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This volume is the result of two international workshops; Infinite Analysis 11 - Frontier of Integrability - held at University of Tokyo, Japan in July 25th to 29th, 2011, and Symmetries, Integrable Systems and Representations held at Université Claude Bernard Lyon 1, France in December 13th to 16th, 2011.

Included are research articles based on the talks presented at the workshops, latest results obtained thereafter, and some review articles. The subjects discussed range across diverse areas such as algebraic geometry, combinatorics, differential equations, integrable systems, representation theory, solvable lattice models and special functions.

Through these topics, the reader will find some recent developments in the field of mathematical physics and their interactions with several other domains.

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A presentation of the deformed W1+1 algebra.- Generating series of the Poincaré polynomials of quasihomogeneous Hilbert schemes.- PBW filtration over Z and compatible bases for VZ(_) in type An and Cn.- On the subgeneric restricted blocks of affine category O at the critical level.- Slavnov determinants, Yang-Mills structure constants, and discrete KP.- Monodromy of partial KZ functors for rational Cherednik algebras.- Category of finite dimensional modules over an orthosymplectic Lie superalgebra: small rank examples.- Monoidal categorifications of cluster algebras of type A and D.- A classification of roots of symmetric Kac-Moody root systems and its application.- Fermions acting on quasi-local operators in the XXZ model.- The Romance of the Ising Model.- A(1) n -Geometric Crystal corresponding to Dynkin index i = 2 and its ultra-discretization.- A Z3-orbifold theory of lattice vertex operator algebra and Z3-orbifold constructions.- Words, automata and Lie theory for tilings.- Toward Berenstein-Zelevinsky data in affine type A, part III: Proof of the connectedness.- Quiver varieties and tensor products, II.- Derivatives of Schur, Tau and Sigma Functions on Abel-Jacobi Images.- Padé interpolation for elliptic Painlevé equation.- Non-commutative harmonic oscillators.- The inversion formula of polylogarithms and the Riemann-Hilbert problem.- Some remarks on the Quantum Hall Effect.- Ordinary differential equations on rational elliptic surfaces.- On the spectral gap of the Kac walk and other binary collision processes on d-dimensional lattice.- A restricted sum formula for a q-analogue of multiple zeta values.- A trinity of the Borcherds _-function.- Sum rule for the eight-vertex model on its combinatorial line.

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