Coordinators
J. Elias, J. M. Giral, R. M. Miró-Roig, and S. Zarzuela
SPEAKERS
L. Avramov (Purdue University)
Title: Structure of free resolutions
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Constructions of resolutions: differential graded algebras, I, II. Yoneda products and the homotopy Lie algebra. Poincaré series and growth of Betti numbers. Resolutions with polynomial growth. |
G. Valla (Università di Genova)
Title: Problems and results on Hilbert functions of graded algebras (G-algebras)
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Generalities on Hilbert functions of standard graded Algebras (G-algebras). The H.F. of reduced or Cohen-Macaulay G-algebras. The perfect codimension two and Gorenstein codimension three case. The H.F. of a set of points in the projective space. The unimodality conjecture. The conjecture on the H.F. of a generic G-algebra. The Eisenbud-Green-Harris conjecture. H.F. of fat points and the Waring problem for homogeneous polynomials. H.F. of Cohen-Macaulay local rings. |
P. Schenzel (Universität Hälle)
Title: The use of local cohomology in algebra and geometry
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A short quide to duality: A few basics on homological algebra. The Cech complex. Another approach to local duality. Quasi-dualizing complexes. Vanishing results. Cohomological annihilators. ideal transforms and asymptotic primes: Chevalley’s theorem on ideal topologies. Grothendieck’s finiteness result. The behaviour of asymptotic prime ideals. Stability of asymptotic primes related to derived functors. Descent from the form ring. Local cohomology and free resolutions: “Tor” and local cohomology. Koszul homology and Green’s modules. An approach to sysygies. Bounds on Betti numbers. A remark on Serre’s intersection theory. |
M. Green (University of California, Los Angeles)
Title: Some interactions between commutative algebra and algebraic geometry
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Generic initial ideals. Macaulay’s bound, Gotzmann’s regularity and Persistence Theorems, hyperplane restriction, special cases. Gins in the exterior algebra. Analogue of Macaulay’s bound, Gotzmann Persistence. Relation to simplicial complexes. Points in $P^2$, numerical character of Gruson-Peskine, curves in $P^3$, Strano’s method. Lexicographic gins. Analogue of Strano’s result. Geomettric interpretation of lexicographic gins. |
C. Huneke (Purdue University)
Title: The theory of tight closure
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General introduction to tight closure and examples. $F$-rationality and Cohen-Macaulay properties. The Briancon-Skoda theorem and rational singularities $F$-rationality implies rationality and $F$-rational Gorenstein rings. Kodaira vanishing and tight closure. |
W. Vasconcelos (Rutgers University)
Title: Rees algebras, Noether normalizations and complexity theory
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The degrees of a graded module with applications to Tees algebras, Noether normalization and complexity theory: degrees of complexity of a graded module. Index of nilpotency. Qualitative aspects of Noether normalization. Homological degrees of a module. Complexity bounds in local rings. |
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