08:30 to 10:00 Registration 10:00 to 11:00 S Sergeev ([Canberra])Classical and quantum three-dimensional lattice field theories We will discuss field-theoretical aspects of q-oscillator/discrete three-wave equations. The classical limit of quantum q-oscillator model is a completely integrable Hamiltonian (Euler-Lagrange-…) system. Physical regimes of classical theories are defined by reality regimes of their lattice action/energy. The real regimes can be classified as three distinct field theories with cone-type dispersion relations and one "classical" statistical mechanics. A principal difference between statistical mechanics and field theory is that in the first case a ground state provides an absolute minimum of energy functional while in the field-theoretical cases there is a class of soliton solutions of equations of motion. Dispersion relations are related to the field-theoretical solitons. We also discuss roughly the quantum field/statistical mechanics theories corresponding to these regimes. 11:00 to 11:30 Coffee 11:30 to 12:30 (Quasi-) exactly solvable Discrete' quantum mechanics This talk is based on the collaboration with Ryu Sasaki. Discrete' quantum mechanics is a quantum mechanical system whose Schr\"{o}dinger equation is a difference equation instead of differential in ordinary quantum mechanics. We present a simple recipe to construct exactly and quasi-exactly solvable Hamiltonians in one-dimensional discrete' quantum mechanics. It reproduces all the known ones whose eigenfunctions consist of the Askey scheme of hypergeometric orthogonal polynomials of a continuous or a discrete variable. An essential role is played by the sinusoidal coordinate, which generates the closure relation and the Askey-Wilson algebra together with the Hamiltonian. We also present the Crum's Theorem for discrete' quantum mechanics. 12:30 to 13:30 Lunch at Wolfson Court 14:00 to 15:00 FA Grunbaum ([UC, Berkeley])Quantum random walks and orthogonal polynomials This is joint work with M.J. Cantero, L. Moral and L. Velazquez from Zaragoza, Spain. We consider quantum random walks (QRW) on the integers, a subject that has been considered in the last few years in the framework of quantum computation. We show how the theory of CMV matrices gives a natural tool to study these processes and to give results that are analogous to those that Karlin and McGregor developed to study (classical) birth-and-death processes using orthogonal polynomials on the real line. In perfect analogy with the classical case the study of QRWs on the set of non-negative integers can be handled using scalar valued (Laurent) polynomials and a scalar valued measure on the circle. In the case of classical or quantum random walks on the integers one needs to allow for matrix valued versions of these notions. We show how our tools yield results in the well known case of the Hadamard walk, but we go beyond this translation invariant model to analyze examples that are hard to analyze using other methods. More precisely we consider QRWs on the set of non-negative integers. The analysis of these cases leads to phenomena that are absent in the case of QRWs on the integers even if one restricts oneself to a constant coin. This is illustrated here by studying recurrence properties of the walk, but the same method can be used for other purposes. 15:00 to 15:30 Tea and Posters 15:30 to 16:30 Manin matrices and quantum spin models We consider a class of matrices with noncommutative entries, first considered by Yu. I. Manin in 1988. They can be defined as noncommutative endomorphisms'' of a polynomial algebra. The main aim of the talk is twofold: the first is to show that quite a lot of properties and theorems of linear algebra (e.g., a natural definition of the determinant, the Cayley-Hamilton theorem, and so on and so forth) have a straightforward and natural counterpart in this case. The second, to show how these matrices appear in the theory of integrable quantum spin models, and present a few applications. (Joint work(s) with A. Chervov and V. Rubtsov). 16:30 to 17:30 Periodicities of T-systems and Y-systems The T and Y-systems originate in conformal field theory and Bethe ansatz studies of solvable lattice models during 80's-90's. They are difference equations of Hirota-Miwa or Toda type that possess a variety of aspects related to dilogarithm identity, Kirillov-Reshetikhin conjecture, q-character of quantum affine algebras and so forth. More recently, there has been a renewed interest in their connection to the cluster algebras of Fomin-Zelevinsky, the cluster category of Buan-Marsh-Reineke-Reiten-Todorov and the periodicity conjecture by Zamolodchikov and others. I shall give an introductory overview on these topics. [joint work with R.Inoue, O.Iyama, B.Keller, T.Nakanishi and J.Suzuki] 17:30 to 18:30 Wine Reception 18:45 to 19:30 Dinner at Wolfson Court (Residents Only)
 10:00 to 11:00 A Isaev (BLTP JINR)Discrete evolution operator for $q$-deformed isotropic top The structure of a cotangent bundle is investigated for quantum linear groups $GL_q(n)$ and $SL_q(n)$. Using a $q$-version of the Cayley-Hamilton theorem we construct an extension of the algebra of differential operators on $SL_q(n)$ by spectral values of the matrix of right invariant vector fields. We consider two applications for the spectral extension. First, we describe the extended Heisenberg double in terms of a new set of generators --- the Weyl partners of the spectral variables. Calculating defining relations in terms of these generators allows us to derive $SL_q(n)$ type dynamical R-matrices in a surprisingly simple way. Second, we calculate discrete evolution operator for the model of $q$-deformed isotropic top introduced by A.Alekseev and L.Faddeev. The evolution operator is not uniquely defined and we present two possible expressions for it. The first one is a Riemann theta function in the spectral variables. The second one is an almost free motion evolution operator in terms of logarithms of the spectral variables. Relation between the two operators is given by a modular functional equation for Riemann theta function. 11:00 to 11:30 Coffee 11:30 to 12:30 Generalized Macdonald-Ruijsenaars systems and Double Affine Hecke Algebras The Double Affine Hecke Algebra (DAHA) is defined by a root system, its basis and by some parameters. The Macdonald-Ruijsenaars systems are known to be obtained from the polynomial representations of DAHAs. We consider submodules in the polynomial representations of DAHAs consisting of functions vanishing on special intersections of shifted mirrors. We derive the generalized Macdonald-Ruijsenaars systems by considering the Dunkl-Cherednik operators acting in the quotient-modules. In the A_n case this recovers Sergeev-Veselov systems, and the corresponding ideals were studied by Kasatani. This is a joint work with M. Feigin. 12:30 to 19:30 Lunch and Excursion 19:30 to 23:00 Conference Dinner at Emmanuel College
 09:30 to 10:00 The Organisers: Opening Remarks 10:00 to 11:00 Quantum geometry of 3-dimensional lattices We study geometric consistency relations between angles on 3-dimensional (3D) circular quadrilateral lattices -- lattices whose faces are planar quadrilaterals inscribable into a circle. We show that these relations generate canonical transformations of a remarkable ultra-local'' Poisson bracket algebra defined on discrete 2D surfaces consisting of circular quadrilaterals. Quantization of this structure leads to new solutions of the tetrahedron equation (the 3D analog of the Yang-Baxter equation). These solutions generate an infinite number of non-trivial solutions of the Yang-Baxter equation and also define integrable 3D models of statistical mechanics and quantum field theory. The latter can be thought of as describing quantum fluctuations of lattice geometry. The classical geometry of the 3D circular lattices arises as a stationary configuration giving the leading contribution to the partition function in the quasi-classical limit. 11:00 to 11:30 Coffee 11:30 to 12:30 Discrete informational models We can easily create simple computational processes, loosely based on the mathematics of physical systems. In one case we model the temporal evolution of simple discrete second order informational processes where we have imposed constraints peculiar to reversible computation. By focusing on systems that share properties with simple physical models we have found a class of such discrete computations that exhibit exact conservation laws despite the absence of continuous symmetries. In particular we have made a surprising observation about the calculation of probabilities from amplitudes that appears to yield a new insight into the nature of the quantum mechanical description of certain physical phenomena. Another simple discrete informational model, the infoton particle, clearly violates local conservation laws while, nevertheless, operationally modeling the process of General Relativity. 12:30 to 13:30 Lunch at Wolfson Court 14:00 to 15:00 Crystalline gravity Matter interacting classically with gravity in 3+1 dimensions usually gives rise to a continuum of degrees of freedom, so that, in any attempt to quantize the theory, ultraviolet divergences are nearly inevitable. We now decided to investigate matter of a form that only displays a finite number of degrees of freedom in compact sections of space-time. In finite domains, one has only exact, analytic solutions. This is achieved by limiting ourselves to straight pieces of string, surrounded by locally flat sections of space-time. We investigate the equations of motion for such forms of matter. Globally, the model is not finite, because solutions tend to generate infinite fractals. The model is not (yet) quantized, but could serve as an interesting setting for analytical approaches to classical general relativity, as well as a possible stepping stone for quantum models. Details of its properties are explained, but some problems remain unsolved, such as a complete description of all possible interactions. 15:00 to 15:30 Tea and Posters 15:30 to 16:30 SP Novikov ([Maryland])New discretization of complex analysis and completely integrable systems New Discretization of Complex Analysis was developed recently by the present author and I.Dynnikov. Our discretization has nothing to do with geometric discretization of conformal maps. We consider complex analysis as a theory of linear Cauchy-Riemann Operator. It is based on the Equilateral Triangle Lattice in the Euclidean Plane (the classical one was based on the quadrilateral lattice). This approach allows us to borrow some crucial ideas from the modern theory of Completely Integrable Systems missing in the case of quadrilateral lattice. New phenomena appear in the case of Equilateral Lattice in Hyperbolic (Lobachevski) Plane: geometric objects became ''stochastic'' requiring to use technic of Symbolic Dynamics. New difficulties appear leading to the unsolved plroblems. 18:45 to 19:30 Dinner at Wolfson Court (Residents Only)