We describe semiuniversal deformation spaces for the noncompact surfaces Z k := Tot(O P 1 (−k)) and prove that any nontrivial deformation Z k (τ ) of Z k is affine.It is known that the moduli spaces of instantons of charge j on Z k are quasi-projective varieties of dimension 2j − k − 2. In contrast, our results imply that the moduli spaces of instantons on any nontrivial deformation Z k (τ ) are empty.

Let A = kQ/I be the path algebra of any finite quiver Q modulo any finitely generated ideal of relations I. We develop a method to give a concrete description of the deformation theory of A via the combinatorics of reduction systems and give a range of examples and applications to deformation quantization and to deformations in commutative and noncommutative algebraic geometry.

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We describe the Fukaya-Seidel category of a Landau-Ginzburg model LG (2) for the semisimple adjoint orbit of sl(2,C). We prove that this category is equivalent to a full triangulated subcategory of the category of coherent sheaves on the second Hirzebruch surface. We show that no projective variety can be mirror to LG(2), and that this remains so after compactification.

Categorification of scattering amplitudes for planar Feynman diagrams in scalar field theories with a polynomial potential is reported. Amplitudes for cubic theories are directly written down in terms of projectives of hearts of intermediate t-structures restricted to the cluster category of quiver representations, without recourse to geometry. It is shown that for theories with φ m+2 potentials those corresponding to m-cluster categories are to be used. The case of generic polynomial potentials is treated and our results suggest the existence of a generalization of higher cluster categories which we call pseudo-periodic categories. An algorithm to obtain the projectives of hearts of intermediate t-structures for these types is presented.