coherent sheaves on rational curves with simple double points and transversal intersections we study the derived categories of coherent sheaves on some singular projective curves and give a complete description of indecomposable objects using the technique of matrix problems

noncommutative extensions of elliptic integrable euler arnold tops and painleve vi equation in this paper we suggest generalizations of elliptic integrable tops to matrix valued variables our consideration is based on the r matrix description which provides lax pairs in terms of quantum and classical r matrices first we prove that for relativistic and non relativistic tops such lax pairs with spectral parameters follow from the associative yang baxter equation and its degenerations then we proceed to matrix extensions of the models and find out that some additional constraints are required for their construction we describe a matrix version of the reduced elliptic top and verify that the latter constraints are fulfilled in this case the construction of matrix extensions is naturally generalized to the monodromy preserving equation in this way we get matrix extensions of the painleve vi equation and its multidimensional analogues written in the form of non autonomous elliptic tops finally it is mentioned that the matrix valued variables can be replaced by elements of noncommutative associative algebra at the end of the paper we also describe special elliptic gaudin models which can be considered as matrix extensions of the reduced elliptic top

quantization and dynamisation of trace poisson brackets the quantization problem for the trace bracket algebra derived from double poisson brackets is discussed we obtain a generalization of the boundary ybe or so called abcd algebra for the quantization of quadratic trace brackets a dynamical deformation is proposed on the lines of gervais neveu felder dynamical quantum algebras

indefinite theta series of signature 1 1 from the point of view of homological mirror symmetry abstract we apply the homological mirror symmetry for elliptic curves to the study of indefinite theta series we prove that every such series corresponding to a quadratic form of signature 1 1 can be expressed in terms of theta series associated with split quadratic forms and the usual theta series we also show that indefinite theta series corresponding to univalued massey products between line bundles on elliptic curve are modular

associative yang baxter equation and fukaya categories of square tiled surfaces abstract we show that all strongly non degenerate trigonometric solutions of the associative yang baxter equation aybe can be obtained from triple massey products in the fukaya categories of square tiled surfaces along the way we give a classification result for cyclic a algebra structures on a certain frobenius algebra associated with a pair of 1 spherical objects in terms of the equivalence classes of the corresponding solutions of the aybe as an application combining our results with homological mirror symmetry for punctured tori cf 17 we prove that any two simple vector bundles on a cycle of projective lines are related by a sequence of 1 spherical twists and their inverses

simple vector bundles on a nodal weierstrass cubic and quasi trigonometric solutions of the classical yang baxter equation in this paper we study the combinatorics of quasi trigonometric solutions of the classical yang baxter equation arising from simple vector bundles on a nodal weierstrab cubic

hypersurfaces quasi invariant by codimension one foliations we present a variant of the classical darboux jouanolou theorem our main result provides a characterization of foliations which are pull backs of foliations on surfaces by rational maps as an application we provide a structure theorem for foliations on 3 folds admitting an infinite number of extremal rays

tilting on non commutative rational projective curves in this article we introduce a new class of non commutative projective curves and show that in certain cases the derived category of coherent sheaves on them has a tilting complex in particular we prove that the right bounded derived category of coherent sheaves on a reduced rational projective curve with only nodes and cusps as singularities can be fully faithfully embedded into the right bounded derived category of the finite dimensional representations of a certain finite dimensional algebra of global dimension two as an application of our approach we show that the dimension of the bounded derived category of coherent sheaves on a rational projective curve with only nodal or cuspidal singularities is at most two in the case of the kodaira cycles of projective lines the corresponding tilted algebras belong to a well known class of gentle algebras we work out in details the tilting equivalence in the case of the weierstrass nodal curve zy2 x3 x2z

birkhoff type decompositions and the baker campbell hausdorff recursion we describe a unification of several apparently unrelated factorizations arising from quantum field theory vertex operator algebras combinatorics and numerical methods in differential equations the unification is given by a birkhoff type decomposition that was obtained from the baker campbell hausdorff formula in our study of the hopf algebra approach of connes and kreimer to renormalization in perturbative quantum field theory there we showed that the birkhoff decomposition of connes and kreimer can be obtained from a certain baker campbell hausdorff recursion formula in the presence of a rota baxter operator we will explain how the same decomposition generalizes the factorization of formal exponentials and uniformization for lie algebras that arose in vertex operator algebra and conformal field theory and the even odd decomposition of combinatorial hopf algebra characters as well as the lie algebra polar decomposition as used in the context of the approximation of matrix exponentials in ordinary differential equations

odd supersymmetric kronecker elliptic function and yang baxter equations we introduce an odd supersymmetric version of the kronecker elliptic function it satisfies the genus one fay identity and supersymmetric version of the heat equation as an application we construct odd supersymmetric extensions of the elliptic r matrices which satisfy the classical and the associative yang baxter equations

on codimension 1 del pezzo foliations on varieties with mild singularities in this paper we extend to the singular setting the theory of fano foliations developed in our previous paper araujo and druel adv math 238 70 118 2013 a mathbb q fano foliation on a complex projective variety x is a foliation fancyscript f subsetneq t x whose anti canonical class k fancyscript f is an ample mathbb q cartier divisor in the spirit of kobayashi ochiai theorem we prove that under some conditions the index i fancyscript f of a mathbb q fano foliation is bounded by the rank r of fancyscript f and classify the cases in which i fancyscript f r next we consider mathbb q fano foliations fancyscript f for which i fancyscript f r 1 these are called del pezzo foliations we classify codimension 1 del pezzo foliations on mildly singular varieties

elliptic functions and equations of modular curves let p ge 5 be a prime we show that the space of weight one eisenstein series defines an embedding into mathbb p p 3 2 of the modular curve x 1 p for the congruence group gamma 1 p that is scheme theoretically cut out by explicit quadratic equations

braid group actions on derived categories of coherent sheaves this paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety x the motivation for this is m kontsevich s homological mirror conjecture together with the occurrence of certain braid group actions in symplectic geometry one of the main results is that when dim x geq 2 our braid group actions are always faithful we describe conjectural mirror symmetries between smoothings and resolutions of singularities which lead us to find examples of braid group actions arising from crepant resolutions of various singularities relations with the mckay correspondence and with exceptional sheaves on fano manifolds are given moreover the case of an elliptic curve is worked out in some detail

classical yang baxter equation and the a constraint abstract we show that elliptic solutions of classical yang baxter equation cybe can be obtained from triple massey products on an elliptic curve we introduce the associative version of this equation which has two spectral parameters and construct its elliptic solutions we also study some degenerations of these solutions

the inhomogeneous toda lattice its hierarchy and darboux backlund transformations the authors show how one can construct hierarchies of nonlinear differential difference equations with n dependent coefficients among these equations they present explicitly a set of inhomogeneous toda lattice equations which are associated with a discrete schrodinger spectral problem whose potentials diverge asymptotically then they derive a new darboux transformation which allows them to get bounded solutions for the equations presented before and apply it in a specially simple case when the solution turns out to be expressed in terms of hermite polynomials