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We use them to understand degenerations of vector bundles on elliptic curves, whereby we allow the curves to pick up singularities. As an application, the construction of solutions of the classical Yang-Baxter equation and the study of degenerations of them will be discussed. This talk reports on joint work with Igor Burban. Composition operators over bounded symmetric domains Michael Mackey UC Dublin We look at topological structure of the composition operators of bounded holomorphic functions on symmetric domains.

Embedding and compact embedding for weighted and abstract Sobolev spaces Seng-Kee Chua. Degree-one, monotone self-maps of Pontryagin surfaces are near-homeomorphisms Robert J. Daverman, Thomas L. Local estimates for Hormander's operators of first kind with analytic Gevrey coefficients and application to the regularity of their Gevrey vectors Makhlouf Derridj. Strongly algebraic realization of dihedral group actions Karl Heinz Dovermann. On commuting billiards in higher-dimensional spaces of constant curvature Alexey Glutsyuk.

The topological biquandle of a knot Eva Horvat. Symplectic and odd orthogonal Pfaffian formulas for algebraic cobordism Thomas Hudson, Tomoo Matsumura. Boundary regularity for asymptotically hyperbolic metrics with smooth Weyl curvature Xiaoshang Jin.

Sums of algebraic trace functions twisted by arithmetic functions Maxim Korolev, Igor Shparlinski. Signature invariants related to the unknotting number Charles Livingston.

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Double graph complex and characteristic classes of fibrations Takahiro Matsuyuki. Gluing Bartnik extensions, continuity of the Bartnik mass, and the equivalence of definitions Stephen McCormick. A generalization of Maloo's theorem on freeness of derivation modules Cleto B. Miranda-Neto, Thyago S. Restricted sum formula for finite and symmetric multiple zeta values Hideki Murahara, Shingo Saito. Fundamental domains and presentations for the Deligne-Mostow lattices with 2-fold symmetry Irene Pasquinelli. We achieve this by generalising the construction of Theorem 4.

Throughout this section we work either in the category of locallyNoetherian algebraic schemes over an algebraically closed field k or in the categoryof complex analytic spaces. The relative residue map. Assume additionally that p has relative dimension one and X itself is smooth. We shall explain our construction in the case of algebraicschemes, whereas its generalisation on the case of complex analytic spaces isstraightforward.

Definition 5. It is easy to see that the morphism res D is C-linear, surjective and does not dependon the choice of a generator of the ideal I. Proposition 5. On the sheaf of relative differential forms of a Gorenstein fibration. Remark 5. From what was said above it follows:Corollary 5. In what follows the morphism cl S will be called the class map. For a Gorenstein projective variety X of dimension n let M X denote the sheaf ofmeromorphic functions on X.

The relationship between this class map and theclass map constructed above will be discussed elsewhere. The following proposition can be shown on the lines of [5, Section II. The reason to introduce the map cl S is explained by the following proposition. In particular, im cl t is a subsheaf of thesheaf of meromorphic differential one-forms on X t regular at smooth pointsof X t. Propositions 5. Geometric triple Massey products.

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The main result of this section is the following theorem. Theorem 5. Lemma 5. Proof of the lemma. By the same argument as in Lemma 5. Summing everything up we obtainthe compatibility of res V,Wh 1with base change, i.

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This finishes the proof ofthe theorem. From Theorem 5. Corollary 5. In the notation of Theorem 5. Geometric associative r—matrixThe main goal of this section is to define the so-called geometric associative r-matrix attached to a genus one fibration. We start with the following geometricdata.

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Let us fix some notation. Definition 6.

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Let divisor D i denote the image of h i. Lemma 6. The constructed section r has the following base-change property. Proposition 6.


This proposition is an immediate consequence of Corollary 5. The following theorem is the main result of this article. The arguments for the remaining two summands of the left-hand side of equation 13 are analogous. By the base-change property of r see Proposition 6. The unitarity of r can be shown in a similarway. Different trivialisations of the universal bundle P lead to equivalentassociative r—matrices in the sense of Definition 2. Proposition is proved. Corollary 6. The function r t v 1 , v 2 ; y 1 , y 2 depends analytically on the parameter t and differentchoices of trivialisations of P n, d lead to equivalent solutions.

The construction of the geometric associative r—matrix can be carriedout in the category of algebraic schemes over C. This result follows from Theorem 7. The following proposition is crucial. Mor M, M. Theorem is proved. Remark 6. In particular, we do notknow whether this result generalises to the relative case.

Vector bundles on degenerations of elliptic curves and Yang-Baxter equations

Moreover, r isholomorphic with respect to the parameters g 1 and g 2. It is natural to conjecture that for any pair of coprime integers n, d the geometric r—matrix corresponding to a cuspidal cubic curve is always of rationaltype. Elliptic solutions of the associative Yang-Baxter equationIn this section we are going to compute the solution of the associative Yang-Baxterequation and the corresponding classical r—matrix, obtained from the universal familyof stable vector bundles of rank two and degree one on a smooth elliptic curve.

In [40, Section 2] Polishchuk computed the corresponding triple Massey products. It is very instructive, however, to carry out a direct computation of the geometrictriple Massey products, independent of homological mirror symmetry for an ellipticcurve.

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In order to carry out the necessary calculations we recall some standard resultsabout holomorphic vector bundles on one-dimensional complex tori, a descriptionof morphisms between them in terms of theta-functions etc. Vector bundles on a one-dimensional complex torus. We start withsome classical results about vector bundles on smooth elliptic curves.

Theorem 7.

Dr. Igor Burban

Let E be a smooth elliptic curve over Cand V a vector bundle on E. In the complex-analytic case one can give an explicit description of the stableholomorphic vector bundles on a one-dimensional complex torus. A very convenient way to carry out calculations with vector bundles on complextori is provided by the theory of automorphy factors, see [34] or [37, Section I. Line bundles of degree zero can be given by constant automorphy factors. They follow by a direct calculation. Corollary 7. Rules to calculate the evaluation and the residue maps. Remark 7. From Lemma 7. Proposition 7. By Corollary 7.