List of Talks

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  • Some thoughts on the jacobian conjecture
    by Shreeram ABHYANKAR, Purdue University
    Abstract: I shall report on some recent progress in this long standing conjecture.

  • The image of the Chern class map for singular varieties
    by Donu ARAPURA, Purdue University
    Abstract: When X is a smooth complete variety over C, its well known that the Chern classes of a vector bundle in rational cohomology are Hodge classes, i. e. classes of type (p,p). When X is singular, the notion of Hodge class can be generalized using mixed Hodge theory, and it is not difficult to see that Chern classes of vector bundles are again of this type. However there are now further constraints on these classes that I want to explain. This phenomenon was first noticed by Barbieri-Viale and Srinivas for line bundles about 10 years ago.

  • Hochschild cohomology and Gorenstein algebras
    by Luchezar AVRAMOV, University of Nebraska at Lincoln
    Abstract: Let K be a noetherian ring and S a commutative algebra, which is essentially of finite type over K and is projective as a K-module. The K-algebra S is said to be Gorenstein if the non-trivial fibers of the structure homomorphism K → S are Gorenstein rings. This property will be characterized in terms of the vanishing of appropriate Hochschild cohomology groups HHn(S|K;S ⊗KS). This is joint work with Srikanth Iyengar.

  • Buchsbaum-Rim multiplicity and Hilbert-Samuel multiplicities
    by C.-Y. Jean CHAN, University of Arkansas
    Abstract: Let $R$ be a regular local ring of dimension $2$ with maximal ideal $\mathfrak m$. We study the Buchsbaum-Rim multiplicity $e_{BR}(M)$ of a finitely generated module $M$ of finite colength in a free module $F$.\par Let $\mathfrak a$ be an $\mathfrak m$-primary ideal in $R$. We first investigate the colength $\ell(R/ \mathfrak a)$ of $\mathfrak a$ and its Hilbert-Samuel multiplicity $e(\mathfrak a)$ using linkage theory. As applications, we establish several multiplicity formulas that express $e_{BR}(M)$ in terms of the Hilbert multiplicities of ideals related to an arbitrary minimal reduction $U$ of $M$. In the special case where the maximal Fitting ideal of $F/U$ is integrally closed, $e_{BR}(M)$ is directly related to all Fitting ideals of $F/U$. \par There exists $\mathfrak m$-primary Bourbaki ideals $I$ and $J$ of the modules $F$ and $M$ respectively such that $F/M \cong I/J$. We also have a formula for $e_{BR}(M)$ in terms of $e(I)$ and $e(J)$. This is related to a graphical interpretation of the multiplicities in the case of monomial ideals. (This is joint work with Jung-Chen Liu and Bernd Ulrich.)

  • On the Cohen-Macaulayness of residual intersections and extended Rees rings
    by Christine CUMMING, Pur-----due University
    Abstract: I will talk about when residual intersections and extended Rees rings are Cohen-Macaulay when the ring is merely Cohen-Macaulay.

  • On Extension of Gabber's Work
    by Sankar DUTTA, University of Illinois at Urbana-Champaign
    Abstract: We will first point out the key steps in Gabber's proof of the non-negativity part of Serre's conjecture on Intersection Multiplicity over regular local rings. Next we will show that his technique cannot be extended to regular schemes of finite type over an excellent discrete valuation ring. We would like to derive some special cases of positivity if time permits.

  • Adjoint ideals
    by Lawrence EIN, University of Illinois at Chicago
    Abstract: Adjoint ideal is a modification of multiplier ideal. We'll describe the properties of adjoint ideals and give some applications of these ideals to extension theorems and singularities of pairs.

  • Matroids arising from certain closure operations on ideals
    by Neil EPSTEIN, University of Michigan
    Abstract: I will discuss some cases where matroids (infinite, with finite basis condition) arise naturally when considering an ideal, a closure operation, and the set of ideals which have the same closure as the given ideal. This opens up a wide range of combinatorial tools for dealing with closure operations for which this situation occurs.

  • The core of ideals
    by Louiza FOULI, Purdue University
    Abstract: Let R be a Noetherian local ring with infinite residue field k and I an R-ideal. We say that J ⊂ I is a reduction of I if I and J have the same integral closure. A reduction is called minimal if it is minimal with respect to inclusion. In general minimal reductions are not unique. To remedy this lack of uniqueness one considers the intersection of all (minimal) reductions, namely the core of I, core(I). The core encodes information about all possible reductions of the ideal. On the other hand reductions are connected with the study of blowup algebras. The core is a mysterious object that appears naturally in the context of Briançon-Skoda kind of theorems. Hyry and Smith have shown that Kawamata's conjecture on the existence of sections of certain line bundles is equivalent to a statement about the core of particular ideals in section rings. Under some technical conditions (which are automatically satisfied in case I is equimultiple) Polini and Ulrich have shown that for a Gorenstein local ring, core(I)=Jn+1:In, for n ≫ 0. We present some recent work on the core of ideals.

  • Comparison of local and global liaison questions
    by Robin HARTSHORNE, University of California at Berkeley
    Abstract: We study the analogy between the global theory of liaison of varieties in projective space and the local theory of liaison of ideals in a local ring. To what extent do the two theories give parallel results? Can one deduce global from local results or vice versa? We give some examples to show this is not always possible.

  • Embedding modules of finite projective dimension
    by Melvin HOCHSTER, University of Michigan
    Abstract: The main result is that over a Noetherian ring finitely generated modules of finite projective dimension can be embedded in in a very special modules with this property. Applications of this result will also be described. (Th is joint work with Yongwei Yao)

  • Integral Closures of Modules and Hyperplane Sections
    by Jooyoun HONG, Purdue University
    Abstract: The theory of the normalization of ideals has been extended to modules and, as in the case of ideals, the Rees algebras of modules serve as a useful tool to detect normality of modules. Let $R$ be a Noetherian ring and $E$ a finitely generated torsionfree $R$-module having a rank. To enable the extension, we consider two technical devices to attach an ideal $I$ of a ring $S$ to the $R$--module $E$ so that the comparison can be made between the Rees algebra of $E$ and the Rees algebra of $I$. One of the ideals that can be used is the ideal generated by the module in the polynomial ring which contains the Rees algebra of the module. The other is known as the generic Bourbaki ideal of $E$. In this joint work with B. Ulrich, we show that integrally closedness of any ideals of height at least $2$ is compatible with a specialization of generic elements using a vanishing theorem of local cohomology of certain degrees. By turning a module into an ideal, we relate the integral closure of a module to that of the ideal. (This is joint work with Bernd Ulrich)

  • Coniveau and the Grothendieck group of varieties
    by Su-Jeong KANG, Purdue University
    Abstract: We prove that if the generalized Hodge conjecture, or some weaker form of it, holds for a Calabi-Yau variety then it holds for any Calabi-Yau varieties birationally equivalent to it. This is joint work with Donu Arapura.

  • The resolution of the Frobenius module
    by Andrew KUSTIN, University of South Carolina
    Abstract: Let R be a ring containing the field of positive characteristic p and let fR to represent the ring R endowed with an R-module structure given by the Frobenius automorphism f:R → R with f(r)=rq, where q=pe for some fixed exponent e. We study the resolution of fR by free R-modules. (This is joint work with C. Miller and A. Vraciu.)

  • Polar syzygies and unirational algebras
    by Aron SIMIS, Universidade Federal de Pernambuco, Brazil
    Abstract: Given a finitely generated subalgebra of a polynomial ring, one considers the syzygies of the transposed Jacobian module of its generators. A good chunk of them is given by the polar syzygies (appeared in disguise in Gordan-Noether). One deals with some special classes of algebras to understand when the polar syzygies generate all syzygies. (Joint work with I. Bermejo and P. Gimenez)

  • Annihilation of elements of local cohomology modules
    by Anurag SINGH, University of Utah
    Abstract: We will discuss some results and questions on the existence of big Cohen-Macaulay algebras, and related big algebras. These can be viewed as statements about the annihilation of elements of local cohomology modules. There are powerful theorems along these lines due to Hochster-Huneke, and to Huneke-Lyubeznik.

  • Asymptotic behavior of multigraded regularity
    by Brent STRUNK, Purdue University
    Abstract: In this talk we discuss asymptotic behavior of multigraded regularity.

  • Indecomposable mixed modules of large rank
    by Roger WIEGAND, University of Nebraska
    Abstract: Let (R,m,k) be a one-dimensional local ring. Our goal is to build indecomposable finitely generated modules with large rank. In most cases, even if R has finite representation type, one can build indecomposable mixed modules whose torsion-free parts have large rank. Exceptions include discrete valuation rings and Dedekind-like rings (essentially, (A1)-singularities, e.g., k[[x,y]]/(xy) and R[[x,y]]/(x2+y2)). We show that, if R is not a homomorphic image of a Dedekind-like ring, then, for each positive integer n, there is an indecomposable finitely generated module M such that MP ≅ RP(n) for every minimal prime ideal P. If char(k) ≠ 2, the existence of such modules is equivalent to wildness of the category of finite-length modules. A key idea in our construction is to produce an indecomposable finite-length module V and a torsion-free module N such that Ext1R(N,V) has big socle dimension. One then uses this fact to produce extensions 0 → V → M → N(t) → 0, with t large and M indecomposable. This is joint work with Wolfgang Hassler, Ryan Karr and Lee Klingler.

  • Ring extensions with trivial generic fiber
    by Sylvia WIEGAND, University of Nebraska
    Abstract: We study examples of domain extensions R ⊆ S such that every nonzero prime ideal of S intersects R in a nonzero prime ideal. In certain circumstances the dimension of S must be less than or equal to 2. This is joint work with Wlliam Heinzer and Christel Rotthaus.

  • Poincaré duality algebras as rings of coinvariants
    by Clarence WILKERSON, Purdue University
    Abstract: If $V$ is a finite dimensional vector space over a field $k$, and $W$ is a finite subgroup of $Aut(V)$, then the symmetric algebra $S(V^\#)$ can be thought of as the algebra of polynomial functions on $V$, and it inherits an action of the group $W$. The algebra of invariants $S(V^\#)^W$ is of particular interest. If it is itself a polynomial algebra, then the quotient algebra $S(V^\#)/I$, where $I$ is the ideal generated by the positive degree elements of $S(V^\#)^W$ form the coinvariants and is Poincare' duality algebra under the induced multiplication.
    However, if the characteristic of $k$ is positive, the coinvariants can be a Poincare duality algebra without $S(V^\#)^W$ being polynomial. The author and W. G. Dwyer give a generic example of this behavior. Moreover, we give a new proof, independent of the previous work of Steinberg, Kane, and T.C. Lin of the following theorem.\par With notation as above, if char($k$) = 0 or $p$ with $p$ relatively prime to the order of $W$, then $S(V^\#)/I$ is a Poincare' duality algebra if and only if $S(V^\#)^W$ is a polynomial algebra.
    This is joint work with W. G. Dwyer, Notre Dame.

  • Resolution of singularities in characteristic zero
    by Jaroslaw WLODARCZYK, Purdue University
    Abstract: We discuss basic ideas of simplified Hironaka desingularization algorithm in characteristic zero.