The Program Associate Seminar Series is a graduate student commutative
algebra seminar at MSRI for the 2012-2013 thematic year. Two 30
minute talks are given at 10:45 and 11:25 on Tuesday mornings in Simons
Auditorium. All are welcome to attend!
Fall 2012
November 6
10:45 Antonio Macchia
Title: The Arithmetical Rank of the Edge Ideals of Whisker Graphs
Abstract: A classical problem in Algebraic Geometry consists in finding
the minimum number of hypersurfaces that define a certain variety. This
problem can be approached from an Algebraic and Combinatorial point of
view.
Given a commutative ring R with identity and an ideal I of R, the
arithmetical rank of I, denoted ara(I), is the minimum number of
elements of R such that the ideal generated by those elements has the
same radical as I. The ideal I is called set-theoretic complete
intersection (STCI) if ara(I)=ht(I).
In general if I is STCI, then I is Cohen-Macaulay, but the converse is
not true. I will show that the converse holds for the edge ideals of
some whisker graphs.
Title: The Geometry of Generic Lagrangian Fibres: An Illustrating
Example
Abstract: The aim of the talk is to give an introduction to a strategy
developed by M. Adler, P. van Moerbeke and P. Vanhaecke to study the
geometry of the generic fibre of a Lagrangian fibration induced by an
integrable system.
Title: Chip-Firing and Binomial Ideals
Abstract: Chip-firing on graphs has been studied for nearly 25 years
since its independent introductions in statistical physics and graph
theory.
Recently, it has received some attention for its connections to the
divisor theory of tropical curves and the combinatorics of lattice
ideals. I will give a quick survey of these relationships and briefly
describe my current research in chip-firing via gluing with emphasis on
the binomial ideal case.
11:25 Justin Chen
Title: Regularity of associated graded modules in dimension one
Abstract: Following a recent paper by Dung (arXiv: 1209.3469v1), a
(sharp) bound for the regularity of the associated graded module of a
one-dimensional module is given, along with characterizations for when
equality is attained. I will outline proofs of these bounds, and the
extremal cases.
Title: Splittings for Rings of Modular Invariants
Abstract: Rings of polynomial invariants of finite group actions are
among the most classical objects in commutative algebra. There are many
beautiful theorems ensuring that the invariant ring has good properties
when the order of the group is invertible. However, if the order of
the group is not a unit (i.e., is divisible by the characteristic of
the ground field), many of these properties become more subtle.
In this talk, I aim to illustrate some of the differences in invariant
theory in this setting, and to describe some of my work in progress in
this
area.
11:25 Jonathan Montaño
Title: j-multiplicity: A survey
Abstract: The j-multiplicity was introduced by Achiles and Manaresi
in 1993 as a
generalization of the Hilbert-Samuel multiplicity for arbitrary
ideals
in a Noetherian ring. Many of the properties and algebraic
applications of the Hilbert-Samuel multiplicity of zero dimensional
ideals have been extended to more general classes of ideals using
the
j-multiplicity.
In this talk, I will review some of these properties and
applications.
At the end of the talk, I will briefly discuss my current research
in
this area.
November 27
10:45 Ali Alilooee
Title: Rees Algebras of Some Classes of Simplicial Complexes
Abstract: In 1995, Villarreal gave a combinatorial description of the
defining ideals of Rees algebras of quadratic square-free monomial
ideals. In this paper we will generalize his results for hypergraphs.
Our approach is based on giving a definition of closed even walks in a
simplicial complex. We apply this combinatorial method to square-free
monomial ideals of higher dimension.
Title: Green's Hyperplane Restriction Theorem
Abstract: In this talk, we will give an introduction to Green's
hyperplane restriction theorem, that gives a bound on the Hilbert
function of the restriction of a symmetric algebra to a generic
linear form. Moreover, we will talk about the generalization of this
theorem to modules.
December 11
10:45 Emma Connon
Title: When do monomial ideals have linear resolutions?
Abstract: In 1990 Fröberg showed that the edge ideal of a graph
has a
linear
resolution if and only if the complement of the graph is chordal. In
this talk we will discuss the generalization of Fröberg's theorem
to
higher dimensions. In particular we will discuss new classes of
simplicial complexes which extend the notion of a chordal graph and
which give rise to a necessary condition for an ideal to have a linear
resolution over any field. We will also provide a necessary and
sufficient combinatorial condition for a square-free monomial ideal to
have a linear resolution over fields of characteristic two.
Title: The combinatorics of toric ideals of hypergraphs
Abstract:
The edge subring of a hypergraph H is the monomial subalgebra
parameterized
by the hyperedges of H. Its defining ideal is a toric ideal which we
can
understand by studying the combinatorics of H. In this talk we will
survey
recent results on the toric ideals of hypergraphs with a particular
focus on
the combinatorics of minimal generators.
Title: Ekedahl invariants for Finite Groups
Abstract:
In 2009 T. Ekedahl introduced some cohomological invariants for
a finite group G. These relate, naturally, to invariant theory for
groups
and, also, to the Noether's Problem (one wonders about the rationality
of
the extension F(G) = F(x_g: g in G)^G over F, for a field F and a finite
group G).
In this talk, we introduce these invariants and we highlight some
results.