We will be taking a look at pseudoreflections and
pseudoreflection groups and discussing some of
their defining properties.

16 Nov.

Sylvia Wiegand (U. Nebraska) "Building
examples using power series"

Abstract.

In ongoing work with William Heinzer and Christel
Rotthaus over the past twenty years we have been
applying a construction technique for obtaining
sometimes baffling, sometimes badly behaved,

sometimes Noetherian, sometimes non-Noetherian
integral domains. This technique of intersecting
fields with power series rings goes back to
Akizuki-Schmidt in the 1930s and Nagata in the
1950s, and since then has also been employed by
Nishimuri, Heitmann, Ogoma, the authors and
others.

We are writing a book about our procedures and
examples. As we write we are streamlining our
results and improving them. We present some of
the theory and techniques we use, and mention
some examples.

12 Nov.

Taylor Dupuy (U. New Mexico)
"Parameterizing Lifts of the Frobenius"

9 Nov.

Luis Garcia-Puente (Sam Houston State
U.) "The control polyhedron of a rational
Bézier surface"

Abstract.

The work presented in this talk lies in the
interplay between geometric modeling and algebraic
geometry. Algebraic geometry investigates the
algebraic and geometric properties of polynomials.
Geometric modeling uses polynomials to build
computer models for industrial design and

manufacture from basic units, called patches, such
as, Bézier curves and surfaces. Bézier curves are
governed by their control points. The polygon
formed by connecting the control points with line
segments is called the control polygon. This
polygon is unique and determines many important
features of the curve, thus validating its name.

A Bézier surface is also intuitively governed by
control points; in particular, the surface lies
within the convex hull of its control points. This
convex hull is often indicated by drawing some
edges between the control points, the resulting
structure is called a "control mesh". Unlike
curves, there is no unique choice of control mesh
for a surface. So it is not clear in which way
these meshes "control" the Bézier surface. In this
talk, we will present one possible answer to this
question. Our results rely upon the geometry of
toric varieties.

This is joint work with Frank Sottile and Chungang Zhu.

2 Nov.

Mara Neusel "Orbit Chern Classes"

Abstract.

This will be an educational lecture about orbit
Chern classes; what they are and how we use them in invariant theory
of finite groups.

26 Oct.

No seminar (Red Raider Mini-Symposium)

Abstract.

This will be an educational lecture about orbit
Chern classes; what they are and how we use them in invariant theory
of finite groups.

19 Oct.

No seminar (public lecture)

12 Oct.

No seminar

5 Oct.

No seminar (faculty meeting)

28 Sep.

Chris Monico "Factoring with large prime variations"

Abstract.

Several modern integer factorization algorithms
including the quadratic sieve and number field
sieve work by producing a large number of perfect
squares whose residues modulo N are smooth. In
practice, squares which almost smooth are used as
well, but until now the behavior of this

variation
has not been carefully studied. After a quick
sketch of the quadratic sieve, we present a graph
theoretic model of the 2-large prime variation and
our analysis, which amounts to determining the
expected dimension of the cycle space of a
particular kind of random graph.

21 Sep.

Arne Ledet "Galois theory for commutative rings II"

Abstract.

The concept of a (finite) Galois extension can be
generalised from fields to arbitrary commutative
rings. While these general Galois extensions are
of course not as well-behaved as the classical
Galois extensions of fields, they are nevertheless
fairly nice. For instance: In the general

case, the Galois
group is not determined by the ring extension, but
has to specified. On the other hand, if \(S/R\) is
a Galois extension with group \(G\), then \(S\) is
a finitely generated faithful projective R-module
of rank \(|G|\). We will go over the basic theory
of Galois extensions of commutative rings, with
various examples to illustrate the ideas.

14 Sep.

Arne Ledet "Galois theory for commutative rings I"

Abstract.

The concept of a (finite) Galois extension can be
generalised from fields to arbitrary commutative
rings. While these general Galois extensions are
of course not as well-behaved as the classical
Galois extensions of fields, they are nevertheless
fairly nice. For instance: In the general

case, the Galois
group is not determined by the ring extension, but
has to specified. On the other hand, if \(S/R\) is
a Galois extension with group \(G\), then \(S\) is
a finitely generated faithful projective R-module
of rank \(|G|\). We will go over the basic theory
of Galois extensions of commutative rings, with
various examples to illustrate the ideas.

7 Sep.

Lars Winther Christensen
"Products in Koszul homology II"

Abstract.

In this talk I will discuss how to classify
quotients of the polynomial algebra in three
variables based on the multiplicative structure of
their Koszul homology. In particular, I will show
an example of an algebra structure that had been
conjectured not to occur.

31 Aug.

Lars Winther Christensen
"Products in Koszul homology I"

Abstract.

In this talk I will discuss how to classify
quotients of the polynomial algebra in three
variables based on the multiplicative structure of
their Koszul homology.

Spring 2012

4 May

Alastair Hamilton
"Differential graded Lie algebras and deformation theory"

Abstract.

In this talk I plan to describe the role that the
cohomology of differential graded Lie algebras
plays in the deformation theory of algebraic
objects.

26 Apr.

No seminar

20 Apr.

Brian Miller
"On the Integration of Algebraic Functions: Computing the Logarithmic Part II"

Abstract.

We provide an algorithm to compute the logarithmic
part of an integral of an algebraic function. The
algorithm finds closed-form solutions that have
been difficult to compute by other means.

We are
able to fully justify the algorithm using
techniques of Gröbner bases, differential
algebra, and algebraic geometry.

13 Apr.

Brian Miller
"On the Integration of Algebraic Functions: Computing the Logarithmic Part I"

Abstract.

We provide an algorithm to compute the logarithmic
part of an integral of an algebraic function. The
algorithm finds closed-form solutions that have
been difficult to compute by other means.

We are
able to fully justify the algorithm using
techniques of Gröbner bases, differential
algebra, and algebraic geometry.

6 Apr.

Lars Winther Christensen
"Koszul complexes in Commutative Algebra"

Abstract.

The Koszul complex on a sequence of elements
\(x_1,\ldots,x_n\) in a
commutative ring is the

exterior algebra on a free
module F of rank n endowed with a differential
defined in terms of the canonical map from F to
the ideal generated by \(x_1,\ldots,x_n\).
However,
the complex also arises in numerous other ways
that may seem more natural to a ring theorist.

In the talk I will discuss the definition in
details and give several examples of natural
constructions in commutative algebra that lead to
the same gadget.

30 Mar.

Arne Ledet
"Lie algebras in algebraic geometry II"

Abstract.

The theory of linear algebraic groups/affine group
varieties in algebraic geometry is an algebraic
analogue of the theory of Lie groups in
differential geometry. In these talks, we will
describe

how Lie algebras show up in
the algebraic setting.

The second talk will be on affine group varieties
and their associated Lie algebras.

23 Mar.

Arne Ledet
"Lie algebras in algebraic geometry I"

Abstract.

The theory of linear algebraic groups/affine group
varieties in algebraic geometry is an algebraic
analogue of the theory of Lie groups in
differential geometry. In these talks, we will
describe

how Lie algebras show up in
the algebraic setting.

The first talk will be mostly a quick run-through
of affine varieties, regular points and tangent
spaces.

16 Mar.

No Seminar; Spring Break

5 Mar.

Amanda Croll (U. Nebraska) "Torsion in
the Poincaré module of a Gorenstein local
ring"

Abstract.

Let R be a local commutative Noetherian ring, and
let F be the free $\mathbb{Z}[t,t^{-1}]$-module
with basis the set of isomorphism classes of
finitely generated R-modules. The Poincaré

module
associated to R is defined to be F/I, where I is
generated by certain relations given by short
exact sequences containing a projective R-module
and split short exact sequences. We give a
characterization of the torsion elements in the
Poincaré module of a Gorenstein local ring and an
example which shows that this characterization
does not extend to the general case.

2 Mar.

Raymond
Hoobler (CCNY) "Deformation theory of Azumaya algebras"

Abstract.

I will discuss the differential Brauer group of a
differential ring R. This requires a procedure for
extending partial derivatives of R to partial
derivatives of an Azumaya algebra A over R.

A
cohomological interpretation is then provided by
piecing together local solutions to differential
equations to get a global cohomology class. how Lie algebras show up in
the algebraic setting.

24 Feb.

No Seminar

17 Feb.

Jeff Lee
"Where do Lie algebras come from? II"

Abstract.

We review various elementary aspects of the way in
which Lie Algebras are associated to Lie
groups. We give a simple rough and ready way to
identify the Lie group associated
to matrix

groups. We will try
to gain some insight into the notion that the Lie
algebra of a Lie group captures information about
the group. We will also say something about the
adjoint representation of the group and the
corresponding adjoint representation of its Lie
algebra. If time permits we will touch on the
phenomenon of "spin".

10 Feb.

Jeff Lee
"Where do Lie algebras come from? I"

Abstract.

We review various elementary aspects of the way in
which Lie Algebras are associated to Lie
groups. We give a simple rough and ready way to
identify the Lie group associated
to matrix

groups. We will try
to gain some insight into the notion that the Lie
algebra of a Lie group captures information about
the group. We will also say something about the
adjoint representation of the group and the
corresponding adjoint representation of its Lie
algebra. If time permits we will touch on the
phenomenon of "spin".

3 Feb.

Lars Winther Christensen
"Abelian vs. triangulated categories II"

Abstract.

An abelian category is "just" a subcategory of a
module category. A triangulated category can be
something much more abstract, but it need not
be. I these lectures I will discuss definitions
and examples of both abelian and triangulated
categories and who how one can naturally lead to
the other.

27 Jan.

Lars Winther Christensen
"Abelian vs. triangulated categories I"

Abstract.

An abelian category is "just" a subcategory of a
module category. A triangulated category can be
something much more abstract, but it need not
be. I these lectures I will discuss definitions
and examples of both abelian and triangulated
categories and who how one can naturally lead to
the other.

20 Jan.

Clyde Martin
"Introduction to Lie Algebras" (joint with the Lie Algebra Reading Seminar)

Abstract.

This is the first meeting of the Lie Algebra
Reading Seminar, which will ordinarily meet on
Mondays and Wednesdays 4–5 pm in
MA 013. Come and see if you are interested
in this reading seminar.

Fall 2011

2 Dec.

No seminar; colloquium

25 Nov.

No seminar; thanksgiving

18 Nov.

Lars Winther Christensen and Arne Ledet
"Lie Algebra reading seminar, organizational
meeting"

Abstract.

In Spring 2012 we propose to organize a reading
seminar on Lie algebras. The purpose of this
organizational meeting is to agree on topics and a
text.

Suppose G is
a finite group. The most basic and interesting
representation of G is the regular

representation, by
left translation lambda on the space C[G] of
complex-valued functions on G. This is the sum of all
the irreducible representations of G, each
representation appearing with multiplicity equal to
its dimension:

(lambda,C[G]) = \sum_{pi irreducible} dim(pi) pi.

Gelfand (on the basis of many interesting examples)
posed the problem of finding a "model representation" of
G: some natural representation (mu,M) with the property
that

(mu,M) = \sum_{pi irreducible} pi.

The first issue is to find a natural space whose
dimension is the sum of the dimensions of the
irreducible representations of G.

Frobenius and Schur showed that every irreducible
representation pi of G falls into exactly one of three
cases:

pi is real (meaning it is the complexification of a
representation on a real vector space)

pi is quaternionic (meaning that the
complex vector space structure can be extended
to a quaternionic vector space structure
respected by G)

4 Nov.

David Weinberg "Symmetry II"

Abstract.

The group theoretic foundations of
symmetry will be described. Included will be a
detailed discussion of isometries of the plane, and
possibly a theorem of Leonardo da Vinci on finite
subgroups of O_{2}.

28 Oct.

David Weinberg "Symmetry I"

Abstract.

The group theoretic foundations of
symmetry will be described. Included will be a
detailed discussion of isometries of the plane, and
possibly a theorem of Leonardo da Vinci on finite
subgroups of O_{2}.

21 Oct.

No seminar

14 Oct.

No seminar

7 Oct.

No seminar

30 Sep.

Valiantsina Laurushchyk "Orbit Chern
Classes: characteristic subalgebra and the splitting principles"

Abstract.

Let G be a Finite group, F be a
Field, and r: G --> GL(n,F) a representation of
G. The group G

acts via r on the algebra
F[V] of polynomial functions on the representation
space V . The main object of the study is the ring
of invariants F[V]^{G}. The orbit Chern
classes are the elementary symmetric polynomials
in the elements of an orbit of G acting on the
space of linear forms V* regarded as elements of
F[V]^{G}. In many cases, the orbit Chern
classes are sufficient to generate the ring of
invariants as an algebra. In this talk I will
discuss the new results related to this
subject. In particular, I will go over the
Stone-Smith Theorem for finite fields and show
that their result is also true for infinite
fields. I will also discuss the equivalence of
Weak and Strong Splitting Principles.

23 Sep.

Arne Ledet "Irreducible and completely reducible modules II"

Abstract.

An irreducible module is a non-trivial module with
only two submodules, namely 0 and
itself.

A completely reducible
module is a module built from irreducible
submodules. It turns out that completely reducible
modules are very well-behaved. We will look at the
properties of completely reducible modules, and
study those rings for which all modules are
completely reducible.

16 Sep.

Arne Ledet "Irreducible and completely reducible modules I"

Abstract.

An irreducible module is a non-trivial module with
only two submodules, namely 0 and
itself.

A completely reducible
module is a module built from irreducible
submodules. It turns out that completely reducible
modules are very well-behaved. We will look at the
properties of completely reducible modules, and
study those rings for which all modules are
completely reducible.

9 Sep.

Lars Winther Christensen "Envelopes and covers III"

Abstract.

Approximations of unknown objects by better known
ones is recurring theme in applications

of
homological algebra within module theory. In these
talks I will discuss what it means to approximate
efficiently or minimally.

2 Sep.

Lars Winther Christensen "Envelopes and covers II"

Abstract.

Approximations of unknown objects by better known
ones is recurring theme in applications

of
homological algebra within module theory. In these
talks I will discuss what it means to approximate
efficiently or minimally.

26 Aug.

Lars Winther Christensen "Envelopes and covers I"

Abstract.

Approximations of unknown objects by better known
ones is recurring theme in applications

of
homological algebra within module theory. In these
talks I will discuss what it means to approximate
efficiently or minimally.

Spring 2011

29 April

Lourdes Juan "Fields with multiple
derivations (continued)"

Abstract.

I will go over the differential Galois theory in
the case when multiple derivations exist and an
interesting problem regarding differential Azumaya
algebras.

22 April

Kosmas Diveris (Syracuse U.) "On the eventual vanishing of self-extensions"

Abstract.

The AC condition concerning the vanishing of
cohomology over a ring originates from the

work of
Auslander. More recently Christensen and Holm
have shown that several longstanding homological
conjectures hold for rings having the AC
condition. In this talk we define a new condition
that generalizes the uniform AC condition. We show
that many of the known results for AC rings hold
for rings having our condition. We will also
discuss some examples of rings for which our
condition holds and some where it fails to hold.

15 April

Alastair Hamilton "Moduli spaces of
Riemann surfaces and the cohomology of Lie
algebras II"

Abstract.

There is a theorem, due to Kontsevich, which
states that the cohomology of the moduli
space

of Riemann surfaces can be
equivalently expressed as the cohomology of a
certain infinite-dimensional Lie algebra. In these
two talks I plan to explain this theorem and how
it may be extended to certain compactifications of
the moduli space. I will also, time permitting,
describe how this theorem can be used to produce
classes in the moduli space from purely algebraic
data, and how the problem of lifting classes to
compactifications of the moduli space may be
formulated in terms of algebraic deformation
theory.

8 April

Alastair Hamilton "Moduli spaces of
Riemann surfaces and the cohomology of Lie
algebras I"

Abstract.

There is a theorem, due to Kontsevich, which
states that the cohomology of the moduli
space

of Riemann surfaces can be
equivalently expressed as the cohomology of a
certain infinite-dimensional Lie algebra. In these
two talks I plan to explain this theorem and how
it may be extended to certain compactifications of
the moduli space. I will also, time permitting,
describe how this theorem can be used to produce
classes in the moduli space from purely algebraic
data, and how the problem of lifting classes to
compactifications of the moduli space may be
formulated in terms of algebraic deformation
theory.

1 April

No seminar

25 Mar.

Lourdes Juan "Fields with multiple derivations"

Abstract.

I will go over the differential Galois theory in
the case when multiple derivations exist and an
interesting problem regarding differential Azumaya
algebras.

We will be discussing Kryptos; A statue at the CIA
building in Langley that has encoded text

written
on it that so far no one has been
able to fully decode. It remains one of the
biggest mysteries for many cryptographers. We will
discuss the history of it and how to decode the
first three parts. Also why the fourth part has
not been decoded, some hints that have been given,
and why some methods couldn't possibly work.

18 Feb.

No seminar;
Texas Geometry and Topology Conference

11 Feb.

Lars Winther Christensen "Tate (co)homology via pinched complexes"

Abstract.

Let R be a ring. For complexes of R-modules we
introduce two constructions, which we call

the pinched tensor product
complex and the pinched Hom complex. Our
motivation for studying these constructions is
their connections to Tate (co)homology, and I will focus on the balancedness
question. The talk is based on joint with David
A. Jorgensen.

4 Feb.

No seminar

28 Jan.

Arne Ledet "Generic polynomials for quaternion groups"

Abstract.

We will consider the preliminary steps in a
construction of generic polynomials for the
family

of quaternion groups over
fields satisfying suitable conditions. (I.e.,
those fields where such a construction is
possible.)

21 Jan.

David Weinberg "The Weierstrass
Preparation Theorem II"

Abstract.

The Weierstrass Preparation Theorem describes how
a power series can be "prepared" for the study of
its zeros. The proof will also be discussed.

14 Jan.

David Weinberg "The Weierstrass
Preparation Theorem I"

Abstract.

The Weierstrass Preparation Theorem describes how
a power series can be "prepared" for the study of
its zeros. The proof will also be discussed.

Fall 2010

3 Dec.

Valiantsina Laurushchyk "Orbit Chern Classes in Invariant Theory"

Abstract.

Let G be a
finite group, F be a field, and r: G ---> GL(n,F) a
representation of G. The group

G acts via r on the
algebra F[V] of polynomial functions on the
representation space V. The main object of the study is
the ring of invariants F[V]^{G}. The orbit Chern classes
are the elementary symmetric polynomials in the elements
of an orbit of G acting on the space of linear forms V*
regarded as elements of F[V]^{G}. In many cases, the orbit
Chern classes are sufficient to generate the ring of
invariants as an algebra. In this talk I will discuss
the role of Chern classes in Invariant theory, main
results and open questions related to the subject.

26 Nov.

No seminar; Thanksgiving.

19 Nov.

Christopher Monico "The discrete log problem
for elliptic curves"

Abstract.

I will
introduce the Diffie-Hellman key exchange protocol,
elliptic curve groups, and ECC

(Elliptic Curve
Cryptography) which is simply the application of the
latter to the former. I'll then discuss the Pollard-rho
method and other techniques that were used to solve the
ECCp-109 instance several years ago.

12 Nov.

Henrik Holm
(U. Copenhagen) "Rings without a Gorenstein analogue of the Govorov-Lazard theorem"

Abstract.

It was proved by Beligiannis and Krause that over
certain Artin algebras, there are

Gorenstein flat
modules which are not direct limits of finitely
generated Gorenstein projective modules. That is,
these algebras have no Gorenstein analogue of the
classical Govorov-Lazard Theorem. We show that, in
fact, there is a large class of rings without such
an analogue. Namely, let R be a commutative local
noetherian ring. Then the analogue fails for R if
it has a dualizing complex, is henselian, not
Gorenstein, and has a finitely generated
Gorenstein projective module which is not free.

The proof is based on a theory of Gorenstein
projective (pre)envelopes. We show, among other
things, that the finitely generated Gorenstein
projective modules form an enveloping class in the
category of finitely generated R-modules if and
only if R is Gorenstein or has the property that
each finitely generated Gorenstein projective
module is free. This is analogous to a recent
result on covers by Christensen, Piepmeyer,
Striuli, and Takahashi, and their methods are an
important input to our work.

This talk is a report on joint work with Peter
Jørgensen.

5 Nov.

No seminar.

29 Oct.

No seminar; Red Raider Mini-Symposium

22 Oct.

Razvan Gelca "Algebras of curves on surfaces"

Abstract.

I will describe some algebras of curves on
surfaces and show how to study them you use
quantum mechanics, algebraic and differential
geometry and quantum groups.

15 Oct.

Lars Winther Christensen "Vanishing of
Tate homology and depth of tensor products"

Abstract.

To infer properties of a tensor product M \otimes
N from properties the factors M and N is

a delicate task. Auslander's
depth formula for pairs of Tor-independent modules
over a regular local ring, depth(M \otimes N) =
depth(M) + depth(N) - depth(R), has been
generalized in several directions over a span of
decades. In the talk I will describe a formula
that holds for every pair of Tate Tor-independent
modules over a Gorenstein local ring. It subsumes
the previous generalizations of Auslander's
formula.

8 Oct.

No seminar

1 Oct.

Jared Painter (UT Arlington)
"Tendencies of Trivariate Monomial
Resolutions"

Abstract.

We will explore some specific properties admitted
by the free resolutions over $S$ of $R=S/I$

where
$S=k[x,y,z]$, $k$ a field and $I$ is a monomial
ideal. The main focus will be resolutions where
$I$ is primary to the homogeneous maximal ideal,
so that $R$ is Cohen-Macaulay. Specifically, we
will identify certain characteristics of the last
matrix of these resolutions. These
characteristics pertain to the question of whether
the first Bass number of $R$ is always larger than
the zeroth.

24 Sep.

Arne Ledet "Skew Fields of Fractions"

Abstract.

For an integral domain R, the field of fractions
Q(R) is the unique `smallest' field containing
R.

For a non-commutative domain
D, it is not as simple: Sometimes, a unique 'skew
field of fractions' exists, sometimes several
non-unique 'smallest' skew fields containing D
exist, and sometimes D cannot be embedded in a
skew field at all.

17 Sep.

Justin DeVries (U. Nebraska) "The Betti
Number of Multi-graded Differential Modules"

Abstract.

A differential module is a module with a
square-zero endomorphism. They have uses in
the

study of non-exact
complexes, but also display interesting behavior
of their own. We investigate the rank of a
differential module when it has finite length
homology. Using the Betti number of a
differential module we prove a lower bound on the
rank for multi-graded differential modules. This
specializes to a bound on the rank of a complex of
multi-graded free modules with finite length
homology.

10 Sep.

No seminar

3 Sep.

Brian Miller "Groebner Bases in Symbolic
Integration II"

Abstract.

The problem of integration in finite terms is to
decide in a finite number of steps whether a
given

integrand has an elementary
integral, and if it exists, compute it. Although
there is a complete algorithmic solution to the
problem, methods for computing the integral are
still being studied. In fact, all of the current
computer algebra systems contain only a partial
implementation of the so-called Risch algorithm.
In recent work, Czichowski has shown that the
logarithmic part of a rational function in Q(x)
may be computed by a Groebner Base. We give a
brief overview of the problem of integration in
finite terms, methods in symbolic integration, and
show that Czichowski's result can be extended to
arbitrary monomials over an arbitrary differential
field.

27 Aug.

Brian Miller "Groebner Bases in Symbolic
Integration I"

Abstract.

The problem of integration in
finite terms is to decide in a finite number of
steps whether a given

integrand has an elementary
integral, and if it exists, compute it. Although
there is a complete algorithmic solution to the
problem, methods for computing the integral are
still being studied. In fact, all of the current
computer algebra systems contain only a partial
implementation of the so-called Risch algorithm.
In recent work, Czichowski has shown that the
logarithmic part of a rational function in Q(x)
may be computed by a Groebner Base. We give a
brief overview of the problem of integration in
finite terms, methods in symbolic integration, and
show that Czichowski's result can be extended to
arbitrary monomials over an arbitrary differential
field.

Spring 2010

30 Apr.

Chris Monico "The cycle space of graphs"

Abstract.

The cycle space of a graph, Z_{1}(G),
is an F_{2}-vector space associated with
the graph.

We'll briefly discuss the
homology of graph embeddings and then give a
linear-algebraic formulation of the well-known
Cycle Double Cover Conjecture.

23 April

David Weinberg "Classification Issues in
Algebraic Geometry"

Abstract.

A variety of equivalence relations will be discussed.

16 April

Manoj Kummini (Purdue U.) "Regularity of
Ext Modules"

Abstract.

Let S be a polynomial ring and I \subseteq
S a homogeneous ideal. We look at
bounds

for the Castelnuovo-Mumford
regularity of the Ext^i(S/I, \omega_S) modules
where \omega_S is the graded canonical module of
S). This provides some information on the growth
of the Hilbert functions the Ext^i(S/I, \omega_S)
modules, which, in turn, describe the growth of
the Hilbert function of the local cohomology
modules of S/I. We will show that
reg Ext^i(S/I, \omega_S) \leq
dim Ext^i(S/I, \omega_S), when I is a
monomial ideal. This is joint work with
S. Murai.

9 April

Inês
B. Henriques (U. Nebraska) "Homological
study of Exact zero divisors"

Abstract.

We consider local rings (R,m,k) containing
an exact zero divisor, that is an
element a in R such that 0 ≠ R/aR ≅ (0:a)
≠ R. We study the behavior of homological
invariants under the change of rings R → R/aR.

2 April

Kristen Beck (UT Arlington) "On the
Hilbert series of m^{4}=0 local
rings admitting non-trivial totally acyclic
complexes"

Abstract.

Totally reflexive modules are a natural
generalization of projective
modules.

They are the building blocks
of Gorenstein dimension, and their existence is
essential to the computation of Tate (co)homology.
In this talk, we will discuss necessary conditions
for certain local rings to admit non-trivial
totally reflexive modules. In particular, we will
characterize the Hilbert series for a local ring
with m^{4}=0 which admits a totally
reflexive module with linear complete resolution.
This characterization will yield interesting
results for complete resolutions with certain
asymmetry. We will illustrate the results with
several examples, and will also consider some open
questions.

Let R be a commutative local noetherian ring. A
finitely generated R-module M is called
totally

reflexive if it is reflexive
and Ext_R^i(M,R) = Ext_R^i(M^*,R) = 0 for all
i>0. It is known that if a non-Gorenstein ring R
admits a non-free totally reflexive module then
there exist infinitely many, pairwise
non-isomorphic, indecomposable totally reflexive
R-modules. We show, in certain cases, how to
construct an infinite family of indecomposable
totally reflexive modules given that a cyclic one
exists.

19 Mar.

No seminar; spring break

12 Mar.

Lourdes Juan "Differential Central
Simple Algebras and Picard-Vessiot Representations
IV"

Abstract.

See first talk in the series.

5 Mar.

Amelia
Taylor (Colorado College) "Primary
Decomposition of Conditional Independence Ideals"

Abstract.

A conditional independence ideal is an ideal
generated by binomials which
describe

the conditional independence
relations of a statistical model. The primary
decomposition of such ideals is interesting as it
can give an alternate description of the
independence relations for the model or illustrate
the failure of the intersection axiom, among other
applications and interpretations. I will define
conditional independence ideals and then describe
what is currently known about their primary
decompositions and discuss some of the
implications of these primary
decompositions.

26 Feb.

Lourdes Juan "Differential Central
Simple Algebras and Picard-Vessiot Representations
III"

Abstract.

See first talk in the series.

19 Feb.

Lourdes Juan "Differential Central
Simple Algebras and Picard-Vessiot Representations
II"

A classical notion in commutative algebra is that
of {\it height}. Let $R$ be a Noetherian
local

ring of dimension $d$. Every
ideal $I$ of $R$ is generated by $d$ elements up
to radical. The question of determining the height
of ideals in general is then the same as that of
determining the height of an ideal generated by
$d$ elements. We say that a sequenece
$\underline{x}=x_1, \ldots, x_d$ is a system of
parameters if it is has maximal height, namely
${\rm ht} (\underline{x})=d$. Let
$\underline{y}=y_1, \ldots, y_d$ be a sequence
such that $(\underline{y}) \subset
(\underline{x})$, where $\underline{x}$ is a
system of parameters. We will discuss necessary
and sufficient conditions for when $\underline{y}$
is a system of parameters. This is joint work with
Craig Huneke.

5 Feb.

Micah Leamer (U. Nebaska) "Tensor
Products and Tor of Artinian modules over
commutative Noetherian rings."

Abstract.

Standard results in commutative algebra are often
restricted to finitely generated
modules.

However, the class of
Artinian modules also has nice properties which
make them easy to work with. We will explore some
of these basic properties over commutative
Noetherian rings and show that Tor_i(M,N) is
Artinian when M is Artinian and N is either
Artinian or finitely generated. A special case
occurs when i=0 ( Tor_0(M,N) equals M tensor N).
An elementary result shows that the tensor product
of two Artinian modules is finite length. Lastly
we will consider what class of modules we should
expect from Tor(M,N) and Ext(M,N) when M and N
vary among the classes of finitely generated
modules, Artinian modules, Matlis reflexive
modules and modules whose quotient by a finitely
generated module is Artinian.

29 Jan.

No seminar; campus closed.

22 Jan.

Lourdes Juan "Differential Central
Simple Algebras and Picard-Vessiot Representations
I"

Abstract.

A central simple algebra over K is a finite
dimensional associative K-algebra A which is
simple

and for which the center is
exactly K. A well known result establishes that if
K is a differential field then the derivation of K
extends to one of A, making it a differential
algebra. In turn, A is trivialized by a
Picard-Vessiot extension of K. In the matrix
algebra case, there is a correspondence between
K-algebras trivialized by E and representations of
the differential Galois group of E over K in
PGLn(C ), which can be interpreted as cocycles
equivalent up to coboundaries. For the most the
first talk will be a friendly introduction to
central simple algebras. The second one will be
about results in joint work with Andy Magid and
will serve as an introduction to future talks on
my work with Ray Hoobler, extending these results
to differential Azumaya algebras.

15 Jan.

No seminar

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 S_{n} 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.

See first talk in the series.

18 Sep.

Alex Wang "Reducing Elliptic Curve
Logarithm to Logarithm of the Field by Weil
pairing and Tate pairing II"

Abstract.

See first talk in the series.

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.

See first talk in the series.

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 "Spin_{7}"

Abstract.

This is a fairly elementary talk, defining the
spin groups and establishing the eight-dimensional
representation of Spin_{7}.

17 Apr.

Zachariah Teitler (Texas
A&M U.) "Ranks of polynomials"

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 obstructionto $\theta$
vanishing.

13 Mar.

Arne Ledet "The spin group
Spin_{7} 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
Spin_{7}, where we describe the cohomology
and use to parametrize
Spin_{7}-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"

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.

Louiza Fouli (UT Austin) "The Core of
Ideals"

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.

Susan Cooper (Cal Poly and U. Nebraska-Lincoln) "Investigating
Macaulay's Theorem"

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.

Gary Birkenmeier (U. Louisiana at Lafayette)
"Hulls of Semiprime Rings with Applications to
C*-algebras"

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.

See first talk in the series.

26 Sep.

Prof. David Weinberg "Singularities of
Algebraic Curves II"

Abstract.

See first talk in the series.

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.

See first talk in the series.

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.