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Let G be a finite group acting linearly on the polynomial ring  with invariant ring R. We assign, to a  linear representation of G, a corresponding quotient scheme over Spec R, and we show how to reconstruct the action from the quotient scheme. This works in particular in the case of a reflection group, where Spec R itself is an affine space, in contrast to the Auslander correspondence, where one has to assume that the basic action is small, i.e. contains no pseudo reflection.  These quotient schemes exhibit rich geometric features which mirror properties of the representation. In order to understand the image of this construction, we encounter module schemes (a forgotten notion of Grothendieck), module schemes up to modification and fiberflat bundles.

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I will discuss certain moduli spaces of A-infinity structures arising naturally in the study of derived categories of coherent sheaves on algebraic curves. One example is related to higher genus curves. Another example arises from genus 1 curves but has interesting noncommutative deformations.

Mahrud Sayrafi: "Splitting of vector bundles on toric varieties"

Abstract: In 1964, Horrocks proved that a vector bundle on a projective space splits as a sum of line bundles if and only if it has no intermediate cohomology. Generalizations of this criterion, under additional hypotheses, have been proven for other toric varieties, for instance by Eisenbud-Erman-Schreyer for products of projective spaces, by Schreyer for Segre-Veronese varieties, and Ottaviani for Grassmannians and quadrics. This talk is about a splitting criterion for arbitrary smooth projective toric varieties, as well as an algorithm for finding indecomposable summands of sheaves and modules in the more general setting of Mori dream spaces.

Feiyang Lin: "Finding special line bundles on special tetragonal curves"

Abstract: There is a canonical way to associate to a degree 4 cover of P^1 two vector bundles E and F, which give rise to a stratification of the Hurwitz space H_{4,g}. It is natural to ask whether the Brill-Noether theory of tetragonal curves is controlled by this data. I will describe a procedure for producing a particular line bundle on tetragonal covers in special strata, which is expected to be special in the Hurwitz-Brill-Noether sense. The main technique is the realization of an inflation of vector bundles on P^1 as a blow-up and blow-down of the associated projective bundle.

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A numerical semigroup S is a cofinite subset of the non-negative integers, containing 0 and closed under addition. They arise as value semigroups of 1-dimensional singularities, as Weierstrass semigroups of points on smooth curves, and the associated semigroup rings form a pleasantly simple family of examples of 1-dimensional domains. 

The smallest nonzero element is called the multiplicity, m(S). Kunz showed that the numerical semigroups of multiplicity m can be represented as the lattice points in a convex rational cone in QQ^(m-1), now called the Kunz cone; and that many properties of the semigroup ring are determined by the face of the Kunz cone on which the semigroup lies.

I'll describe the Kunz cone and some of the still-open problems about semigroup rings that might be studied using it.

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