Commutative Algebra Seminar

Seminars will be inAvery 351, and will be held twice a week: Wednesdays 3:30 - 4:20 pm Thursdays 2:30 - 3:20 pm There is a mailing list (unlcas) of the participants of the Commutative Algebra Seminar: If you want to subscribe or unsubscribe visit the unlcas info page, and follow the instructions.

This week's seminar

12th December (Wednesday) at 3:30pm. Speaker: Title:Abstract:

13th December (Thursday) at 2:30pm. Speaker: , Title: Abstract:

5th December (Wednesday) at 3:30pm. Speaker:Frank Moore, University of Nebraska Title:On a conjecture of Boij and SoederbergAbstract: I will introduce a conjecture of Boij and Soederberg on the form of Betti diagrams of Cohen-Macaulay modules over a polynomial ring, and its relationship to the multiplicity conjecture of Herzog, Huneke and Srinivasan.

6th December (Thursday) at 2:30pm. Speaker:Daniel Katz, University of Kansas Title:Beyond Cramer's RuleAbstract: In this talk we follow a path from Cramer's rule to one of the speaker's favorite open questions. Along the way we encounter some important ideas in commutative algebra. This talk is accessible to a general mathematical audience.

28th November (Wednesday) at 3:30pm. Speaker:Srikanth Iyengar, University of Nebraska-Lincoln Title:Modules of finite projective dimension and of finite injective dimensionAbstract:

29th November (Thursday) at 2:30pm. Speaker:Ovidiu Pasarescu, Institute of Mathematics of the Romanian Academy in Algebraic Geometry, Fulbright scholar at Colorado State University Title:Classification of the degenerate Del Pezzo Surfaces after the Singular LocusAbstract: It is well known that the smooth, non-degenerate, surfaces of degree d in P^d exists only if d is between 3 and 9. These surfaces are called Del Pezzo surfaces, and they are all rational. Gorenstein surfaces of degree d in P^d with isolated singularities also exists only if d is between 3 and 9 (Hidaka-Watanabe). They are either cones or they are rational and have only A-D-E singularities (in this last case they are called degenerate Del Pezzo surfaces). I will present a strategy giving, for any d, all the possible configurations of singularities which appear on the degenerate Del Pezzo surfaces of degree d. For d=3 (resp. d=4) we recover the classical case of cubic surfaces (resp. of complete intersections of type (2,2) from P^4.

14th November (Wednesday) at 3:30pm. Speaker:Jeffrey Mermin, University of Kansas. Title:Progress on the Lex-Plus-Powers ConjectureAbstract:Let S be the polynomial ring in n variables over a field, and let F=(f_1,...f_r) be a homogeneous regular sequence. Let P be the ideal generated by pure powers of the variables (the powers given by the degrees of the f_i). Inspired by applications in algebraic geometry, Eisenbud, Green, and Harris made the following conjecture (the "Eisenbud-Green-Harris Conjecture"): Let I be any ideal containing F. Then there exists a lex ideal L such that L+P has the same Hilbert function as I. This was later extended by Evans (the "Lex-Plus-Powers Conjecture"): Furthermore, all of the graded Betti numbers of L+P are greater than or equal to those of I. I will define most of the words above, and discuss recent independent work of Murai and myself proving the Lex-Plus-Powers Conjecture in the case that F consists of monomials.

15th November (Thursday) at 2:30pm. Speaker:Tyler Lemberg, University of Nebraska Title:Numerical MonoidsAstract: Let $S$ be a numerical monoid (i.e., an additive submonoid of $\mathbb{N}_0$) with minimal generating set $\langle n_1, \ldots , n_t \rangle$. For $m \in S$, if $m = \sum_{i=1}^t x_in_i$, then $\sum_{i=1}^t x_i$ is called a {\it factorization length} of $m$. We denote by $\mathcal{L}(m) = \{ m_1, \ldots , m_k\}$ (where $m_i < m_{i+1}$ for each $1 \le i < k$) the set of all possible factorization lengths of $m$. The delta set of $m$ is defined by $\Delta (m) = \{ m_{i+1} - m_i \mid 1 \le i < k \}$ and the delta set of S by $\Delta (S) = \bigcup_{m \in S} \Delta (m)$. Let $r_1, \ldots , r_t$ be an increasing sequence of positive integers and $M_n = \langle n, n+r_1, \ldots, n+r_t \rangle$ a numerical monoid where $n$ is some positive integer. We prove that there exists a positive integer $N$ such that if $n > N$ then $\lvert \Delta (M_n) \rvert = 1$. If $M_n$ has 3 generators, and $r_1$ and $r_2$ are relatively prime, then we determine a value for $N$ which is sharp.

31st October (Wednesday) at 3:30pm. Speaker:Jinjia Li, Syracuse University Title:Abstract:

1st November (Thursday) at 2:30pm. Speaker:Jinjia Li, Syracuse University Title:Abstract:

24th October (Wednesday) at 3:30pm. Speaker:David Pitts, University of Nebraska (joint seminar with the Operator Algebra group) (Avery 118) Title:A tour of Choquet Theory'Abstract:

25th October (Thursday) at 2:30pm (usual seminar room) Speaker:William Arveson, Title:The noncommutative Choquet boundaryAbstract:

17th October (Wednesday) at 3:30pm. Speaker: , Title: Abstract:

18th October (Thursday) at 2:30pm. Speaker:Ananth Hariharan, University of Kansas Title:The Gorenstein Colength of an Artinian Local RingAbstract:In this talk, I will make the concept of approximating an Artinian local ring R by an Artinian Gorenstein local ring precise by using the notion of Gorenstein colength, denoted g(R). We will discuss some tools and techniques that can be used to compute g(R). Huneke-Vraciu and Teter characterize Artinian local rings R for which g(R) \leq 1. We will also see how one can use the idea of a connected sum to extend their result.

10th October (Wednesday) at 3:30pm. Speaker:Livia Miller, University of Nebraska Title:A Tale of non-Noetherian Rings: Defining Non-Noetherian Gorenstein RingsAbstract:Motivated by a question of Glaz, Hamilton and Marley recently introduced a theory of non-Noetherian Cohen-Macaulay rings equivalent to the usual definition of Noetherian Cohen-Macaulay rings. In my talk I will introduce this definition and use it to develop a theory of non-Noetherian Gorenstein rings. I plan to give a global picture of this theory by explaining how non-Noetherian regular, Gorenstein and Cohen-Macaulay rings relate. If time allows, will also discuss additional characterizations and properties of non-Noetherian Gorenstein rings.

11th October (Thursday) at 2:30pm. Speaker:Javid ValidashtiUnivesity of Kansas Title:Multiplicities of Graded Algebras and Integral Dependence of Modules.Abstract: Abstract: Let R be a universally catenary, locally equidimensional Noetherian ring. We give a multiplicity based criterion for integral dependence of an arbitrary finitely generated R-module over a submodule.

3rd October (Wednesday) at 3:30pm. Speaker:Frank Moore, University of Nebraska Title:Cohomology of connected sums of artinian Gorenstein ringsAbstract: We will give the (topologically motivated) definition of a connected sum of artinian Gorenstein rings, as well as give some results on its Ext algebra. Time permitting, we will also discuss some results on its Koszul homology algebra. In order to perform these calculations, we will use the theory of Golod homomorphisms, and so the talk will also contain the relevant definitions and some basic examples.

4th October (Thursday) at 2:30pm. Speaker:Brian Harbourne, University of Nebraska Title:Comparing powers and symbolic powers of idealsAbstract:Craig Huneke has asked: is it true for any radical homogeneous ideal I defining a 0-dimensional subscheme of P2 that I^2 contains I^(3)? More generally, given a homogeneous ideal I in a polynomial ring k[x0,...,xn], for precisely what r and m does I^r contain I^(m)? Recent joint work of Harbourne with C. Bocci has begun to address such problems.

26th September (Wednesday) at 3:30pm. Speaker:Mark Walker, University of Nebraska Title:The K-theory of toric varietiesAbstract: This is joint work with Chuck Weibel, Christian Haesemeyer, and Willie Cortinas. Our main result is a new and (it is hoped) simpler proof of Gubeladeze's "Nilpotence Conjecture", proven by him a few years ago. Our proof uses the "cdh-topology", Hochschild homology, and Andre-Quillen homology. My goal in these talks is to explain what all these terms mean and to sketch our proof.

27th September (Thursday) at 2:30pm. Speaker:Mark Walker, University of Nebraska Title:The K-theory of toric varietiesAbstract: This is joint work with Chuck Weibel, Christian Haesemeyer, and Willie Cortinas. Our main result is a new and (it is hoped) simpler proof of Gubeladeze's "Nilpotence Conjecture", proven by him a few years ago. Our proof uses the "cdh-topology", Hochschild homology, and Andre-Quillen homology. My goal in these talks is to explain what all these terms mean and to sketch our proof.

12 September (Wednesday) at 3:30pm. Speaker:Andrew Crabbe, University of Nebraska Title:Hilbert Functions of Ext-Modules and ApplicationsAbstract:Let M, N be modules over a local ring (R,m). The function sending a natural number n to the length of the R-module Ext^i(M,N/m^n N) is given by a polynomial for n large enough. We give the degree of this polynomial for i = 0; and, with some assumptions on the Betti numbers of M, we are able to give lower bounds on the degree for i > 0. Following the work of R.Wiegand and Hassler, we use these results to build indecomposable modules which are MCM (modulo a finite part), having arbitrarily large rank on the punctured spectrum.

13 September (Thursday) at 2:30pm. Speaker:Andrew Crabbe, University of Nebraska Title:Hilbert Functions of Ext-Modules and Applications(Part II)

5th September (Wednesday) at 3:30pm. Speaker:Janet Striuli, University of Nebraska Title:Number of indecomposable totally reflexive modulesAbstract: This is a practice talk for KuMuNu.

29th August (Wednesday) at 3:30pm. Speaker:Peter Symonds, University of Manchester Title:Group actions on polynomial rings.

8th August (Wednesday) at 3:30pm. Speaker:Tim Römer, Universität Osnabrück Title:On the regularity over positively graded algebrasAbstract: We study the relationship between the Tor-regularity and the local-regularity over a positively graded algebra defined over a field which coincide if the algebra is a standard graded polynomial ring. In this case both are characterizations of the so-called Castelnuovo-Mumford regularity. Moreover, we can characterize a standard graded polynomial ring as an algebra with extremal properties with respect to the Tor- and the local-regularity. For modules of finite projective dimension we get a nice formula relating the two regularity notions.

2nd August (Thursday) at 9:30 am. Speaker:Sandra Spiroff, Seattle University. Title:An algebraic proof of the commutativity of intersection with divisorsAbstract:

Past Seminars.

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