 Tight Closure
I. Swanson
New Mexico State University, USA
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Date:  Jan. 1216, 2002  
 Jan. 1923, 2002  Time:  15:0018:00 
Prof. I. Swanson will conduct a short course in "Tight Closure" for about ten days and will present two lectures on "Primary Decomposition".
Abstract
The theory of tight closure was started by Hochster and Huneke in the early 1980s, and since then it has proved to be a powerful theory. It has been used to prove the BrianconSkoda theorems, the vanishing of
Tors, the Kodaira vanishing theorem, the uniform ArtinRees theorems, the classification and properties of rational and logcanonical singularities, asymptotic properties of powers of ideals, and so on.
This minicourse will start with the basics of tight closure, proceed to the early results which proved the power of the theory, and end up with the recent developments on the primary decompositions of ideals.
Tight closure is defined for modules over rings containing fields. For rings of positive prime characteristic p one uses (the repeated applications of) the Frobenius morphism, and for rings in characteristic 0 one used the reduction to characteristic p. All this will be explained in detail. An important but technical ingredient are test elements, and at least one talk will be devoted to the existence and utility of test elements. The last few talks will be devoted to the asymptotic theory of powers of ideals.
Prof. Swanson's lectures are proceeded by the following talks.
Information:
Place:  Seminar Hall, School of Mathematics, Niavaran Bldg., Niavaran Sqr.  Tel:  +98 21 2290928  Fax:  +98 21 2290648  Email:  ipmmath@ipm.ir 
Seminars of Commutative Algebra
Time: 16:3017:30
Jan. 14, 2002  The ideals of minors of pluriciculant matrices,, R. ZaareNahandi, University of Tehran
Abstract  Jan. 15, 2002  Cohomological dimension of certain algebraic varieties,, K. DivaaniAzar, School of Mathematics & AzZahra University
Abstract  Jan. 16, 2002  Generalized CohenMaculay modules,, K. Khashyarmanesh, School of Mathematics & Damghan University
Abstract  Jan. 19, 2002  Some related notions to tight closure in algebraic geometry,, H. Haghighi, K.N. Toosi University of Technology  Jan. 20, 2002  Gorenstein dimensions,, L. Khatami, School of Mathematics & University of Tehran  Jan. 21, 2002  On the minimal flat resolution of modules,, J. Asadollahi, Tarbiat Modarres University of Technology
Abstract  Jan. 22, 2002  On the number of minimal generators of modules,, T. Sharif, School of Mathematics & University of Tehran
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