Fall 2009
4 Dec. |
Lars Winther Christensen "A curious problem on
injective modules"
|
| Abstract. |
Let R be a commutative ring
and S be a
flat R-algebra. Let E be
an R-module and assume that
Hom(S,E) is an
injective S-module; is then E an
injective R-module? I will discuss
partial answers to this question, but only after
explaining why one would even ask such a
question.
|
20 Nov. |
No seminar
|
13 Nov. |
Jesse Burke (U. Nebraska) "Vanishing of self-extensions over a complete intersection"
|
| Abstract. |
Let $R$ be a local complete intersection ring and
$M$ a finite $R$-module. Avramov and Buchweitz
showed that when $\operatorname{Ext}^{2n}_R(M, M)
= 0$ for some $n \geq 1$ then $M$ has finite
projective dimension. We generalize this result by
showing that finiteness is not neccessary. This
follows from a more general result for
complexes. The proof is new and in particular
makes use of Bousfield localization.
|
6 Nov. |
Arne Ledet "Real closed fields"
|
| Abstract. |
A real closed field is a field that behaves very
much like the field of real numbers. I will give a
quick overview over what they are, and in what
sense their behaviour is "very much like" that of the real numbers.
|
30 Oct. |
Petros Hadjicostas "A re-examination of the Diaconis-Graham inequality on the symmetric group"
|
| Abstract. |
Right-invariant metrics on the set of
permutations Sn of the first n positive
integers were introduced by Diaconis and Graham in
1977. Each such metric provides a distance between
two permutations of the first n positive integers in
such a way that, if one changes the order of the
numbers in the two permutations in exactly the same
way, then the distance between the two permutations
stays the
same ... more
|
23 Oct. |
Chris Monico "Introduction to the Number Field Sieve"
|
| Abstract. |
The Number Field Sieve (NFS)
is the best known algorithm for factoring integers. It's
been used to successfully factor integers having more
than 170 digits. It's a mathematically beautiful
algorithm which exploits the Dedekind domain structure
of number fields. In this talk, we'll give an overview
of the NFS.
|
16 Oct. |
No Seminar.
|
9 Oct. |
Hamid Rahmati "Standard systems of parameters and rings with finite local cohomology."
|
| Abstract. |
Let $(R, \mathfrak m)$ be a local commutative
noetherian ring. It is known that the local
cohomology modules $\operatorname H^i_{\mathfrak
m} (R)$, for $i < \operatorname {dim} R$, are
finitely generated if and only if there exists an
integer $n$ such that every system of parameters
$\underline x = x_1, \dots, x_d$ in $\mathfrak
m^n$ is standard, that is to say $\underline x$
satisfies $$(\underline x)\operatorname H
^i_{\mathfrak m}(R/{(x_1,\dots,x_j)}) =0$$ for all
non-negative integers $i,j$ with $i + j < d$. We
give an upper bound for the smallest $n$ with this
property.
|
2 Oct. |
Hamid Rahmati "Artinian Gorenstein rings
and infinite syzygies"
|
| Abstract. |
Let $R$ be a commutative local ring and $M$ be an
$R$-module. We say that $M$ is an infinite syzygy
if there is an exact sequence $0 \rightarrow M
\rightarrow F_1 \rightarrow F_2 \rightarrow \cdots
\rightarrow F_{n-1} \rightarrow F_n \rightarrow
\cdots$, where $F_i$ is free for all $i \geq
1$. The ring $R$ is artinian Gorenstein if and
only if every finitely generated module is an
infinite syzygy. We show that if the embedding
dimension of $R$ is small, one only needs to
verify that the residue field is an infinite syzygy.
|
25 Sep. |
Alex Wang "Reducing Elliptic Curve
Logarithm to Logarithm of the Field by Weil
pairing and Tate pairing III"
|
| Abstract. |
The discrete logarithm for the multiplicative
group of a finite field can be solved in
subexponential time. However currently there is no
subexponential time algorithm to solve the
elliptic curve logarithm generally. Weil pairing
and Tate pairing are two methods which can reduce
the elliptic curve logarithm to logarithm of the
field. In this talk, we will use a very
elementary way to introduce Weil and Tate pairings.
|
18 Sep. |
Alex Wang "Reducing Elliptic Curve
Logarithm to Logarithm of the Field by Weil
pairing and Tate pairing II"
|
| Abstract. |
The discrete logarithm for the multiplicative
group of a finite field can be solved in
subexponential time. However currently there is no
subexponential time algorithm to solve the
elliptic curve logarithm generally. Weil pairing
and Tate pairing are two methods which can reduce
the elliptic curve logarithm to logarithm of the
field. In this talk, we will use a very
elementary way to introduce Weil and Tate pairings.
|
11 Sep. |
Alex Wang "Reducing Elliptic Curve
Logarithm to Logarithm of the Field by Weil
pairing and Tate pairing I"
|
| Abstract. |
The discrete logarithm for the multiplicative
group of a finite field can be solved in
subexponential time. However currently there is no
subexponential time algorithm to solve the
elliptic curve logarithm generally. Weil pairing
and Tate pairing are two methods which can reduce
the elliptic curve logarithm to logarithm of the
field. In this talk, we will use a very
elementary way to introduce Weil and Tate pairings.
|
4 Sep. |
Lars Winther Christensen "Totally
reflexive modules II"
|
| Abstract. |
Totally reflexive modules are manifestations of
solutions to infinite families of coupled systems of
linear equations. Even when their abundance is
assured by abstract arguments, it can be hard to get
ones hands on them. In these talks I will focus on
concrete examples and constructions and advertise
some open problems.
|
28 Aug. |
Lars Winther Christensen "Totally
reflexive modules I"
|
| Abstract. |
Totally reflexive modules are manifestations of
solutions to infinite families of coupled systems of
linear equations. Even when their abundance is
assured by abstract arguments, it can be hard to get
ones hands on them. In these talks I will focus on
concrete examples and constructions and advertise
some open problems.
|
Spring 2009
23 Apr. |
Arne Ledet "Spin7" |
| Abstract. |
This is a fairly elementary talk, defining the
spin groups and establishing the eight-dimensional
representation of Spin7.
|
17 Apr. |
|
| Abstract. |
The Waring problem for polynomials asks how to
write a homogeneous polynomial of degree d as a sum of
dth powers of linear polynomials. The rank of a
polynomial is the least number of terms in such an
expression. The problem of finding the rank of a given
polynomial and studying rank in general has been a
central problem of classical algebraic geometry,
related to secant varieties; in addition, there are
applications to signal processing and computational
complexity. In 1916, Macaulay gave a lower bound for
rank in terms of catalecticant matrices. In the almost
100 years since, there has been relatively little
progress on the problem of determining or bounding
rank (although related questions have proved very
fruitful). I will describe new upper and lower bounds,
with especially nice results for some examples
including monomials and cubic polynomials. This is
joint work with J.M. Landsberg. |
10 Apr. |
Lars Winther Christensen
"Floor plans in local algebra" |
| Abstract. |
In a paper from 2003, Schoutens proved that every
module over a commutative local ring can be built from
the prime ideals in the singular locus using a few
simple constructions, or "moves". Schoutens'
work provides an upper bound for the number of moves
required to build the entire module category. In the
talk I will show exactly how many moves are required.
|
3 Apr. |
No seminar |
27 Mar. |
Frank Moore (Cornell U.) "Hochster's theta function and graded isolated singularities" |
| Abstract. |
In 1981, Hochster introduced the $\theta$ function in
his study of the Direct Summand Conjecture. The
$\theta$ function measures asymptotic behavior of
$\Tor_i^R(M,N)$ for a pair of modules $M$, $N$ over an
isolated hypersurface singularity. Work of Hochster
as well as Dao has shown that the vanishing of
$\theta(M,N)$ provides information on questions
regarding dimension and $\Tor$-rigidity.
In joint work with Greg Piepmeyer, Sandy Spiroff and
Mark Walker, we study properties of $\theta$ in the
case of a graded local hypersurface with an isolated
singularity. We find conditions that allows one to
define $\theta$ on some components of the Chow group,
as well as a formula for $\theta$ when the dimensions
of $M$ and $N$ are complimentary. In characteristic
zero, these results allow us to show that a component
of the Chow group is the obstruction to $\theta$
vanishing.
|
13 Mar. |
Arne Ledet "The spin group Spin7 as a differential Galois group" |
| Abstract. |
If M/K is a Picard-Vessiot extension with
differential Galois group G, then M is the function
field over K of a G-torsor. These torsors are
classified by the cohomology H^1(K,G). For some G,
describing this cohomology is fairly trivial. In this
talk, we look at a non-trivial case, namely the spin
group Spin7, where we describe the cohomology and use
to parametrize Spin7-extensions. |
6 Mar. |
Jesse Burke (U. Nebraska) "Connectedness of support varieties" |
| Abstract. |
Support varieties were defined by Carlson for
representations of finite groups, originating out of
work of Quillen in the early 1970's. They have since
spread to many different areas of mathematics, serving
key roles in proofs of several important
theorems. Recently Benson, Iyengar and Krause have
defined support varieties in a very general setting that
in addition to specializing to many contexts gives new
tools. In this talk we will give a brief exposition of
different guises of support varieties and then show how
the theory of Benson, Iyengar and Krause specializes to
the case of support varieties for commutative complete
intersection rings. In particular showing how the new
tools can be used to give a conceptually straightforward
proof of a recent connectivity result of Bergh. |
27 Feb. |
Inês Henriques (U. Nebraska) "Cohomology over short Gorenstein rings" |
| Abstract. |
We identify a class of local rings $(R,\mathfrak
m,k)$ with ${\mathfrak m}^4=0$, exhibiting the Koszul
like property that
$\operatorname{H}_{R}(-t)\operatorname{P}_{M}^{R}(t)$
is a polynomial in $\mathbb{Z}[t]$, for all finite
$R$-modules $M$. This class includes generic graded
Gorenstein algebras of socle degree $3$. |
20 Feb. |
Hai Long Dao (U. Kansas) "On geometric and homological properties of algebraic sets" |
| Abstract. |
In 1890, Hilbert proved that any graded module over
a polynomial ring over a field (the coordinate ring of
an affine space) has a finite free resolution. This
was later extended to all regular local rings by
Auslander-Buchsbaum-Serre. Such results hint at a
broader pattern: varieties with nice geometric
properties also enjoy nice homological properties (and
vice versa). This point of view motivates questions
and conjectures which have been studied in Commutative
Algebra and Algebraic Geometry over the last 50
years. In this talk we will survey the history of some
of these questions, as well as recent
developments. |
13 Feb. |
Raymond Dick "An additive characterization of quadratic residues" |
| Abstract. |
In 2006, Monico and Elia gave an additive characterization of quadratic residues in fields of prime order. In
this talk we will discuss our progress in generalizing this result to
all finite fields. |
6 Feb. |
Lourdes Juan "A normal basis theorem in
differential Galois theory II" |
| Abstract. |
In joint work with T. Chinburg and A. Magid we
address the problem of recogizing a Picard-Vessiot
extension E of a differential field F from weaker
information than the structure of E as a differential
field. Our work includes a differential counterpart to the
Normal basis theorem in polynomial Galois theory and the
construction of an invariant that depends on the
differential Galois group. In these talks we will prensent
the main results and provide some examples. |
30 Jan. |
Lourdes Juan "A normal basis theorem in
differential Galois theory I" |
| Abstract. |
In joint work with T. Chinburg and A. Magid we
address the problem of recogizing a Picard-Vessiot
extension E of a differential field F from weaker
information than the structure of E as a differential
field. Our work includes a differential counterpart to the
Normal basis theorem in polynomial Galois theory and the
construction of an invariant that depends on the
differential Galois group. In these talks we will prensent
the main results and provide some examples. |
23 Jan. |
Aaron Lauve (Texas A&M U.) "Noncommutative
invariants and coinvariants of the symmetric
group" |
| Abstract. |
In this talk, classical results on invariants for
the symmetric group S(n) will be extended in two
relatively modern directions. First, I describe the
S(n)-invariants inside the ring of noncommutative
polynomials in n variables. Second, I describe new work
(joint with F. Bergeron) connecting the noncommutative
and commutative invariants for S(n). A surprising
Chevalley-type decomposition occurs that also holds in a
number of related noncommutative settings. Time
permitting, we mention some Hopf algebra connections
tying these different settings together. |
16 Jan. |
Mara Neusel "Representation
theory of symmetric groups II" |
| Abstract. |
See first talk in the series. |
9 Jan. |
Mara Neusel "Representation
theory of symmetric groups I" |
| Abstract. |
The main goal of this series of talks is to describe
all irreducible $\mathbb{C} \Sigma_n$-modules.
For any finite group the number of irreducible complex
representations is equal to the number of conjugacy classes. Since
the conjugacy classes of the symmetric group $\Sigma_n$ are given by
cycle type, they are characterized by the associated partition
$\lambda $ of $n$ and thus can be visualized in Ferrers Diagrams.
From there it is not hard to construct certain permutation modules
$M^{\lambda}$ (for each partition $\lambda$ exactly one) with the
following very nice property: For a suitable ordering $\leq$ on the
set of partitions the modules $M^{\lambda}$ decompose into irreducible
modules $S^{\lambda}$ \[ M^{\lambda} = \oplus_{\mu \leq \lambda}
m_{\mu \lambda} S^{\mu} \] with multiplicities $m_{\mu \lambda}$ where
$m_{\lambda \lambda}=1$. In other words, a module $M^{\lambda}$ is a
direct sum of {\it one} copy of $S^{\lambda}$ and some copies of
$S^{\mu}$ for $\mu < \lambda$. So, for example \[
M^{\lambda}=S^{\lambda} \] is irreducible if (and only if) $\lambda$
is the minimal partition. In case $\lambda$ is the maximal partition
in our ordering $M^{\lambda}$ is the regular representation.
Now, the irreducible modules $S^{\lambda}$ appearing in the direct sum
decomposition above are called {\bf Specht modules}. They form a {\it
complete} set of irreducible $\mathbb{C} \Sigma_n$-modules. We will
describe them explicitly.
If time permits we will discuss the Branching Rule that relates the
Specht modules of $\Sigma_n$ and $\Sigma_{n+1}$. |
Fall 2008
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. |
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. |
|
