$B:G=*99?7F|(B: 2017/06/23 13:02 $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

### $BBh#6#82s!'(B $B6e=#Bg3X0KET%-%c%s%Q%9%&%'%9%H#19f4[(B 5$B3,(B C-503$B9f<<(B)$B!#0#1#7G/#77n#2#0F|!JLZ!K#1#6!'#3#0!A#1#7!'#3#0(B • $B9V1i
• $B%?%$%H%k!'(B Yang-Baxter equation, elliptic hypergeometric integrals, and ABS equations.
• $B35MW(B: The Yang-Baxter equation is a key equation for integrability of two-dimensional models of statistical mechanics. Particularly, for some lattice models, the Yang-Baxter equation takes a special form known as the "star-triangle relation". The most general known forms of the Yang-Baxter equation for lattice models were recently found, that are expressed in terms of the elliptic gamma function, and are equivalent to transformation formulas of elliptic hypergeometric integrals. This discovery has lead to new elliptic hypergeometric "sum/integral" transformation formulas, which involve a mixture of complex and integer valued variables, and contain the well known (e.g. A_n, BC_n) integral transformation formulas as special cases. Furthermore, the quasi-classical asymptotics of the aforementioned Yang-Baxter equations, are directly associated to discrete integrable equations in the classification of Adler, Bobenko, and Suris (ABS). This talk will give an overview of these results, based on some of the recent works of the speaker. ### $BBh#6#72s!'(B

$B6e=#Bg3X0KET%-%c%s%Q%9%&%'%9%H#19f4[(B 5$B3,(B C-503$B9f<<(B)$B!#0#1#7G/#67n#2#9F|!JLZ!K#1#6!'#3#0!A#1#7!'#3#0(B

• $B9V1i>1:K>(B ($BJ!2,Bg3X(B)
• $B%?%$%H%k!'(B $BN%;6Hs8GM-%"%U%#%s5eLL$NI=8=8x<0(B
• $B35MW(B: $B%"%U%#%s5eLL$O$U$D$&$N5eLL$N%"%U%#%sHyJ,4v2?E*$JN;wJ*$H$7$FDj5A$5$l$k6JLL$G!"2D@QJ,7OE*$J4QE@$+$i$O%D%#%D%'%$%+J}Dx<0$d%j%&%t%#%kJ}Dx<0$,BP1~$7$^$9!#$3$N$&$A%j%&%t%#%kJ}Dx<0$K$h$C$F5-=R$5$l$k%?%$%W$N%"%U%#%s5eLL$rHs8GM-%"%U%#%s5eLL$H$^$9!#Hs8GM-%"%U%#%s5eLL$r9=@.$9$kJ}K!$O$$/D+CNilF$$$^$9$,!"K\9V1i$G$O!V$U$?$D$NJ?LL6J@~$rMQ$$k9=@.J}K!!WH!V6u4V6J@~NJQ7AKhk9=@.J}K!!WKCmL\7!"=l>lN>l9gKD$$$FN%;62=$7$?8x<0$r>R2p$7$^$9!#$3$l$O>.NS??J?;a!JKL3$F;Bg3X!K$H$N6&F18&5f$G$9!#(B

### $BBh#6#62s!'(B $B6e=#Bg3X0KET%-%c%s%Q%9%&%'%9%H#19f4[(B 5$B3,(B C-501$B9f<<(B)$B!#0#1#7G/#67n#2#2F|!JLZ!K#1#6!'#0#0!A#1#7!'#3#0(B • $B9V1i
• $B%?%$%H%k!'(B Integrable discrete models for one-dimensional soil water infiltration
• $B35MW(B: I will describe some integrable discrete models for one-dimensional soil water infiltration, developed through collaborative research at the IMI Australia Branch. The discrete models are based on the continuum model by Broadbridge and White, which takes the form of nonlinear convection-diffusion equation with a nonlinear flux boundary condition at the surface. It is transformed to the Burgers equation with a time-dependent flux term by the hodograph transformation. We construct discrete models parallel to the continuum model and preserving the underlying integrability. These take the form of self-adaptive moving mesh schemes. The discretizations are based on linearizability of the Burgers equation to the linear diffusion equation, however the naive Euler discretization that is often used in the theory of discrete integrable systems does not necessarily produce a good numerical scheme. Taking desirable properties of a numerical scheme into account, we propose an alternative discrete model with reasonable stability and accuracy. $B:#2s$N%;%_%J!<$O(B$B8=>]?tM}%;%_%J!<(B$B$H9gF1$G9T$o$l$^$9!%(B

### $BBh#6#52s!'(B $B6e=#Bg3X0KET%-%c%s%Q%9%&%'%9%H#19f4[(B 7$B3,(B C-716$B9f<<(B)$B!#0#1#7G/#17n(B18$BF|!J?e!K#1#6!'#3#0!A#1#7!'#3#0(B

• $B9V1i • $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.

$B:#2s$N%;%_%J!<$O(B$B8=>]?tM}%;%_%J!<(B$B$H9gF1$G9T$o$l$^$9!%(B ### $BBh#6#42s!'(B

$B6e=#Bg3X0KET%-%c%s%Q%9%&%'%9%H#19f4[(B $B#43,(B IMI$B%+%s%U%!%l%s%9%k!<%(B(D-414$B9f<<(B)$B!#0#1#6G/#1#07n#2#1F|!J6b!K#1#5!'#0#0!A#1#6!'#0#0(B • $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. ### $BBh#6#32s!'(B

$B6e=#Bg3X0KET%-%c%s%Q%9%&%'%9%H#19f4[(B $B#43,(B $B%*!<%G%#%H%j%"%!#0#1#6G/#97n#9F|!J6b!K#1#6!'#0#0!A#1#7!'#0#0(B • $B9V1i
• $B%?%$%H%k!'(B Raney distribution and random matrix theory
• $B35MW(B: The Raney numbers are a generalisation of the Fuss-Catalan numbers, which occur in ballot type problems. Recently they have been shown to occur in random matrix theory as the moments of eigenvalue probability densities. Some themes resulting from this interpretation will be developed. ### $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.

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• $B%?%$%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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