3 Dec. |
Lourdes Juan
"A normal basis theorem in differential Galois theory" |
| Abstract. |
We will discuss a differential counterpart of the
normal basis theorem in classical Galois theory and
see how this result leads to a new characterization of
differential Galois extensions. |
21 Nov. |
|
| Abstract. |
Let $R$ be a Noetherian local ring with infinite
residue field $k$ and $I$ an $R$-ideal. The ideal $J$
is a \textit{reduction} of $I$ if $J \subset I$ and
$I^{r+1}=JI^{r}$ for some positive integer $r$. A
reduction can be thought of as a simplification of the
ideal $I$. The notion of a reduction for an ideal was
introduced by D. Northcott and D. Rees in order to
study multiplicities. Reductions are connected to the
study of blowup algebras such as the Rees ring
$\mathcal{R}(I)=R[It]$ of $I$, and the associated
graded ring ${\rm{gr}}_{I} (R)=R[It]/IR[It]$ of $I$.
In general minimal reductions are not unique. To
remedy this lack of uniqueness, one considers the
intersection of all reductions, namely the
\textit{core} of the ideal, ${\rm{core}}(I)$. This
object, that appears naturally in the context of the
Brian\c con-Skoda theorem, encodes information about
all possible reductions. We present an introduction as
well as some recent work on the shape of the core of
ideals. |
14 Nov. |
|
| Abstract. |
In this talk we consider homogeneous ideals I in a
polynomial ring over a field. The Hilbert function of
I is a sequence of non-negative integers which gives
the dimensions of the graded pieces of I
degree-by-degree. Hilbert functions have played a
central role in many algebraic problems. Indeed, many
people have obtained methods to extract non-trivial
information about an ideal from its Hilbert function.
Thus it is natural to ask what a Hilbert function
looks like. A famous theorem due to Macaulay has
characterized which sequences arise as Hilbert
functions of homogeneous ideals. In this talk we will
survey some of the generalizations and related
conjectures of Macaulay's Theorem. |
7 Nov. |
|
| Abstract. |
For a ring R, we investigate "minimal" right
essential overrings (called right ring hulls)
belonging to certain classes of rings which are
generated by R and subsets of the central idempotents
of Q(R), where Q(R) is the maximal right ring of
quotients of R. We show the existence of a quasi-Baer
hull and a right FI-extending hull for every semiprime
ring and explicitly describe these hulls. Our results
are used to obtain a complete characterization for an
intermediate C*-algebra between a C*-algebra A and its
local multiplier C*-algebra M_loc(A) to be boundedly
centrally closed. (This research was done jointly
with Jae K. Park and S. Tariq Rizvi). |
31 Oct. |
Aaron Adcock "Vector Invariants of Elementary Abelian p-Groups" |
| Abstract. |
Consider a faithful representation G --> GL(n,F) of
a finite group G over a field F. It induces an action
of the group on the vector space V = F^n, thus on the
dual space, and hence on the symmetric algebra on the
dual space, denoted by F[V ]. The subring of invariant
polynomials is denoted by F[V ]^G. If n = 2 and F a
finite field of characteristic 2 and order q = 2^s,
then a 2-Sylow subgroup G of GL(2; F) consists of all
upper triangular matrices with 1's on the diagonal.
This is then an elementary abelian 2-group of rank
s. Its invariants form a polynomial ring. We are
interested in the n-fold vector invariants of this
representation. |
24 Oct. |
Jason Parker "The Transfer Homomorphism" |
| Abstract. |
The transfer is useful in constructing invariants
in the nonmodular case, but it is less nice in the
modular case, and not as much is known about its
image. The conjecture I am working on is that, for an
arbitrary P-group, the image of the transfer is a
principle ideal if and only if the ring of invariants
is polynomial. To lead up to this conjecture, much of
the talk will focus on introducing the very basics,
such as the ring of invariants and the transfer
homomorphism, with examples and basic properties, and
a particular example that illustrates the conjecture
nicely. |
17 Oct. |
Hamid Rahmati (U. Nebraska-Lincoln) "Contracting endomorphisms and Gorenstein modules" |
| Abstract. |
A finite module $M$ over a noetherian local ring
$(R, \mathfrak m, k)$ is said to be Gorenstein if $\rm
{Ext}_R^i(k,M)=0$ for all $i \ne \dim R$. An
endomorphism $\varphi\colon R \to R$ of rings is
called contracting if $\varphi ^i(\mathfrak m)
\subseteq \mathfrak m^2$ for some $i \geq 1$. Letting
$S $ denote the $R$-module $R$ with action induced by
$\varphi$, we prove: A finite $R$-module M is
Gorenstein if and only if $\rm {Hom}_R(S,M) \cong M$
and $\rm {Ext}_R^i(S,M) = 0$ for $1 \leq i \leq \depth
R$. |
10 Oct. |
Arne Ledet "Puiseux series" |
| Abstract. |
We cover the basic theory of Puiseux series, up to
and including the theorem that the field of all
complex Puiseux series is algebraically closed. The
talk is elementary. |
3 Oct. |
Prof. David Weinberg "Singularities of Algebraic Curves III" |
| Abstract. |
An equivalence relation, based on Puiseux
expansions, for classifying singular points of plane
algebraic curves will be described. The
classification of singular points of quartic and
quintic curves will then be described. |
26 Sep. |
Prof. David Weinberg "Singularities of Algebraic Curves II" |
| Abstract. |
An equivalence relation, based on Puiseux
expansions, for classifying singular points of plane
algebraic curves will be described. The
classification of singular points of quartic and
quintic curves will then be described. |
19 Sep. |
Prof. David Weinberg "Singularities of Algebraic Curves I" |
| Abstract. |
An equivalence relation, based on Puiseux
expansions, for classifying singular points of plane
algebraic curves will be described. The
classification of singular points of quartic and
quintic curves will then be described. |
12 Sep. |
Seminar canceled |
5 Sep. |
Mara Neusel "Classical representation theory of finite groups II" |
| Abstract. |
In this sequence of lecture I will present an
overview over the classical representation theory of
finite groups. I will start with explaining what a
representation is, then why group theory IS
representation theory, then I will classify the
representations, explain characters, and their
properties. |
29 Aug. |
Mara Neusel "Classical representation theory of finite groups I" |
| Abstract. |
In this sequence of lecture I will present an
overview over the classical representation theory of
finite groups. I will start with explaining what a
representation is, then why group theory IS
representation theory, then I will classify the
representations, explain characters, and their
properties. |
