University of Kansas Algebra Seminar

Fall 2011

All seminar talks will take place from 2:30pm to 3:30pm in Snow Hall 306

Tuesday, August 30, 2011

Name: Osamu Iyama, Nagoya University
Title: Calabi-Yau reduction for hypersurface singularities
Abstract :

Tuesday, September 20, 2011

Name: David Fu, National Security Agency
Title: Cyclic groups in cryptography
Abstract :

At the heart of modern cryptography lies an almost trivial mathematical object, the humble finite cyclic group. We will discuss the Diffie-Hellman key agreement, the related computational problem, and the mathematics involved in solving said problem in varying amounts of time. To enjoy this talk one should have knowledge of cyclic groups and a non-disdain for things finite.

Tuesday, October 18, 2011

Name: Paolo Mantero, Purdue University
Title: Minimal representatives of even linkage classes
Abstract :

In linkage, most of the work has been done to understand, classify and describe the properties of ideals lying in the linkage class of a complete intersection (licci and glicci ideals). Licci and glicci ideals have very clear minimal representatives of their linkage class (i.e. ideals having, in some sense, the best possible homological properties): complete intersection ideals. However, if an ideal is not licci or glicci, it is not even clear if one could define a minimal representative of its even linkage class and how to define it. In this talk (where we will focus only on the CI-linkage case) we will explore homological properties that can make an ideal a minimal of its even linkage class and provide a definition of this minimality. We will then discuss upper bounds to the number of steps needed to link an ideal to a minimal representative of its even linkage class. In general these bounds are pretty large (as in the licci case). However, for special classes of ideals, smaller and more interesting bounds will be presented.

Tuesday, October 25, 2011

Name: Satya Mandal, University of Kansas
Title: Excision in Algebraic Obstruction Theory Abstract :

Spring 2011

All seminar talks will take place from 2:30pm to 3:30pm in Snow Hall 306

Thursday, March 24

Name: Kevin Tucker, University of Utah/Princeton
Title: Generalized F-signature
Abstract :

In positive characteristic, the existence of a splitting of the Frobenius map has strong algebraic and geometric consequences. In this talk, we will consider a local numerical invariant called the F-signature which roughly gives an asymptotic measure of the number of splittings of the iterates of Frobenius. This invariant was first formally defined by C. Huneke and G. Leuschke, and until recently has been shown to exist only in special cases. In joint work with M. Blickle and K. Schwede, we generalize the F-signature to incorporate divisors and ideal pairs. After discussing these generalizations, we shall describe how they can be used to show the positivity of a related numerical invariant called the F-splitting ratio. This answers an open question of I. Aberbach and F. Enescu.

Tuesday, April 26

Name: Ngo Viet Trung, Institute of Mathematics, Hanoi
Title: Cohen-Macaulayness of monomial ideals
Abstract :

A combinatorial criterion for the Cohen-Macaulayness of monomial ideals will be the presented. This criterion helps to explain allmost all previous results on this topics. As applications, I will show that there are some striking relationships between the Cohen-Macaulayness of powers of Stanley-Reisner ideals and special classes of simplicial complexes (or hypergraphs).

Thursday, April 28

Name: Bangere Purnaprajna, University of Kansas
Title: Why study fundamental groups of algebraic varieties?
Abstract :

We all seem to have fun with fundamental groups. Fun without meaning is usually equated with hedonism. In this talk we show that fun with fundamental groups is not hedonism after all, for studying fundamental groups has some serious consequences in the world of algebraic geometry and that includes connections with holomorphic convexity, a notion in several complex variables, structure and classification of algebraic varieties, diffeomorphism types among other things.

Tuesday, May 3

Name: Russ Woodroofe, Washington University
Title: Matchings, coverings, and Castelnuovo-Mumford regularity
Abstract :

I will use a theorem of Kalai and Meshulam to calculate (or at least bound from above) the Castelnuovo-Mumford regularity of the edge ideal of a graph. The bounds will come from covers of the edges of the graph by subgraphs whose complements are chordal. I'll also show how such a covering can be obtained from, for example, any maximal matching, and review related results from the graph theory literature.

Fall 2010

All seminar talks will take place from 2:30pm to 3:30pm in Snow Hall 306

Thursday, September 9

Name: Bangere Purnaprajna, University of Kansas
Title: On moduli problems in algebraic geometry
Abstract :

This will be a general talk accessible to students. Moduli space is one of the premier topics in algebraic geometry due to its enormous applications to various problems in algebraic geometry. The moduli of curves are well studied. There are still many open questions in the topic of moduli of curves. But the moduli of surfaces of general type (think of surfaces of general type as two dimensional analogues of curves of genus greater than or equal to 2) is a totally different ball game altogether. In this talk various aspects of these moduli spaces will be discussed.

Tuesday, September 14

Name: Huy Tai Ha, Tulane University
Title: Linearity of regularity and the a*-invariant
Abstract :

Let X = Proj R be a projective scheme over a field k. Let I be an ideal generated by forms of degree d, and let Y --> X be the blowing up of X along the subscheme defined by I. We shall discuss how the asymptotic linearity of the regularity and a*-invariant of powers of I can be related to a set of local data associated to fibers of a projection map from Y.

Thursday, September 16

Name: Huy Tai Ha, Tulane University
Title: Pure O-sequences and h-vectors of matroids
Abstract :

A long-standing conjecture of Stanley states that the h-vectors of matroid simplicial complexes are pure O-sequences. We shall discuss a new conjecture on pure O-sequences that implies, in particular, Stanley's conjecture for matroids of rank 3.

Tuesday, September 21

Name: Roger Wiegand, University of Nebraska
Title: Brauer-Thrall Theorems in Various Contexts
Abstract :

Let (R,m,k) be a local ring (commutative and Noetherian). Roughly speaking, the Brauer-Thrall theorems, transplanted to commutative algebra, say this: If a class C of finitely generated R-modules contains infinitely many non-isomorphic indecomposable modules, then it contains arbitrarily large indecomposables. Moreover, if k is infinite, then there are infinitely many n for which C contains |k| non-isomorphic indecomposables of size n. (Here "size" might be number of generators, length, multiplicity, ... .) I will survey some situations where these conjectures hold, situations where they do not, and situations where we don't know the answer. I will also indicate how one can build large indecomposable modules.

Thursday, September 23

Name: Yi Zhang, University of Minnesota
Title: A property of local cohomology modules of polynomial rings
Abstract :

Let R=k[x_1,...,x_n] be a polynomial ring over a field k of characteristic p, and let I=(f_1,...,f_s) be an ideal of R. Then every associated prime P of H^i_I(R) satisfies dim R/P is greater than or equal to n-deg f_1 - deg f_2 -...-deg f_n.

Thursday, September 30

Name: Bangere Purnaprajna, University of Kansas
Title: Deformation theory and moduli spaces
Abstract :


Tuesday, October 5

Name: Olgur Celikbas, University of Kansas
Title: Vanishing of Ext for modules over complete intersection rings
Abstract :

Let R be a local complete intersection and let M and N be finitely generated R-modules. We will discuss, under certain conditions, a pairing that generalizes Buchweitz's notion of the Herbrand difference. We use this pairing to examine the number of consecutive vanishing of Ext^{i}_{R}(M,N) needed to ensure that Ext^{i}_{R}(M,N)=0 for all postive i$. This is joint work with Hailong Dao.

Thursday, October 7

Name: Bangere Purnaprajna, University of Kansas
Title: Syzygies and geometry
Abstract :

Thursday, October 21

Name: Bangere Purnaprajna, University of Kansas
Title: Fundamental groups of surfaces and two conjectures in algebraic geometry
Abstract :

Thursday, October 28 (postponed)

Name: Hailong Dao, University of Kansas
Title: On the structure of Hom(M,N)
Abstract :

Let R be a regular local ring and M,N be reflexive, non-free R-modules. We describe restrictions on the direct summands of Hom(M,N). We will relate our results to earlier ones by Auslander, Auslander-Goldman, and Griffith.

Tuesday, November 16

Name: Bangere Purnaprajna, University of Kansas
Abstract :

We continue to explore the state of the trans-marriage that has taken place between commutative algebra and algebraic geometry, officiated many decades ago. The word trans comes from the fact that each new idea connecting the two fields calls for a ceremony resembling the original matrimony. We note that this has indeed been a robust one with several healthy offspring growing well. One such offspring, is the notion of Castelnuovo-Mumford regularity. This makes its head in the form of a vanishing theorem in geometry. In this talk we relate how such vanishings give raise to some beautiful algebra of free resolutions. These results connect algebra and geometry and give very effective uniform bounds towards some conjectures in Syzygy of algebraic surfaces. The officiating chair of the seminar noted that the previous talk was X rated. This talk will be only PG 21, so kids should come with adult supervision.

Thursday, December 2

Name: Emily Witt, University of Michigan
Title: Local cohomology with support in ideals of maximal minors
Abstract :

Suppose that k is a field of characteristic zero, X is an r x r matrix of indeterminates, where r is strictly less than s, and R = k[X] is the polynomial ring over k in the entries of X. Let I be the ideal of R generated by the maximal (r x r) minors of X. Using the structure induced on the local cohomology modules H^i_I(R) by the action of SL_r(k) on R, as well as Lyubeznik's theory of local cohomology modules as D-modules, we provide information about the modules H^i_I(R). In particular, we find their associated primes, compute H^i_I(R) at the highest nonvanishing index, i = r(s-r) +1, identify the indices i for which these modules vanish, and characterize the nonzero ones as submodules of certain indecomposable injective modules. Moreover, these results are consequences of a more general theorem regarding local cohomology modules with actions of linearly reductive groups.

Spring 2010

Tuesday, January 26

Name: H. Dao, University of Kansas
Title: Endomorphism rings of finite global dimension and cluster tilting objects over Gorenstein rings, I
Abstract :

Let M be a reflexive modules over a local ring R and A = HomR(M,M). If A has finite global dimension, then it has been proposed to serve as a non-commutative analogue of desingularizations of Spec(R). In these talks, we will survey what is known about when such module M exists, and discuss various connections with birational geometry and representation theory of commutative rings. Part of the new results are joint work with Craig Huneke.

Tuesday, February 2

Name: H. Dao, University of Kansas
Title: Endomorphism rings of finite global dimension and cluster tilting objects over Gorenstein rings, II

Tuesday, February 9

Name: D. Katz, University of Kansas
Title: The e1 coefficient in the Hilbert polynomial of a parameter ideal
Abstract :

We will discuss recent work of Goto, Vasconcelos, et. al. concerning the e1 coefficient in the Hilbert polynomial of an ideal generated by a system of parameters.

Tuesday, February 16

Name: B. Purnaprajna, University of Kansas
Title: Geometry and syzygies of an algebraic surface
Abstract :

In the last 25 years or so, after the work of M. Green, the topic of syzygies of algebraic varieties has attracted special attention. The relation between the geometry of an embedding and the algebra of the associated free resolution is to some extent clear for curves but mostly open for an algebraic surface and higher dimensional varieties. I will speak about my recent work with Krishna Hanumanthu that relates the syzygies and geometry of an algebraic surface. This lecture will be accessible to graduate students as well.

Thursday, February 18

Name: S. Cooper, University of Nebraska
Title: Invariants Related to Symbolic Powers of Ideals of Points
Abstract :

Many algebraic tools have been developed to gain insight into 0-dimensional schemes. In particular, the Hilbert function and graded Betti numbers of the homogeneous ideal of the scheme have played a central role in many intriguing problems. Hilbert functions of ideals defining reduced 0-dimensional schemes have a well-known characterization. However, we are left perplexed when trying to characterize Hilbert functions of symbolic powers of such ideals (which define non-reduced schemes called fat point schemes). In this talk we'll compare the two situations and provide some insight into the case for fat points. By "trimming down" fat point schemes as a sequence of residuals with respect to lines, we will obtain upper and lower bounds for the Hilbert function of any fat point scheme in projective 2-space. This reduction procedure can be applied to also obtain bounds on Betti numbers, regularity and initial degree. This is joint work with B. Harbourne and Z. Teitler.

Tuesday, February 23

Name: G. Leuschke, Syracuse University
Title: Wild Hypersurfaces
Abstract :

In the representation theory of finite-dimensional algebras over a field, Drozd's dichotomy theorem says that an algebra has either tame module type or wild module type. Loosely, these two possibilities correspond to: hoping for a classification theorem, or throwing up our hands in dismay. We'd very much like a similar dichotomy result in other representation-theoretic contexts, specifically for maximal Cohen--Macaulay modules over a Cohen--Macaulay local ring. I'll give a little background on the problem, including definitions of tame and wild CM type, and talk about recent work with Andrew Crabbe (Syracuse) which shows that hypersurfaces of multiplicity four or more, in three or more variables, have wild CM type.

Tuesday, March 2

Name: Bangere Purnaprajna, University of Kansas
Title: Geometry and syzygies of an algebraic surface, continued
Abstract : see above

Thursday, March 11

Name: Bhargav Bhatt, Princeton University
Title: Derived direct summands
Abstract :

A scheme satisfies the direct summand condition if its structure sheaf is a summand of that of a finite cover; in characteristic 0, this is equivalent to normality. We will discuss a derived category analogue of this condition. The two conditions diverge in characteristic 0 (the latter characterises rational singularities by a theorem of Kovacs), but turn out to be very closely related in positive and in mixed characteristic. Using this relation, I will also discuss some applications to extensions of standard cohomological vanishing results beyond the ample cone in positive characteristic algebraic geometry.

Tuesday, March 23

Name: Ryan Kinser, University of Connecticut
Title: Rank functions and tensor products of quiver representations
Abstract :

We'll start with an overview of quiver representations, giving the basic definition and examples, then provide a brief summary their use in algebra, geometry and combinatorics. Next we'll discuss rank functions, which assign a nonnegative integer to any quiver representation, generalizing the notion of the rank of a linear map. They have algebraic properties similar to the classical rank function (additive with respect to direct sum, multiplicative with respect to tensor product, invariant under duality). A "global rank function" can be constructed for any quiver and then maps between directed graphs can be used to derive more rank functions. These can be applied to study the structure of the representation ring (i.e., split Grothendieck ring) of a quiver. Time permitting, we'll also discuss the constructibility of rank functions on representation spaces of quivers.

Thursday, March 25

Name: P. Blass
Abstract :

Tuesday, March 30

Name: Ben Williams, Stanford University
Title: Motovic cohomology and problems in commutative algebra
Abstract :

Let R=k[x1, ... , xm] be a graded polynomial ring over a field, and let C be a chain complex of graded free, finite rank R-modules having only finite length homology. There are many open problems in commutative algebra relating to such C, including a special case of a conjecture of Carlsson's with application to the theory of transformation groups, which itself generalizes a weak form of the Horrocks conjecture.

We try to understand the space of complexes C by constructing a moduli space X of exact sequences, and then exhibiting C as a map from a punctured affine space to such X. Our space X has a presentation as a homogeneous space of linear groups. Using methods stolen from classical algebraic topology, we can compute the motivic cohomology of the space X, and consequently find numerical restrictions on the existence of such complexes C, including a mild generalization of the Herzog-Kuehl equations.

Thursday, April 1

Name: S. Sather-Wagstaff, North Dakota State University
Title: Extension and Torsion Functors for Artinian Modules
Abstract :

Let R be a commutative noetherian ring. It is well known that if N and N' are noetherian R-modules, then the modules Ext^i_R(N,N') and Tor^R_i(N,N') are also noetherian. Similarly, if N is a noetherian R-module and A is an artinian R-module, then the modules Ext^i_R(N,A) and Tor^R_i(N,A) are artinian. We will discuss the properties of Ext and Tor modules when applied to other combinations of noetherian modules, artinian modules, and Matlis reflexive modules. This is joint work with Bethany Kubik (NDSU) and Micah Leamer (UNL).

Thursday, April 8

Name: Bangere Purnaprjana, University of Kansas
Title: Deformation of maps and moduli space of surfaces of general type
Abstract :

Thursday, April 22

Name: A. Hariharan, University of Nebraska
Title: Connected Sums of Gorenstein Local Rings
Abstract :

This talk will introduce a new construction of local rings called the connected sum and study its properties. In particular, we will obtain conditions for the connected sum to be Gorenstein. Finally, we will see some applications of this construction. This is joint work with W. Frank Moore (Cornell) and Luchezar L. Avramov (UNL).

Thursday, April 29

Name: Y. Kachi, University of Kansas
Title: Wallis type formula for the special value of Riemann's zeta function at positive odd integers.
Abstract :

As in the title, I will talk about Wallis type formulas for ζ(3) and also Catalan's number. This is joint work with Joel Font.

Fall 2009

Thursday, September 3

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.

Tuesday, September 8

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.

Tuesday, September 15

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.

Tuesday, September 22

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.

Tuesday, October 13

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.

Thursday, October 22

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.

Tuesday, October 27

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 Teissierís Plücker formula to hypersurfaces with non-isolated singularities.

Thursday, November 12

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.

Thursday, November 19

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.

Spring 2009

Tuesday, February 10

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.

Tuesday, February 24

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.

Thursday, February 26

Name: C. Huneke, University of Kansas
Title: Inequalities between multiplicities in graded rings, cont

Tuesday, March 17

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.

Thursday, March 19

Name: C. Francisco, Oklahoma State University
Title: Graph colorings via commutative algebra, cont

Tuesday, March 24

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.

Thursday, March 26

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.

Thursday, April 2

Name: B. Purnaprajna, University of Kansas
Title: Local algebra and global geometry of canonical covers
Abstract :

Fall 2008

Thursday, September 4

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.

Tuesday, September 9

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.

Thursday, September 11

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.

Tuesday, September 16

Name: K. Hanumanthu, University of Kansas
Title: Toroidalization of locally toroidal morphisms, continued

Tuesday, October 7

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

Thursday, October 9

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.

Thursday, October 23

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.

Tuesday, October 28

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.

Thursday, October 30

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.

Thursday, November 6

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.

Tuesday, November 4

Name: H. Long, University of Kansas
Title: Almost Cohen Macaulay modules, almost regular sequences and the monomial conjecture: a survey, II

Thursday, November 13

Name: S. Sane, Tata Institute of Fundamental Research
Title: TBA

Tuesday, November 18

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.

Thursday, November 20

Name: D. Katz, University of Kansas
Title: Multiplicities and Rees valuations, II

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