UIUC Dept. of Mathematics Seminar Calendar

       May 2012              June 2012              July 2012      
 Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
        1  2  3  4  5                   1  2    1  2  3  4  5  6  7
  6  7  8  9 10 11 12    3  4  5  6  7  8  9    8  9 10 11 12 13 14
 13 14 15 16 17 18 19   10 11 12 13 14 15 16   15 16 17 18 19 20 21
 20 21 22 23 24 25 26   17 18 19 20 21 22 23   22 23 24 25 26 27 28
 27 28 29 30 31         24 25 26 27 28 29 30   29 30 31

Thursday, January 26, 2012

Commutative Ring Theory Seminar
3:00 pm in 243 Altgeld Hall, Thursday, January 26, 2012

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Organizational Meeting

Thursday, February 2, 2012

Thursday, April 5, 2012

Commutative Ring Theory Seminar
3:00 pm in 243 Altgeld Hall, Thursday, April 5, 2012

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Yu Xie (Notre Dame Math)
An extension of a Lemma of Huneke to non $m$-primary ideals and formulas for the generalized Hilbert coefficients
Abstract: Let $(R,m)$ be a Cohen-Macaulay local ring and $I$ an $m$-primary ideal. In 1996, Huckaba provided a $d$-dimensional version of a 2-dimensional formula due to Huneke. This formula relates the length $\lambda(I^{n+1}/JI^n)$ to the difference $P(n+1)-H(n+1)$, where $J$ is a minimal reduction of $I$, and $P(n+1)$ and $H(n+1)$ are Hilbert polynomial and Hilbert function of $I$ respectively. We extend the formula further to non $m$-primary ideals and use it to compute the generalized Hilbert coefficients defined by Polini and Xie recently.

Thursday, April 12, 2012

Commutative Ring Theory Seminar
3:00 pm in 243 Altgeld Hall, Thursday, April 12, 2012

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Paulo Mantero (Purdue Math)
Liaison classes of non-licci ideals
Abstract: A vast part of literature on liaison via complete intersections adresses questions relative to licci ideals, that is, ideals that are in the linkage class of a complete intersection. For a non-licci ideal I, there are few results describing the structure of the linkage class of I, most of them dealing with the case of height 2 ideals (Rao, Lazarsfeld, Ballico-Bolondi-Migliore, Perrin, Martin-Descamps, Nagel, etc.). In particular one would like to find distinguished elements in every linkage class. In this talk we introduce a theoretical definition for `minimal' ideals in any even linkage class. We show that, under reasonable assumptions, these ideals exist and are essentially unique. Among all ideals in an even linkage class, these ideals minimize homological invariants (e.g. Betti numbers, multiplicity). We provide several concrete situations where one can identify these minimal elements (e.g. determinantal ideals or ideals with homogeneous linear resolutions are minimal in their respective even linkage classes). If time permits, I will show an application to producing more evidence towards the Buchsbaum-Eisenbud-Horrocks Conjecture.

Thursday, October 11, 2012

Thursday, October 18, 2012

Thursday, November 8, 2012

Commutative ring theory seminar
3:00 pm in 243 Altgeld Hall, Thursday, November 8, 2012

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Youngsu Kim (Purdue University)
On the Equality of Ordinary and Symbolic Powers of Ideals
Abstract: Symbolic powers of ideals are central objects in commutative algebra and algebraic geometry. For example, in a polynomial ring over an algebraically closed field, the $n$-th symbolic power $P^{(n)}$ of a prime ideal P is the set of functions that vanish to order at least $n$ along $V(P)$. In this talk, we discuss criteria for the equality of ordinary and symbolic powers. In particular, we are interested to know if the equality of $P^n$ and $P^{(n)}$ for all $n \leq $ a certain bound implies the equality for all $n$ (joint work with A. Hosry and J. Validashti).

Thursday, November 15, 2012

Commutative ring theory seminar
3:00 pm in 243 Altgeld Hall, Thursday, November 15, 2012

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Winfried Bruns [email] (Universitat Osnabruck)
Relations of minors
Abstract: We report on joint work with Aldo Conca and Matteo Varbaro. It is a classical theorem that the Grassmannian $G(m,n)$ of $m$-subspaces of an $n$-dimensional vector space is defined by the degree $2$ Plucker relations. These are obtained as polynomial relations of the Plucker coordinates, which, in their turn, are the $m$-minors ($m\times m$-subdeterminants) of an $m\times n$-matrix $X$ of indeterminates, generating the homogeneous coordinate ring of $G(m,n)$. The algebra $A_t$ generated by the $t$-minors of $X$ for $t\lneq m$ is well understood both from the representation-theoretic viewpoint (by work of De Concini-Eisenbud-Procesi) and in its structural properties, as well as the corresponding variety of exterior powers of linear maps (by work of Bruns-Conca). By contrast, the relations of the $t$-minors for $t\lneq m$ have not yet been determined, and already in the first non-trivial case relations of degree $3$ appear (and for $t \gneq 3$ there are non-Plucker relations of degree $2$). We want to describe a representation-theoretic approach that leads to a conjectural description of the relations by which their ideal is generated in degrees $2$ and $3$. The conjecture is supported by computational results for small cases.