07n#6F|!JLZ!K#1#5!'#3#0!A#1#7!'#0#0(B

• $B9V1i • $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$^$?!$$3No2rd?J9TGH2r*hSBg0h2rrM?(k!%(B B3liN2rO85NJ}Dx<0N2rHN;w7F$$$k$3$H$,J,$+$k!%(B

### $BBh#5#22s!'(B $B6e=#Bg3X0KET%-%c%s%Q%9?tM}3X8&5f650iEo(B($B0KET?^=q4[(B) $BCf%;%_%J!<<<(B7$B!#0#1#1G/#67n#3F|!J6b!K#1#5!'#0#0!A#1#8!'#1#5(B 1. 15:00-16:30 • $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

$B6e=#Bg3X0KET%-%c%s%Q%9?tM}3X8&5f650iEo(B($B0KET?^=q4[(B) $BCf%;%_%J!<<<(B7$B!#0#1#1G/#57n#2#3F|!J7n!K#1#5!'#3#0!A#1#7!'#0#0(B

• $B9V1i • $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.

• $B%?%$%H%k!'(B $BGzH/$9$k2r$r$b$DHs@~7AGHF0J}Dx<0$NN%;62=(B • $B35MW(B: $B35MW!'Hs@~7A9$,N_>h4X?t$N7A$r$7$?Hs@~7AGHF0J}Dx<0$NN%;62=$rJs9p$9$k!%(B $B$3$NN%;6J}Dx<0$K$O85$NJPHyJ,J}Dx<0$N2r$NGzH/$KBP1~$7$?@-r7o$HF1MM$N7k2L$,F@$i$l$?$3$H$b<($9!%(B ### $BBh#4#92s!'(B

$B6e=#Bg3X0KET%-%c%s%Q%9?tM}3X8&5f650iEo(B($B0KET?^=q4[(B) $BCf%;%_%J!<<<(B2$B!#0#1#0G/#1#17n#1#2F|!J6b!K#1#5!'#3#0!A#1#7!'#0#0(B