A famous theorem of D. Orlov describes the derived bounded category of coherent sheaves on projective hypersurfaces in terms of an algebraic construction called graded matrix factorizations. In this article, I implement a proposal of E. Segal to prove Orlov's theorem in the Calabi-Yau setting using a globalization of the category of graded matrix factorizations (graded D-branes). Let X ⊂ P be a projective hypersurface. Segal has already established an equivalence between Orlov's category of graded matrix factorizations and the category of graded D-branes on the canonical bundle K P to P. To complete the picture, I give an equivalence between the homotopy category of graded D-branes on K P and D b coh(X). This can be achieved directly, as well as by deforming K P to the normal bundle of X ⊂ K P and invoking a global version of Knörrer periodicity. We also discuss an equivalence between graded D-branes on a general smooth quasiprojective variety and on the formal neighborhood of the singular locus of the zero fiber of the potential.

We present the first (polynomial-time) algorithm for reducing a given deterministic finite state automaton (DFA) into a hyperminimized DFA, which may have fewer states than the classically minimized DFA. The price we pay is that the language recognized by the new machine can differ from the original on a finite number of inputs. These hyper-minimized automata are optimal, in the sense that every DFA with fewer states must disagree on infinitely many inputs. With small modifications, the construction works also for finite state transducers producing outputs. Within a class of finitely differing languages, the hyper-minimized automaton is not necessarily unique. There may exist several non-isomorphic machines using the minimum number of states, each accepting a separate language finitely-different from the original one. We will show that there are large structural similarities among all these smallest automata.

We study autoequivalences of the derived category of coherent sheaves of a variety arising from a variation of GIT quotient. We show that these autoequivalences are spherical twists, and describe how they result from mutations of semiorthogonal decompositions. Beyond the GIT setting, we show that all spherical twist autoequivalences of a dg-category can be obtained from mutation in this manner.Motivated by a prediction from mirror symmetry, we refine the recent notion of "grade restriction rules" in equivariant derived categories. We produce additional derived autoequivalences of a GIT quotient and propose an interpretation in terms of monodromy of the quantum connection. We generalize this observation by proving a criterion under which a spherical twist autoequivalence factors into a composition of other spherical twists.

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Let I be a finite set and C{I} be the algebra of functions on I. For a finite dimensional C algebra A with C{I} ⊂ A we show that certain moduli spaces of finite dimsional modules are isomorphic to certain Grassmannian (quot-type) varieties. There is a special case of interest in representation theory. Lusztig defined two varieties related to a quiver and gave a bijection between their C-points, [4, Theorem 2.20]. Savage and Tingley raised the question [7, Remark 4.5] of whether these varieties are isomorphic as algebraic varieties. This question has been open since Lusztig's original work. It follows from the result of this note that the two varieties are indeed isomorphic.Date: November 2, 2018.