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Yobuko defined recently the notion of a quasi-F-splitting which shares many properties with F-splittings but is satisfied by a broader class of varieties and rings. In my talk I will discuss how to generalise strong F-regularity to the quasi-setting

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There are many situations in which the derived category of coherent sheaves on a smooth projective variety can be decomposed into smaller pieces that reflect something interesting about its geometry. I will present a new unifying framework for studying these semiorthogonal decompositions using Bridgeland stability conditions. There is a partial compactification of the space of stability conditions, constructed jointly with Alekos Robotis, whose boundary points correspond to a new homological structure called a multiscale decomposition, which generalizes a semiorthogonal decomposition. From this perspective, I will formulate some conjectures about canonical flows on the space of stability conditions that imply several important conjectures on the structure of derived categories, such as the D-equivalence conjecture and Dubrovin's conjecture.

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We present an effective global generation result for direct images of pluricanonical bundles in mixed characteristic. This is a mixed characteristic analog of Ejiri's theorem in positive characteristic and the theorem of Popa and Schnell regarding their Fujita-type conjecture in characteristic zero. As an application, we also discuss a weak positivity statement for relative canonical bundles in mixed characteristic.

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Symmetric homology is a natural generalization of cyclic homology, in which symmetric groups play the role of cyclic groups. For associative algebras, the symmetric homology theory  was introduced by Z. Fiedorowicz (1991) based on his earlier work with J.-L. Loday. In this talk, after giving a survey of known results on symmetric homology, we show that, for algebras over a field of characteristic 0, this homology theory is naturally equivalent to the (one-dimensional) representation homology. Then, using known results on representation homology, we compute symmetric homology explicitly for basic algebras, such as polynomial algebras and universal enveloping algebras of (DG) Lie algebras. As an application, we prove two conjectures of Ault and Fiedorowicz (2007), including their main conjecture on topological realization of symmetric homology of polynomial algebras.

Frank Sottile: "Galois groups in Enumerative Geometry"

Abstract:  In 1870 Jordan explained how Galois theory can be applied to problems from enumerative geometry, with the group encoding intrinsic structure of the problem. Earlier Hermite showed the equivalence of Galois groups with geometric monodromy groups, and in 1979 Harris initiated the modern study of Galois groups of enumerative problems. He posited that a Galois group should be `as large as possible' in that it will be the largest group preserving internal symmetry in the geometric problem. I will describe this background and discuss some work of many to compute, study, and use Galois groups of geometric problems, including those that arise in applications of algebraic geometry.

Aldo Conca: "Two bounds on Castelnuovo-Mumford regularity"

Abstract:  I will report on bounds on the Castelnuovo-Mumford regularity for ideals with polynomial parametrization  (joint work with F.Cioffi) and for ideals associated with general subspace arrangements (joint work with M.Tsakiris).

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Tensor triangular geometry in its simplest reincarnation associates a Zariski space to a (small) tensor triangulated category momentarily pretending that such a category is just a ring. I’ll describe - a very familiar - basic construction and mention how it applies to some examples from stable homotopy theory, algebraic geometry and representation theory leaving numerous other tensor triangulated categories to the audience’s imagination. 

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The theory of rings of finite Frobenius representation type was developed by Smith and Van den Bergh, as part of an attack on the conjectured simplicity of rings of differential operators on invariant rings of characteristic zero.  We will discuss
connections with differential operators, and also several other applications that the theory has since found. 

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In this third installment of my introduction to CEI, I will present the actual definition of CEI, and discuss various computational aspects involved in the definition. I will also present known cases where CEI compute the "correct" answer -- some involving mirror symmetry for Calabi-Yau and LG models.

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The cohomology ring of a smooth projective complex variety has the strong Lefschetz property, i.e., the multiplication maps given by powers of a general linear forms all have maximal rank. For other artinian Gorenstein algebras (commutative algebras with Poincaré duality) it is well known that this needs not be true, not even for multiplication by linear forms, which would be the weak Lefschetz property. However, there are lots of results and some conjectures about when we have the strong or the weak Lefschetz property. I will explain why I find the problems in this area interesting, and I will highlight some of the main results. At the end I will report on some recent results obtained in joint work with Juan Migliore, Rosa Maria Miró-Roig and Uwe Nagel.

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The Kahler, Noether and Dedekind differents are ideals that arise in the study of ramification loci. In this talk we compute these ideals explicitly for determinantal rings in the case of submaximal minors of square generic matrices using DG-algebra structures and formulas founded by Polini and Ulrich.

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Ilya Smirnov has proposed a precise relationship between the Hilbert-Kunz multiplicity of powers of an ideal and the Hilbert coefficients of its Frobenius powers. We shall discuss the known results about this question for ideals in pseudo-rational local rings, the maximal ideal of the face ring of a simplicial complex, and certain ideals in hypersurface rings. We shall also point out a possible relation between the Hilbert-Kunz multiplicity of power products of ideals and mixed multiplicities of Frobenius powers of ideals.

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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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