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Andrea Tobia Ricolfi

Senior assistant professor (fixed-term)

Department of Mathematics

Academic discipline: MAT/03 Geometry


Keywords: Moduli of sheaves Donaldson-Thomas invariants Virtual classes, localisation Semiorthogonal decompositions Grothendieck rings of varieties

  1. Refined invariants in moduli theory. Moduli spaces of stable sheaves on complex algebraic 3-folds Y have a rich geometric structure which can be used to define enumerative invariants (numbers) that virtually count the number of points in the moduli space; in some cases (when Y is Calabi-Yau), these numbers can be refined to more sophisticated invariants, e.g. in the K-theoretic or motivic setting. These are called Donaldson-Thomas invariants. The computation of these refined invariants can be performed in various ways, depending on the nature of the refinement: for instance one might use virtual torus localisation for the K-theoretic invariants, Hall algebras and the theory of quiver representations for the motivic invariants.
  2. Geometry of Hilbert and Quot schemes and their virtual structures, such as existence of virtual fundamental classes, but also their very structure as algebraic schemes.
  3. Semiorthogonal decompositions. A semiorthogonal decomposition of a triangulated category T allows one to "decompose" T in smaller pieces, hopefully easier to study. The deformation theory of semiorthogonal decompositions has still many interesting aspects to be explored.
  4. Rings of motives, such as the Grothendieck ring of varieties, are interesting and mysterious. It is sometimes possible to compute the generating function of the motives of a given family moduli spaces and observe its behaviour, good or bad. It is a challenge to find a family of moduli spaces for which such generating function has a nice intrinsic expression.

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