We consider moduli spaces of Azumaya algebras on K3 surfaces and construct an example. In some cases we show a derived equivalence which corresponds to a derived equivalence between twisted sheaves. We prove if A and A are Morita equivalent Azumaya algebras of degree r then 2r divides c 2 (A) − c 2 (A ). In particular this implies that if A is an Azumaya algebra on a K3 surface and c 2 (A) is within 2r of its minimal bound then the moduli stack of Azumaya algebras with the same underlying gerbe, if non-empty, is a proper algebraic space.

Abstract. Let X be a K3 surface of degree 8 in P 5 with hyperplane section H. We associate to it another K3 surface M which is a double cover of P 2 ramified on a sextic curve C. In the generic case when X is smooth and a complete intersection of three quadrics, there is a natural correspondence between M and the moduli space M of rank two vector bundles on X with Chern classes c 1 = H and c 2 = 4. We build on previous work of Mukai and others, giving conditions and examples where M is fine, compact, non-empty; and birational or isomorphic to M . We also present an explicit calculation of the Fourier-Mukai transform when X contains a line and has Picard number two.

In graph pegging, we view each vertex of a graph as a hole into which a peg can be placed, with checker-like "pegging moves" allowed. Motivated by well-studied questions in graph pebbling, we introduce two pegging quantities. The pegging number (respectively, the optimal pegging number) of a graph is the minimum number of pegs such that for every (respectively, some) distribution of that many pegs on the graph, any vertex can be reached by a sequence of pegging moves. We prove several basic properties of pegging and analyze the pegging number and optimal pegging number of several classes of graphs, including paths, cycles, products with complete graphs, hypercubes, and graphs of small diameter.

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We study Lagrangian points on smooth holomorphic curves in TP 1 equipped with a natural neutral Kähler structure, and prove that they must form real curves. By virtue of the identification of TP 1 with the space L(E 3 ) of oriented affine lines in Euclidean 3-space E 3 , these Lagrangian curves give rise to ruled surfaces in E 3 , which we prove have zero Gauss curvature.Each ruled surface is shown to be the tangent lines to a curve in E 3 , called the edge of regression of the ruled surface. We give an alternative characterization of these curves as the points in E 3 where the number of oriented lines in the complex curve Σ that pass through the point is less than the degree of Σ. We then apply these results to the spectral curves of certain monopoles and construct the ruled surfaces and edges of regression generated by the Lagrangian curves.