# TTU Algebra Seminar

TTU Department
of Mathematics
and Statistics

## Fall 2012

### Jack Byers  "Pseudoreflection Groups"

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

### 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,

### 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

### 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.

### 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.

### 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

### 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

### 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

### 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.

### 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

### 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.

### 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.

### 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.

### 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

### 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

### 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

### 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

### 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.

### 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

### 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

### 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.

### 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.

### 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

### 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.

### David Vogan (MIT)  "Model representations of finite Coxeter groups"

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

### 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 O2.

### 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 O2.

### 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

### 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.

### 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.

### Lars Winther Christensen  "Envelopes and covers III"

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

### Lars Winther Christensen  "Envelopes and covers II"

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

### Lars Winther Christensen  "Envelopes and covers I"

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

## Spring 2011

### 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.

### 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

### 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

### 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

### 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.

### Kristine Seaman  "Kryptos"

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

### 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

### Arne Ledet  "Generic polynomials for quaternion groups"

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

### 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.

### 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

### 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

### Christopher Monico  "The discrete log problem for elliptic curves"

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

### 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

### 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.

### 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

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

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

### 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.

### 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

### 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

### 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

## Spring 2010

### Chris Monico  "The cycle space of graphs"

Abstract. The cycle space of a graph, Z1(G), is an F2-vector space associated with the graph.

### David Weinberg  "Classification Issues in Algebraic Geometry"

Abstract. A variety of equivalence relations will be discussed.

### 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

### 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.

### Kristen Beck (UT Arlington)  "On the Hilbert series of m4=0 local rings admitting non-trivial totally acyclic complexes"

Abstract. Totally reflexive modules are a natural generalization of projective modules.

### Hamid Rahmati  "Constructing totally reflexive modules"

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

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

Abstract. See first talk in the series.

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

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

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

Abstract. See first talk in the series.

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

Abstract. See first talk in the series.

### Louiza Fouli (New Mexico State U.)  "Systems of Parameters in Noetherian Local Rings"

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

### 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.

### 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

## Fall 2009

### 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

### 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

### 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.

### 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

### 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

### 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

### 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

### 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.

### 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.

### 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

### Lars Winther Christensen  "Totally reflexive modules II"

Abstract. See first talk in the series.

### Lars Winther Christensen  "Totally reflexive modules I"

Abstract. Totally reflexive modules are manifestations of solutions to infinite families of coupled systems

## Spring 2009

### Arne Ledet  "Spin7"

Abstract. This is a fairly elementary talk, defining the spin groups and establishing the eight-dimensional representation of Spin7.

### 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

### 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

### 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.

### 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

### Jesse Burke (U. Nebraska)  "Connectedness of support varieties"

Abstract. Support varieties were defined by Carlson for representations of finite groups, originating out

### 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

### 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

### 13 Feb.

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.

### 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

### 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

### Mara Neusel  "Representation theory of symmetric groups II"

Abstract. See first talk in the series.

### 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.

## Fall 2008

### 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.

### 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 $J$

### 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

### 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

### 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

### Jason Parker  "The Transfer Homomorphism"

Abstract. The transfer is useful in constructing invariants in the nonmodular case, but it is less nice in

### 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

### 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.

### Prof. David Weinberg  "Singularities of Algebraic Curves III"

Abstract. See first talk in the series.

### Prof. David Weinberg  "Singularities of Algebraic Curves II"

Abstract. See first talk in the series.

### Prof. David Weinberg  "Singularities of Algebraic Curves I"

Abstract. An equivalence relation, based on Puiseux expansions, for classifying singular points of plane

### Mara Neusel  "Classical representation theory of finite groups II"

Abstract. See first talk in the series.

### 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

Organizer:  Lars Winther Christensen