University of Kansas Algebra Seminar
All seminar talks will take place from 2:30pm to 3:30pm in Snow Hall 306
Name: H. Dao, University of Kansas
Title: On weak lifting of modules
Abstract :
Let (S, m) be a local ring and f an element in m. Let R=S/(f). The lifting question asks whether given a finite R-module M, one can find an S-module N such that M=N/fN and f is a nonzerodivisor on N. The point is that M would inherit nice homological properties of N if such a lifting exists. In this talk we will discuss various questions and some answers on lifting and a weaker version of the above question: whether M is a direct summand of a liftable module.
Name: S. Ramanan, Chennai Mathematical Institute
Title: Cohomology of Lie groups
Abstract :
Heinz Hopf studied the structure of the cohomology of a Lie group treated as a topological space. The result was a surprisingly simple and elegant description of this graded algebra in terms of l natural numbers m1, ... , ml called exponents of the Lie group, where l is the rank of the compact part of the Lie group. Later these exponents were explicitly determined for all simple goups. Kostant gave an interpretation of these exponents in terms of a special homomorphism of SL(2) into the Lie group. There has been a renewed interest in this circle of ideas, thanks to the work of Hitchin and the so-called `geometric Langlands programme'. I will give an elementary and non-technical account of the classical theory and indicate how the new ideas may throw light on fundamental questions regarding the classification of Lie groups, etc.
Name: K. Hanumanthu, University of Kansas
Title: Toroidalization of locally toroidal morphisms
Abstract :
The toroidalization conjecture of D. Abramovich, K. Karu, K. Matsuki, and J. Wlodarczyk asks whether any given morphism of nonsingular varieties over an algebraically closed field of characteristic zero can be modified into a "toroidal morphism". In these talks, we will try to understand the origins and the significance of this conjecture. Apart from discussing its status, a related local notion will be defined along with a new question and some answers.
Name: K. Hanumanthu, University of Kansas
Title: Toroidalization of locally toroidal morphisms, continued
Name: T. Puthenpurakal, IIT Bombay
Title: Properties of Koszul homology modules
Abstract :
We investigate various module-theoretic properties of Koszul homology under mild conditions. These include their depth, S2-property and their Bass numbers This is joint work with Uwe Nagel
Name: T. Marley, University of Nebraska
Title: Coherent Gorenstein rings
Abstract :
The concept of Gorenstein dimension, defined by Auslander and Bridger for finitely generated modules over a Noetherian ring, is studied in the context of finitely presented modules over a coherent ring. A generalization of the Auslander-Bridger formula is established and is used as a cornerstone in the development of a theory of coherent Gorenstein rings.
Name: T. Dinh, University of Utah
Title: Growth of primary decomposition of Frobenius powers
Abstract :
The linear growth property of primary decompositions of Frobenius powers has strong connection to the localization problem in tight closure theory. The localization problem has recently been settled in the negative, but the linear growth question is still open. I will introduce the linear growth problem and discuss some recent results.
Name: H. Long, University of Kansas
Title: Almost Cohen Macaulay modules, almost regular sequences and the monomial conjecture: a survey
Abstract :
In these talks we will give a survey of some new developments in the study of the monomial conjecture. The talk will be based on recent works by Roberts and others on the existence of almost Cohen-Macaulay modules and its consequences.
Name: J. Validashti, University of Kansas
Title: Numerical Criteria For Integral Dependence.
Let R be a Noetherian local ring and A ⊆ B standard graded Noetherian R-algebras. We define a few notions of multiplicity for the pair A ⊆ B and we describe numerical criteria for integrality of the extension A ⊆ B, especially when A and B are arising from Rees algebras of a pair of modules.
Name: N. Mohan Kumar, Washington University
Title: Reducedness of generalized quadrics
An ubiquitous equation that arises in the study of polynomials is that of a quadric. For definiteness, let fi,gi, 1 ≤ i ≤ n, be 2n homogeneous polynomials in 2n variables such that these polynomials have no non-trivial common zeroes and let Q=Σ figi, also homogeneous, called the generalized quadric. We will show that if n ≥ 2, over the field of complex numbers, such a quadric is necessarily reduced-that is they have no multiple factors.
Name: H. Long, University of Kansas
Title: Almost Cohen Macaulay modules, almost regular sequences and the monomial conjecture: a survey, II
Name: S. Sane, Tata Institute of Fundamental Research
Title: TBA
Name: D. Katz, University of Kansas
Title: Multiplicities and Rees valuations
Abstract :
Let (R,m) be a local, Noetherain ring. In these talks we will show that a number of standard relations between multiplicities and Rees valuations of m-primary ideals carry over to Rees valuations and more general multiplicities for ideals that are not necesarily m-primary. In particular, we show that the j-multiplicity of an ideal with maximal analytic spread is determined by the Rees valuations of the ideal centered on m.
Name: D. Katz, University of Kansas
Title: Multiplicities and Rees valuations, II
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