University of Kansas Algebra Seminar
All seminar talks will take place from 2:30pm to 3:30pm in Snow Hall 306
Name: K. Schwede, University of Michigan
Title: Discreteness and rationality of F-jumping numbers
Abstract :
The jumping numbers of a multiplier ideal are an important set of invariants in algebraic geometry. Since the test ideal is a characteristic p > 0 analog of the multiplier ideal, it is natural to define and study the associated F-jumping numbers. Two basic questions about these invariants follow: 1. Are they always rational? (rationality) 2. Are there always no limit points? (discreteness) Several groups have partially answered these questions in the past. I will discuss joint work with M. Blickle, S. Takagi and W. Zhang, where we affirmatively answer these questions in the same generality as answers are known for multiplier ideals.
Name: K. Lee, Purdue University
Title: Hilbert schemes of points
Abstract :
The famous n! conjecture can be stated in an elementary language. In fact it asserts that the dimension of the vector space spanned by all derivatives of a certain bivariate analogue of the n x n Vandermonde determinant is equal to n!. Earlier results of Haiman and Garsia had shown that the n! conjecture implied the Macdonald positivity conjecture. Later Haiman proved the n! conjecture, and the proof is closely related to the algebraic and geometric properties of isospectral Hilbert schemes of points on the plane. I'll discuss how some of the results in the plane case can or cannot be generalized to the higher dimensional case.
Name: K. Hanumanthu, University of Kansas
Title: Koszul rings in geometry
Abstract :
Koszul rings appear naturally in algebra and geometry. We will discuss a basic question regarding the Koszulness of section rings of powers of a line bundle on a projective variety. This is studied in connection to the regularity of the line bundle in question. The relation to the $N_1$ property will be discussed. We will talk also about an analogous question for adjoint line bundles on certain nice surfaces. An attempt will be made to explore the relevance of and interest in these questions from an algebraic perspective.
Name: Z. Teitler, Texas A&M University
Title: Ranks of polynomials
Abstract :
The Waring rank of a polynomial of degree d is the least number of terms in an expression for the polynomial as a sum of dth powers. The problem of finding the rank of a given polynomial and studying rank in general has been a central problem of classical algebraic geometry, related to secant varieties; in addition, there are applications to signal processing and computational complexity. Other than a well-known lower bound for rank in terms of catalecticant matrices, there has been relatively little progress on the problem of determining or bounding rank for a given polynomial (although related questions have proved very fruitful). I will describe new upper and lower bounds, with especially nice results for some examples including monomials and cubic polynomials. The talk will be very elementary. This is joint work with J.M. Landsberg.
Name: N. Epstein, University of Osnabrück
Title: Homogenous equational tight closure
Abstract :
The title refers to a new variant of tight closure for ideals and modules over locally excellent rings in positive characteristic and in equal characteristic zero, defined in terms of persistence from `homogeneous instances' of tight closure. Unlike the original notion, it commutes with arbitrary localization! In all cases, it sits between plus closure and tight closure. It agrees with the original notion of tight closure for parameter ideals, and for all ideals and modules over finitely generated positively-graded algebras over any algebraic field extension of a prime field. It captures colons, yields a `Brian\,con-Skoda' type theorem, and gives a theory of phantom homology similar to ordinary tight closure. The new notion works well with module-finite extensions and smooth base change, and generally acts the way one expects a "tight closure theory" to behave. It raises a host of new questions, and provides new criteria for when ordinary tight closure commutes with localization. This is joint work with Mel Hochster.
Name: D. Murfet, University of Bonn
Title: Duality in singularity categories and the Kapustin-Li formula
Abstract :
We will discuss the derivation, from Grothendieck duality, of a trace formula for the Serre functor in the singularity category of an isolated hypersurface singularity, first obtained by the string theorists Kapustin and Li.
Name: Y. Xie, University of Notre Dame
Title: Formulas for the multiplicity of graded algebras
Abstract :
For simplicity, we will assume that A = k[A1] ⊆ B = k[B1] is a homogeneous inclusion of standard graded Noetherian domains over the field k. We want to express the multiplicity of A in terms of that of B and local multiplicities along Proj(B). One of the applications is to find the multiplicity of the special fibre ring of an ideal generated by forms of the same degree in a standard graded Noetherian k-algebra.
Observe that dim A ≤ dim B and they are equal if and only if their quotient field extension is algebraic of degree r. If B is integral over A, i.e. dim B/A1B = 0, then e(B) = re(A). In 2001, Simis, Ulrich and Vasconcelos gave a formula when dim A = dim B and dim B/A1B = 1. We generalize their formula to arbitrary dimensions of B/A1B. We also provide the formula for the case when dim A < dim B. Thus we give a complete answer to the original question. The techniques we use are j-multiplicities and filter-regular sequences.
The formulas we obtain can be used to find the degree of dual varieties for any hypersurfaces without any restrictions on its dual varieties and singularities. In particular, it gives a generalization of Teissier’s Plücker formula to hypersurfaces with non-isolated singularities.
Name: B. Purnaprajna, University of Kansas
Title: Fundamental groups of fibrations and conjectures of Nori and Shafarevich
Abstract :
This is joint work with Rajendra Gurjar.
Name: L. Christensen, Texas Tech University
Title: The depth formula revisited
Abstract :
Let R be a commutative noetherian local ring. A formula that expresses the depth of the tensor product of two R-modules in terms of their individual depths is traditionally called a depth formula. In the talk I will discuss one such formula that specializes to previous versions by Huneke and Wiegand and by Choi and Iyengar. The talk is based on joint work with Dave Jorgensen.
Name: M. Das, Indian Statistical Institute, Kolkata
Title: On a conjecture of Nori
Abstract :
Motivated by a topological result, which he proved in an appendix to a paper of Mandal, Nori posed the "homotopy conjecture" for sections of projective modules over a smooth affine domain. The conjecture has been settled in the affirmative by Bhatwadekar-Keshari. In this talk we will outline the impact of Nori's conjecture in the development of Euler class theory. We will also analyze the conjecture in the non-smooth set up, where it is no longer true.
Name: C. Huneke, University of Kansas
Title: Inequalities between multiplicities in graded rings
Abstract :
In this talk we discuss a conjecture which related the multiplicites of two systems of parameters in a local or graded Noetherian ring. We prove a recent result of Mustata, Takagi, Watanabe and myself which proves the conejcture for graded rings. The proof uses reduction to characteristic p, and needs the existence of graded big Cohen-Macaulay algebras.
Name: C. Huneke, University of Kansas
Title: Inequalities between multiplicities in graded rings, cont
Name: C. Francisco, Oklahoma State University
Title: Graph colorings via commutative algebra
Abstract :
I'll discuss joint work with Tai Ha and Adam Van Tuyl in which we explore how powers of the cover ideal of a graph G and their associated primes encode information about the chromatic number of G and its induced subgraphs.
Name: C. Francisco, Oklahoma State University
Title: Graph colorings via commutative algebra, cont
Name: A. Hariharan, University of Kansas
Title: 3-Standardness of the maximal ideal
Abstract :
Let R be a Cohen-Macaulay local ring with infinite residue field.
Let J be a minimal reduction of the maximal ideal m. P. Valabrega and
G. Valla show that the condition
mn ∩ J = J m n-1 holds for all
n if and only if the associated graded ring of the maximal ideal is Cohen-Macaulay.
We investigate conditions under which the equalities J ∩ m2 = J m and
J ∩ m3 = J m2 hold i.e., m is 3-standard. (Note that
m is n-standard if J ∩ mk = J mk-1, for all
k ≤ n.) We also give some applications when the 3-standard condition holds.
Name: J. Validashti, University of Kansas
Title: Relative multiplicities of graded algebras
Abstract :
Let R be a Noetherian local ring and A ⊆ B be standard graded Noetherian R-algebras. We define a sequence of relative multiplicities for the pair A ⊆ B and we study the properties of these numbers to give numerical criteria for integrality and birationality of the extension A ⊆ B, specially when A and B are arising from Rees algebras of a pair of modules.
Name: B. Purnaprajna, University of Kansas
Title: Local algebra and global geometry of canonical covers
Abstract :
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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