## Fall 2007



Seminars will be in  Avery 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,

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 Soederberg

Abstract: 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 Rule

Abstract: 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 dimension

Abstract:

29th November (Thursday) at 2:30pm.

Speaker: Ovidiu Pasarescu, Institute of Mathematics of the
State University

Title: Classification of the degenerate Del Pezzo Surfaces after
the Singular Locus

Abstract: 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 Conjecture

Abstract: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 Monoids
Astract: 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 boundary

Abstract:

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 Ring

Abstract: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 Rings

Abstract: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 Validashti Univesity 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 rings

Abstract: 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 ideals

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

26th September (Wednesday) at 3:30pm.

Speaker: Mark Walker, University of Nebraska

Title: The K-theory of toric varieties

Abstract: 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 varieties

Abstract: 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 Applications

Abstract: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 modules

Abstract: 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 algebras

Abstract: 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 divisors

Abstract:

Past Seminars.

Maintained by Janet Striuli

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