TTU Algebra Seminar

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Friday 4 December, 4–5 pm  in MATH 114

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.

TTU Department
of Mathematics
and Statistics

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.

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 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.

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. 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.