Abstract
Categories of matrix factorisations have become centrally important in topics ranging from commutative algebra and algebraic geometry, where they incarnate as singularity categories, to mathematical physics, where they emerge from Landau-Ginzburg models as mirror partners in Kontsevich's Homological Mirror Symmetry. The present project is devoted to the study of the deformation theory of these objects from different angles: geometrical, algebraic, and categorical. For this, we will make use of various cohomology theories inspired upon the classical Hochschild cohomology (HH) of algebras, in particular singular HH (as studied by Keller, Wang among others) and HH of the second kind (as studied by Positselski, Holstein among others). These we will in turn relate to the recent theory of categorical extensions initiated by Lehmann and Lowen. Throughout, we will focus on cases that can be computed using representation theory of quivers. Finally, we will build on our findings to establish an entirely novel take on the generalised Deligne conjecture from the deformation perspective.
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