Algebra & Discrete Mathematics Seminar

Spring 2017 Schedule

25 April 2017

Jessica Striker:
Promotion and rowmotion, revisited

Abstract: We extend results in dynamical algebraic combinatorics connecting the actions of promotion on (standard Young and increasing) tableaux and rowmotion on order ideals to a more general setting. This is joint work with Kevin Dilks and Corey Vorland.

25 April 2017

Ben Noteboom:
Betti Splittings of Monomial Ideals

Abstract: Betti numbers are a widely studied invariant of homogeneous ideals, but can be difficult to calculate, since to find them one must usually construct a minimal free resolution of the ideal. A Betti splitting of an ideal is a tool which allows us to calculate the Betti numbers of an ideal by breaking it up into smaller pieces and finding the Betti numbers of those pieces. In this talk I will discuss how to determine if a splitting of an ideal is a Betti splitting, then I will go over some applications of these splittings. This will mostly follow a paper by Christopher Francisco, Huy Tai Ha, and Adam Van Tuyl.

11 April 2017

Brandon Allen:
Category Theory: String Diagrams

Abstract: One hard concept to handle in the beginning of Category Theory is the natural transformation. However, thanks to the Poincare Dual, we have an efficient way to visualize the relations that the natural transformation has with our categorical diagram known as a string diagram. We shall delve into some basic proofs using string diagrams. The talk we based mostly on about the first 25 pages of the following document listed below.

https://arxiv.org/pdf/math/0307200v3.pdf

4 April 2017

Sara Solhjem:
Semistandard Young Tableaux Polytopes

Abstract: Motivated by the study of the polytopes formed by the convex hull of permutation matrices and alternating sign matrices, we will define two new families of polytopes. The first as the convex hull of certain  $$\{0,1,−1 \}$$ matrices in bijection with semistandard Young tableaux. The second are defined as the convex hull of m by n sign matrices. We investigate various properties of these polytope families, including their inequality descriptions, vertices, and facets.

28 March 2017

Corey Vorland:
Homomesy for $$J([2] \times [a] \times [b])$$ and multidimensional recombination

Abstract: The actions of rowmotion and promotion on order ideals of a poset have generated a significant amount of interest in recent algebraic combinatorics research. One property associated to rowmotion and promotion that might suggest a poset is "nice" is the homomesy property. In this talk, we will discuss homomesy on the product of chains poset using a beautiful technique called recombination.

21 March 2017

Jim Coykendall
(Clemson University): CK-Domains and a Strong Noetherian Property

Abstract: A CK (Cohen-Kaplansky) domain is an atomic integral domain that contains only finitely many irreducible elements. Of course, PIDs with only finitely many primes are examples, but there are myriad other examples (e.g. if $$F$$ is a finite field then any proper subring of $$F[x]$$ generated over $$F$$ by powers of $$x$$ is a CK domain that is not a PID). We will consider some examples and give some relevant (and reasonably new) results on CK-domains. From there we will see how this class of rings naturally segues into a class of domains with a condition that is stronger than the Noetherian property. And in fact, this property lends itself nicely to (certain aspects of) topology.

14 February 2017

Jessica Striker:
Juggling and the stationary state distribution

Abstract:
In a discrete dynamical system, such as a random juggling pattern, a natural question is to find the probability that the system will be in a given state after running for a long time. This is called the stationary state distribution. In this expository talk, I’ll show how to compute the stationary state distribution for random juggling patterns, using only linear algebra, and then mention an idea for a related research project.

Spring 2016 Schedule

26 April 2016

Chelsey Morrow:
Factorization

Abstract: We will cover basic factorization properties and how these properties give "nice" factorization in rings.

12 April 2016

Jessica Striker:
Combinatorial dynamics of monomial ideals

Abstract:
We introduce the notion of combinatorial dynamics on algebraic ideals by translating combinatorial results involving rowmotion and other toggle group actions on order ideals of posets to the setting of monomial ideals. This is joint work with David Cook.

23 February 2016

Trevor McGuire: Resolutions of $$k[M]$$-modules

Abstract: Resolutions of $$k[x_1,...x_n]$$-modules have been widely studied, and in particular, combinatorial methods have been applied to large classes of modules. If we replace the variables with more general objects, the problems get proportionally more difficult. Specifically, we will investigate $$k[M]$$-modules where $$M$$ is a monoid. There are two equally realistic avenues to take with $$M$$, and we will discuss both avenues. The presentation will begin with a review of aforementioned combinatorial methods for the traditional case.

16 February 2016

Cătălin Ciupercă: Reduction numbers of equimultiple ideals (Part II)

9 February 2016

Cătălin Ciupercă: Reduction numbers of equimultiple ideals

Abstract: Let $$(A,\mathfrak{m})$$ be an unmixed local ring containing a field. If $$J$$ is an $$\mathfrak{m}$$-primary ideal with Hilbert-Samuel multiplicity $$\operatorname{e}(J)$$, a recent result of Hickel shows that every element in the integral closure of $$J$$ satisfies an equation of integral dependence over $$J$$ of degree at most $$\operatorname{e}(J)$$. We extend this result to equimultiple ideals $$J$$ by showing that the degree of such an equation of integral dependence is at most $$c(J)$$, which is one of the elements of the so-called multiplicity sequence introduced by Achilles and Manaresi.  As a consequence, if the characteristic of the field contained in $$A$$ is zero, it follows that the reduction number of an equimultiple ideal $$J$$ with respect to any minimal reduction is at most $$c(J)-1$$.

Fall 2015 Schedule

24 November 2015

Susan Cooper:  The Waldschmidt Constant for Monomial Ideals

Abstract:  Understanding the differences between symbolic and regular powers of a homogeneous ideal is quite a challenge, even for monomial ideals.  One technique experts have employed is to consider a special limit called the Waldschmidt constant.  In this talk we will investigate the Waldschmidt constant for monomial ideals using two approaches:  the symbolic polyhedron and a linear program.

3 November 2015

Trevor McGuire: Gröbner Bases, Part II

27 October 2015

Trevor McGuire: Gröbner Bases, Part I

Abstract: In this talk, we will begin only with the idea of an ideal in the ring of polynomials in n variables over a field of characteristic 0. We will build up the terminology needed to define a  Gröbner basis of a given ideal. Topics covered will be term orders, a multivariable division algorithm, and the Buchberger algorithm. This is an expository presentation that is suitable for graduate and undergraduate students.

13 October 2015

Emily Gunawan (University of Minnesota): Cluster algebras from triangulations of surfaces

Abstract: The notion of cluster algebra, introduced by Fomin and Zelevinsky in 2000, links together diverse fields of study, e.g. discrete dynamical systems, Riemann surfaces, representation theory of quivers, knot theory, etc.

Cluster algebras are commutative algebras which are generated by a distinguished set of (usually infinitely many) generators, called cluster variables. Starting from a finite set $$x_1, x_2, \ldots, x_n$$, the cluster variables can be computed by an iterated elementary process. They miraculously turn out to always be Laurent polynomials in $$x_1, x_2,\ldots, x_n$$, with positive coefficients. Finding a closed-form formula for the cluster variables is one of the main problems in the theory of cluster algebra.

In this talk, I will discuss such a closed-form formula for the class of cluster algebras which can be modeled after triangulations of orientable Riemann surfaces with marked points (my running examples will be a pentagon and an annulus). The formula is given in terms of paths (called T-paths) along the edges of a fixed triangulation. The T-paths can be used to give a combinatorial proof for a natural basis (consisting of elements which are indecomposable and positive in some sense) for some types of cluster algebras.

29 September 2015

Jessica Striker: Partition and plane partition promotion and rowmotion

Abstract: In this talk, we discuss promotion and rowmotion actions on partitions and plane partitions, along with their bijective and dynamical properties. We give a previous result on the cyclic nature of rowmotion on partitions inside a box or staircase. We present a recent analog of this result, relating actions on plane partitions and increasing tableaux and exploring a new dynamical pseudo-periodicity phenomenon we call resonance. We welcome comments as to how this work may relate to monomial ideals of two and three variables.

15 September 2015

Nursel Erey: Resolutions of Monomial Ideals

Abstract: The problem to construct the minimal free resolution of a monomial ideal was raised in 1960s. While this problem remains open, much work has been done to understand invariants arising from minimal free resolutions. Thanks to polarization method, one can restrict to squarefree monomial ideals for the purpose of studying minimal free resolutions of monomial ideals.

In this talk, I will discuss the relation of squarefree monomial ideals to combinatorics. In particular, I will explain some combinatorial properties of simplicial complexes which can be used to describe the graded Betti numbers of associated ideals.

8 September 2015

Kevin Dilks: Horn's Conjecture and Related Mathematics

Abstract: A natural question in linear algebra one can ask is, if we only know the eigenvalues for two Hermitian matrices $$A$$ and $$B$$, then what possible eigenvalues can $$A+B$$ have? Trace gives us one equality that has to hold, and the min-max principle gives us some inequalities that have to be satisfied. Horn conjectured the trace condition and a certain finite set of inequalities were both necessary and sufficient, and gave a recursive formula for constructing the inequalities in arbitrarily high dimension. Horn's conjecture was later proven to be true, and it was shown that the inequalities that arise are very closely related to the structure constants that arise in the Grassmannian, symmetric functions, and representation theory of the symmetric group.

In this talk, we will go over the history of Horn's conjecture, describe its connection to the Grassmannian, and discuss how it comes up in other areas of mathematics.

Spring 2015 Schedule

Location: ABEN 215
Time:
Tuesday, 10:00-10:50 a.m.
Organizer: Cătălin Ciupercă

May  2015

Sean Sather-Wagstaff, NDSU:
Do Gorenstein injective filtrations give rise to Gorenstein injective decompositions?

Abstract: No.

31 March  2015

Trevor McGuire, NDSU: Lamé's Theorem and Quadratic Integers

Abstract:
Lamé's theorem is a little-known theorem that relates the Euclidean algorithm on the integers to the Fibonacci sequence.  In this talk, we will rediscover Lamé's result and talk about the inherent shortcomings, including an error in the traditional statement of the theorem.  From there, we will discuss recent work in extending the result to rings of quadratic integers that are Euclidean domains.  Specifically, we will look at the case of Gaussian integers.  If time permits, we will also drop as many more famous mathematicians' names as possible; possibilities include Euler, Noether an Gröbner.

10 March  2015

Jessica Striker, NDSU:
Toggle group actions, applications, and abstraction

Abstract:
The toggle group, a permutation group acting on order ideals of a finite partially ordered set, was defined in 1995 by P. Cameron and D. Fon-der-Flaass and further studied in 2012 by the speaker with N. Williams. In this talk, we explore toggle group actions on many combinatorial objects which can be encoded as order ideals of a poset, such as subsets, partitions, Catalan objects, plane partitions, and alternating sign matrices. We express known actions on these objects as toggle group actions and use conjugacy of toggle group elements to show several remarkable orbit structures.

We apply this perspective to model gyration on fully-packed loops, an action whose analysis by L. Cantini and A. Sportiello let to their proof of the Razumov-Stroganov conjecture, an intriguing link between combinatorics and statistical physics. We prove an instance of the homomesy phenomenon of J. Propp and T. Roby, which when specialized to the setting of the Razumov-Stroganov conjecture, yields a key component of this proof.

Finally, we abstract the notion of the toggle group to any finite set of subsets. Interesting special cases of this general setting include chains and antichains of a poset, independent sets, acyclic subgraphs, and vertex or edge covers of a graph. We conclude with many accessible open problems.

17 February  2015

Sean Sather-Wagstaff, NDSU:
Supports and Direct Sum Decompositions (part II)

10 February  2015

Sean Sather-Wagstaff, NDSU:
Supports and Direct Sum Decompositions (part I)
Abstract: The small support of a module M was introduced by Foxby.  I will discuss a general strategy for showing how conditions on small supports allow one to transform certain filtrations into direct sum decompositions. In a sense, this is a continuation of my talks from last semester and the recent talk by Aaron Feickert; however, I will not assume much (if anything) from these previous talks. This is joint work with Aaron Feickert.

3 February  2015

Aaron Feickert, NDSU: Gorenstein injective decompositions over Cohen-Macaulay rings with a dualizing module
Abstract: Over a noetherian ring, it is a classic result of Matlis that injective modules admit direct sum decompositions into injective hulls of quotients by prime ideals. In this talk, we will show an analogous result using Gorenstein injective modules over a Cohen-Macaulay ring admitting a dualizing module. In particular, we will follow the example of Enochs and Huang to construct filtrations involving these modules, and describe how these filtrations give rise to decompositions. Along the way, we will provide examples to motivate our constructions.

Fall 2014 Schedule

Location: Minard 214
Time:
Tuesday, 10:00-10:50 a.m.
Organizer: Cătălin Ciupercă

9 December  2014

2 December  2014

25 Nobember  2014

18 Nobember  2014

4 Nobember  2014

28 October  2014

7 October  2014

30 September  2014

23 September  2014

16 September  2014

9 September  2014

Spring 2014 Schedule

Location: Minard 210
Time:
Tuesday, 10:00-10:50 a.m.
Organizer: Cătălin Ciupercă

6 May  2014

Cătălin Ciupercă, NDSU:
The concept of multiplicity of a point on an algebraic variety (Part II)

29 April  2014

Cătălin Ciupercă, NDSU:
The concept of multiplicity of a point on an algebraic variety (Part I)

22 April  2014

15 April  2014

25 March  2014

11 March  2014

4 March  2014

25 February  2014

18 February  2014

Fall 2013 Schedule

Location: Minard 308
Time:
Tuesday, 10:00-10:50 a.m.
Organizer: Cătălin Ciupercă

10 December  2013

26 Nobember  2013

19 Nobember  2013

12 November  2013

29 October  2013

Jason Boynton, NDSU

22 October  2013

15 October  2013

1 October  2013

24 September  2013

17 September  2013

10 September  2013

Spring 2013 Schedule

Location: Minard 210
Time:
Tuesday, 10:00-10:50 a.m.
Organizer: Cătălin Ciupercă

2 April  2013
Sean Sather-Wagstaff, NDSU: Rings of Prime Characteristic (part III)

26 March  2013
Sean Sather-Wagstaff, NDSU: Rings of Prime Characteristic (part II)

19 March  2013
Sean Sather-Wagstaff, NDSU: Rings of Prime Characteristic
Abstract:  Let (R,m,k) be a commutative noetherian local ring. A famous result of Auslander, Buchsbaum, and Serre says that R is regular if and only if the residue field k has finite projective dimension. Other ring-theoretic properties (e.g., complete intersection and Gorenstein) have similar characterizations. If R contains a field of characteristic p > 0, then the Frobenius module R' has a similar ability to characterize properties of R. Essentially, R' is a copy of R viewed as an R-module via the ring homomorphism R \to R given by r \mapsto r^p. We will discuss various aspects of this parallel world, beginning with basic computations, and ending with a recent characterization of dualizing complexes which is joint work with Saeed Nasseh.

26 February  2013
Azer Akhmedov, NDSU:  Left-orderable groups with almost free actions

19 February  2013
Azer Akhmedov, NDSU:  Left-orderable groups with almost free actions
Abstract: I will prove a classical result that any countable left-orderable group is isomorphic to a subgroup of Homeo(R). If the action is free, then the group is Archimedean and therefore Abelian. It is also a classical result (Barbot-Kovacevic) that if every non-identity element has at most one fixed point then the group is metaabelian. We'll discuss almost free actions in a more relaxed sense and study the algebraic consequences.

12 February  2013
Azer Akhmedov, NDSU: Left-orderable groups
Abstract: This talk is aimed at a very general audience, and will serve as a preparation for the next talk. We will study basic properties of left-orderable and bi-orderable groups. Many examples will be provided.

Fall 2012 Schedule

Location: Minard 340
Time:
Tuesday, 10:00-10:50 a.m.
Organizer: Cătălin Ciupercă

4 September  2012
Sean Sather-Wagstaff, NDSU: Applications of semidualizing modules
Abstract: In these talks, I will present some applications of semidualizing modules. The point is to explain some motivation for studying these gadgets. Applications will include the following: compositions of ring homomorphisms of finite G-dimension, growth rate bounds for Bass numbers, and structure results for quasi-deformations. I will include plenty of background. In particular, I will not assume any prior background knowledge about semidualizing modules. These talks will be geared toward graduate students.

11 September  2012
Sean Sather-Wagstaff, NDSU: Applications of semidualizing modules II
Abstract: see above.

18 September  2012
Sean Sather-Wagstaff, NDSU: Applications of semidualizing modules III
Abstract: see above.

25 September 2012
Sean Sather-Wagstaff, NDSU: Applications of semidualizing modules IV
Abstract: see above.

2 October 2012  no seminar

9 October  2012
Jessica Striker, University of Minnesota: Toggle group actions on posets
Abstract: In this talk, we introduce the toggle group, which acts on the order ideals of a partially ordered set, or poset. We use the toggle group to model actions on various objects important in combinatorics, including partitions, plane partitions, Dyck paths, and Young tableaux. This perspective allows us to find easy proofs of the cyclic sieving phenomenon, in which the number of objects invariant under a cyclic action is given by specializing the generating function at a certain root of unity. This is based on joint work with Nathan Williams.

16 October  2012
Rich Wicklein, NDSU:  Codualizing Modules
Abstract: We'll define the concepts of semidualizing and quasidualizing modules. We'll then define a codualizing module, which is a common framework for discussing both ideas. We'll look at some examples and some natural questions that arise.
Mark Batell, NDSU: A note on factorization in polynomial rings

30 October  2012
Tom Dunn, NDSU: A Linear Formula for the Generalized Multiplicity Sequence

6 November  2012

Kosmas Diveris,  St. Olaf College: Vanishing of self-extensions over symmetric algebras
Abstract: The Auslander-Reiten (AR) quiver of an Artin algebra is a combinatorial device for organizing the indecomposable modules over the algebra.  In the case of symmetric algebras, the combinatorial structure of this quiver is well suited for investigating modules with eventually vanishing self-extensions.  In fact, one can determine when the vanishing of self-extensions must begin for any such module based on its position in the AR quiver.  In this talk, we will explain how one can use the combinatorial data of the AR quiver to prove such a fact and discuss connections with a conjecture of Auslander and Reiten.  This is based on joint work with Marju Purin of St. Olaf College.

Spring 2012 Schedule

Location: Morrill 109
Time: Tuesday, 10:00-10:50 a.m.
Organizer: Cătălin Ciupercă

1 May 2012
Josef Dorfmeister, NDSU:
Calc III meets Homological Algebra
Abstract: I will show what Green's Theorem, Stokes' Theorem and Gauss' Theorem of Calc III fame have to do with chain maps and (co-)homology. I will define DeRham cohomology, singular homology and show that Stokes' Theorem (not the Calc III version) shows that integration of forms on simplices is a chain map between DeRham cohomology and (singular homology)*.

10 April 2012
Pye Aung, NDSU:
Nagata’s Idealization and the Amalgamated Duplication of a Ring along an Ideal
Abstract: If R is a commutative ring with identity and E is an R-module, then the idealization R \ltimes E, called "Nagata’s idealization" of E, is a new ring, containing R as a subring. Marco D’anna and Marco Fontana introduced in 2007 a new general construction, denoted R \bowtie E; it is called the ""amalgamated duplication" of a ring R along an R-submodule E of T(R), the total ring of fractions of R. When E^2=0, this new construction coincides with R \ltimes E. I will present definitions and some basic properties of these two constructions, and briefly discuss the case when E is an ideal in R and E is semi-dualizing as an R-module.

27 March 2012

Sean Sather-Wagstaff, NDSU:
Factorizations of local ring homomorphisms (Part II)

20 March 2012
Sean Sather-Wagstaff, NDSU:
Factorizations of local ring homomorphisms
Abstract:
Let f: R --> S be a homomorphism of commutative rings. Many techniques for studying R-modules focus on finitely generated modules. As a consequence, these techniques are not well-suited for studying S as an R-module. However, a technique of Avramov, Foxby, and Herzog sometimes allows one to replace the original homomorphism with a surjective one f': R' --> S where R and R' are tightly connected. In this setting, S is a cyclic R'-module, so one can study it using finitely generated techniques. I will give a general introduction to such factorizations, followed by a discussion of some new results on "weakly functorial properties" of such factorizations and applications. The new results are joint with Saeed Nasseh.

6 March 2012
Jason Boynton, NDSU:
An introduction to the ring Int(D) (part II)

Abstract: PDF

28 February 2012
Jason Boynton, NDSU:
An introduction to the ring Int(D)

Abstract: PDF

21 February 2012

Azer Akhmedov, NDSU: Hamiltonian cycles in some homogeneous graphs (part III)

14 February 2012

Azer Akhmedov, NDSU: Hamiltonian cycles in some homogeneous graphs (part II)

7 February 2012
Azer Akhmedov, NDSU: Hamiltonian cycles in some homogeneous graphsAbstract: L.Lovasz has conjectured (1970) that all vertex transitive graphs, except 5 of them, are Hamiltonian. We discuss/prove this conjecture for some examples of vertex transitive graphs; these examples turn out to be useful in musical theory.

31 January 2012Cătălin Ciupercă, NDSU: Kronecker (general) extensions II

 24 January 2012Cătălin Ciupercă, NDSU: Kronecker (general) extensions

Fall 2011 Schedule

Location: Minard 336
Time: Tuesday, 10:00-10:50 a.m.
Organizer: Cătălin Ciupercă

22 November 2011

Rocío Blanco, Universidad de Castilla-La Mancha: Combinatorial resolution of binomial idealsAbstract: In this talk we will construct an algorithm of resolution of singularities for binomial ideals in arbitrary characteristic. To resolve binomial ideals we define a modified order function, E-order, as the order along a normal crossing divisor E. With this E-order function we construct a resolution function that drops after blowing up and which provides only combinatorial centers. This kind of centers preserve the binomial structure of the ideal after blowing up. The output of this procedure is a locally monomial ideal that can be easily resolved to achieve a log-resolution.

25 October 2011
Kristen Beck, University of Arizona: Asymmetric linear complete resolutions over a short local ringAbstract: Let (R,m) be a local ring satisfying m^4=0.  The goal of this talk is to investigate the existence of a certain class of totall reflexive R-modules which are characterized by asymmetry in their complete resolutions.  Such a phenomenon is known to occur, by work of Jorgensen and Șega (2005).

18 October  2011

Sean Sather-Wagstaff, NDSU: Totally reflexive modules, or How to resolve freely in both directionsAbstract: I will present an introduction to the concept of totally reflexive modules. In particular, I hope to prove that a module over a noetherian ring is totally reflexive if and only if it has a complete resolution. I will define these new terms, and present several examples. This talk is a pre-seminar, in preparation for Kristen Beck's talk on October 25.

11 October 2011

Thomas Robinson, NDSU:
A classical non-trivial example of a vertex operator algebra constructed in full from scratch (part II)

27 September 2011
Thomas Robinson, NDSU: A classical non-trivial example of a vertex operator algebra constructed in full from scratchAbstract: I will begin by giving a (very) brief history of the classical algebraic theory of vertex (operator) algebras and why algebraists began studying them. It is somewhat difficult to construct even one interesting example of a vertex algebra. There are two main classical algebraic approaches to do this. I will focus on one of these approaches first developed by Frenkel, Lepowsky and Meurman. Some new techniques streamlining the original approach will allow me to give from scratch a complete construction of one non-trivial example of a vertex algebra in a reasonable amount of time. Then finally this example can be easily used to demonstrate one of the classical applications of vertex algebras, the construction of certain infinite dimensional Lie algebras.

20 September 2011

Jim Coykendall, NDSU: A survey of Factorization (part II)

13 September 2011

Jim Coykendall, NDSU: A survey of FactorizationAbstract:  Since about 1990, there has been much attention paid to the study of factorization in integral domains. Factorization is classically fundamental in number theory and algebra and has myriad applications (perhaps the most familiar of which is the application to coding theory). The general study of factorization in integral domains is the study of the multiplicative structure of a domain. Familiar examples include Euclidean domains, PIDs, and UFDs, but more exotic examples include finite factorization domains (FFDs), bounded factorization domains (BFDs), and atomic domains (the largest class of domains where irreducible factorizations exist for an arbitrary nonzero nonunit).In this sequence of two talks, we will review some of these interesting domains (there will be a number of examples for illumination purposes) and some of their fundamental properties and pathologies. We will also explore some natural questions about stability of these factorizations in polynomial, power series, and other extensions. We will also review some very recent developments concerning Kaplansky conditions and the contrast with monoid factorizations.This talk will be mostly survey and will be aimed at a beginning graduate student audience. All interested parties are encouraged to attend!

Spring 2011 Schedule

Location: Minard 304A (Seminar Room)
Time: Tuesday, 10:00-10:50 a.m.
Organizer: Cătălin Ciupercă

3 May 2011

Pye Aung, NDSU: Baer's Criterion and Whitehead Problem
Abstract:  This talk is an expository talk about tests for injectivity and projectivity of modules, and their connection to the Whitehead Problem. Projectivity and injectivity can be viewed as dual concepts, hence many definitions and theorems are possible to be "dualized" including some equivalent conditions (or tests) of projectivity and injectivity. Those facts will be formulated using the functor Ext. However, when one attempts to find a projective counterpart of the Baer's criterion for injectivity, difficulties arise leading to the first-known undecidable statement outside set theory.

26 April 2011

Saeed Nasseh, NDSU: Lifting of DG modules over DG algebras
Abstract:  We are mainly concerned with the generalizations of a result of Auslander, Ding and Solberg about lifting of modules. In particular, we apply DG techniques to investigate lifting properties for DG-modules over DG algebras.

19 April 2011

Rich Wicklein, NDSU: A connection between the Koszul complex and Tor
Abstract:  Let (R, m, k) be a local ring and M an R-module. A result of Lescot relates vector space dimension over k of H_i(M \otimes_R K(x)), where K(x) is the Koszul complex, and vector space dimension of Tor^R_i(k,M). I will discuss an alternate proof of Lescot’s result. This talk should be accessible to anyone who has completed 721 or has a basic understanding of modules.

12 April 2011

Tom Dunn, NDSU: Multiplicities in Local Rings

1 March 2011

Jim Coykendall, NDSU: Norms in Rings of Algebraic Integers (Part II)

Abstract:  We will present from the beginnings the concept of a norm in a ring of algebraic integers. Some basic number theory will be reviewed to demonstrate this concept. After the general concept is introduced, we will concentrate on the utility of the norm in gleaning factorization information of the ring that can be obtained from the factorization properties of the multiplicative monoid of norms. Many examples will be presented to (hopefully) provide clarity. Our aim is to present this from a basic and intuitive point of view.

22 February 2011

Jim Coykendall, NDSU: Norms in Rings of Algebraic Integers

Abstract:  We will present from the beginnings the concept of a norm in a ring of algebraic integers. Some basic number theory will be reviewed to demonstrate this concept. After the general concept is introduced, we will concentrate on the utility of the norm in gleaning factorization information of the ring that can be obtained from the factorization properties of the multiplicative monoid of norms. Many examples will be presented to (hopefully) provide clarity. Our aim is to present this from a basic and intuitive point of view.

15 February 2011

Sean Sather-Wagstaff, NDSU: Nakayama's Lemma for Ext and ascent of module structures II

8 February 2011

Sean Sather-Wagstaff, NDSU: Nakayama's Lemma for Ext and ascent of module structures

Abstract:
Let f: (R,m,k) -> (S,mS,k) be a flat local ring homomorphism, and let M be a finitely generated R-module. We show that the following are equivalent:
(i) M has an S-module structure compatible with its R-module structure;
(ii) Ext^i_R(S,M)=0  for i>0;
(iii) Ext^i_R(S,M) is finitely generated over R for i=1,...,dim_R(M);
(iv) Ext^i_R(S,M) is finitely generated over S for i=1,...,dim_R(M);
(v) Ext^i_R(S,M) satisfies Nakayama's Lemma over R for i=1,...,dim_R(M).

This improves upon recent results of Frankild, Sather-Wagstaff, and Wiegand and results of Christensen and Sather-Wagstaff. This is joint work with Ben Anderson and Jim Coykendall.

Fall 2010 Schedule

Location: Minard 304A (Seminar Room)
Time: Tuesday, 10:00-10:50 a.m.
Organizer: Cătălin Ciupercă

16 November 2010

Jason Boynton, NDSU: The D+M Construction and a Generalization
Abstract: In 1976 J.W. Brewer and E.A. Rutter investigated certain ring and ideal theoretic properties that behave nicely in the now well-studied D+M construction. In more recent years, it has become fashionable to consider such pullback constructions in greater generality. We will survey some past and present results concerning the transfer of ring and ideal theoretic properties in a special case of a pullback diagram called a conductor square.

9 November 2010

Sean Sather-Wagstaff, NDSU:
Bass Numbers and Semidualizing Modules (part III)
Abstract: Let (R,m,k) be a Cohen-Macaulay local ring. We are interested in a question of Huneke on the rate of growth of the sequence of Bass numbers m^i = rank_k(Ext^i(k,R)). In previous work, we showed that, if R has a semidualizing module that is not free and not dualizing, then the sequence {m^i} is unbounded. This reduces the original question of Huneke down to the case where R has only trivial semidualizing modules. We discuss a method for reducing generalizations of Huneke's question down to the same case.

2 November 2010

Sean Sather-Wagstaff, NDSU:
Bass Numbers and Semidualizing Modules (part II)
Abstract: Let (R,m,k) be a Cohen-Macaulay local ring. We are interested in a question of Huneke on the rate of growth of the sequence of Bass numbers m^i = rank_k(Ext^i(k,R)). In previous work, we showed that, if R has a semidualizing module that is not free and not dualizing, then the sequence {m^i} is unbounded. This reduces the original question of Huneke down to the case where R has only trivial semidualizing modules. We discuss a method for reducing generalizations of Huneke's question down to the same case.

26 October 2010

Sean Sather-Wagstaff, NDSU:
Bass Numbers and Semidualizing Modules
Abstract: Let (R,m,k) be a Cohen-Macaulay local ring. We are interested in a question of Huneke on the rate of growth of the sequence of Bass numbers m^i = rank_k(Ext^i(k,R)). In previous work, we showed that, if R has a semidualizing module that is not free and not dualizing, then the sequence {m^i} is unbounded. This reduces the original question of Huneke down to the case where R has only trivial semidualizing modules. We discuss a method for reducing generalizations of Huneke's question down to the same case.

12 October 2010

Cătălin Ciupercă, NDSU:
Constructing prime ideals

5 October 2010

Azer Akhmedov, NDSU:
Some examples of finitely generated infinite simple groups

28 September 2010

Azer Akhmedov, NDSU:
Some examples of finitely generated infinite simple groups

Spring 2010 Schedule

Location: Minard 304A (Seminar Room)
Time: Tuesday, 10:00-10:50 a.m.
Organizer: Cătălin Ciupercă

29 April 2010

Micah Leamer, University of Nebraska, Lincoln:
Torsion in tensor products over commutative rings

Abstract: Let R be a commutative local domain. We are interested in finding conditions under which the tensor product of two torsion free modules is torsion free.  In particular when R is one dimensional and M is a torsion free R-module, which is not free, does M tensored with Hom(M,R) always have torsion. We explore the special case where R is a subring of a discrete valuation domain and show that at least for monomial ideals the problem can be simplified to working with submonoids of the natural numbers.  This work is inspired by an attempt to make progress on the following conjecture:  Let M be a maximal Cohen-Macaulay R-module. If M tensored with Hom(M,R) is maximal Cohen-Macaulay then M is free. When R is one dimensional being maximal Cohen-Macaulay is equivalent to being torsion free.  The one dimensional case is relevant since it has been  shown that proving the conjecture for one dimensional Gorenstein rings is equivalent to proving the conjecture for Gorenstein rings of arbitrary dimension.

6 April 2010

Azer Akhmedov, NDSU:
On Shreier Graphs of Groups (II)

30 March 2010

Azer Akhmedov, NDSU:
On Shreier Graphs of Groups

2 March 2010

Saeed Nasseh, NDSU:
Symmetry in the Vanishing of Ext (II)

23 February 2010

Saeed Nasseh, NDSU:
Symmetry in the Vanishing of Ext

16 February 2010

Bethany Kubik, NDSU:
Evaluation Homomorphisms

Abstract:
R is a local noetherian ring and A, N, and I are R-modules. The Hom evaluation homomorphism is the map \theta_{ANI}:A\Otimes\Hom{N,I}\rightarrow\Hom{\Hom{A,N},I}.  This map is known to be an isomorphism only under certain conditions placed upon the modules.  We will expand the conditions under which the Hom evaluation is an isomorphism.  In particular we will show that when A is artinian, N is noetherian and Matlis reflexive, and I is injective, the Hom evaluation homorphism is an isomorphism.

9 February 2010

Sean Sather-Wagstaff, NDSU:
Extension and Torsion Functors for Artinian Modules (III)

2 February 2010

Sean Sather-Wagstaff, NDSU:
Extension and Torsion Functors for Artinian Modules (II)

26 January 2010

Sean Sather-Wagstaff, NDSU:
Extension and Torsion Functors for Artinian Modules

Abstract:
Let R be a commutative noetherian ring. It is well known that if N and N' are noetherian R-modules, then the modules Ext^i_R(N,N') and Tor^R_i(N,N') are also noetherian. Similarly, if N is a noetherian R-module and A is an artinian R-module, then the modules Ext^i_R(N,A) and Tor^R_i(N,A) are artinian. We will discuss basic properties of artinian modules. Then we will state and prove some properties of Ext and Tor modules when applied to other combinations of noetherian modules, artinian modules, and Matlis reflexive modules. This is joint work with Bethany Kubik (NDSU) and Micah Leamer (UNL).

Fall 2009 Schedule

Location: Minard 304A (Seminar Room)
Time: Tuesday, 10:00-10:50 a.m.
Organizer: Cătălin Ciupercă

8 December 2009

Azer Akhmedov, NDSU:
On linear groups

Abstract:
A group is linear if it admits a faithful representation in GL(n,k) for some natural number n and some field k. In this talk I will present some (well known) results about linear groups. The talk is aimed at a general audience.

24 November 2009

Cătălin Ciupercă, NDSU: Integral closure modulo generic elements (III)

17 November 2009

Cătălin Ciupercă, NDSU: Integral closure modulo generic elements (II)

10 November 2009

Cătălin Ciupercă, NDSU:  Integral closure modulo generic elements

27 October 2009

Bethany Kubik, NDSU:
Quasidualizing Modules and their relationship to Semidualizing Modules

Abstract: Let R be a local complete noetherian ring. A noetherian R-module C is semidualizing if Hom_R(C,C)is isomorphic to R and Ext_R^i(C,C)=0 for all i greater than or equal to 1. We introduce and study the artinian counterpart which we call a quasidualizing module. We explore the relationship between these two concepts through Matlis Duality.

20 October 2009

Sean Sather Wagstaff, NDSU: Semidualizing modules for rings of codimension 2 (part II)

13 October 2009

Sean Sather Wagstaff, NDSU: Semidualizing modules for rings of codimension 2

Abstract: Semidualizing modules are algebraic objects that are objects for the study of several aspects of commutative noetherian rings. However, the program of completely understanding the structure of the collection of such modules is still far from complete. We will provide a criterion for characterizing the semidualizing modules over Cohen-Macaulay rings of codimension 2, and we will prove that several classes of rings satisfy this criterion: generically Gorenstein rings (e.g., reduced rings), rings arising from fat point schemes, and rings that are obtained as quotients by monomial ideals. This is joint work with Susan Cooper.

6 October 2009

Azer Akhmedov, NDSU: On the girth of groups

Abstract: I'll introduce the notion of girth of a finitely generated group, and will mention examples of groups with finite as well as infinite girth. It is a classic theorem of J.Tits that every finitely generated linear group is either virtually solvable or contains non-abelian free subgroup. This result is called Tits Alternative. I'll introduce the so-called Girth Alternative, and compare it with Tits Alternative.

29 September 2009

Stacy Trentham, NDSU: MCD (maximal common divisor) Rings

Abstract: In this talk, we will be looking at MCD domains. In particular, we will examine some properties of polynomial extensions of MCD domains. We will end by generalizing the MCD property to include rings with zero divisors to see if polynomial extensions of these rings possess properties similar to their domain counterparts.

15 September 2009
Sean Sather-Wagstaff, NDSU:
Hilbert-Kunz multiplicities

8 September 2009
Cătălin Ciupercă, NDSU:
Structure theorems for certain integrally closed ideals

Algebra & Discrete Mathematics Seminar
Spring 2009 Schedule

Location: Minard 304A (Seminar Room)
Time: Thursday, 10:00-10:50 a.m.
Organizer: Cătălin Ciupercă

30 April 2009
Bethany Kubik, NDSU
: Quasidualizing modules

23 April 2009
Sean Sather-Wagstaff, NDSU
: Semidualizing modules: Some background, an application, and some structure (part III)

16 April 2009
Sean Sather-Wagstaff, NDSU
: Semidualizing modules: Some background, an application, and some structure (part II)

9 April 2009
Sean Sather-Wagstaff, NDSU
: Semidualizing modules: Some background, an application, and some structure
Abstract:
Semidualizing modules were "discovered" independently by Foxby, Golod, Vasconcelos and Wakamatsu. I learned about them through some work of Avramov and Fozby where semidualizing modules are used to study local ring homomorphisms of finite G-dimension. I plan to give three lectures on this subject. In the first lecture, I will present some background information on these modules. In the second lecture, I will discuss an application of semidualizing modules to a question of Huneke on the rate of growth of the Bass numbers of a local ring. In the third lecture, I will discuss some recent progress on the question of whether a given local ring has exactly 2^n semidualizing modules for some integer n.

12 March 2009

Travis Trentham, NDSU
: A generalization of Krull dimension (part III)

5 March 2009

Travis Trentham, NDSU
: A generalization of Krull dimension (part II)

26 February 2009
Travis Trentham, NDSU: A generalization of Krull dimension
Abstract: In this talk we will look at a generalization of our present notion of Krull dimension. It will be shown that this definition is well-defined in the sense that every ring admits a unique Krull dimension. Further, it wil be shown how Krull dimension is preserved in all ring extensions that are INC and GU. We will also be looking at some interesting pathologies that have presented themselves. If time allows, we will compare the Krull dimensions of R and R[x], where R is a ring having infinite Krull dimension.

22 January 2009
Azer Akhmedov, NDSU: Groups without big tiles and tiles in symmetric spaces with arbitrarily big Heesch number
Abstract: I will discuss the following property of a discrete group G:

(P) Given any finite subset K of G, there exists a finite subset F of G such that F contains K and and F tiles G.

The main question is, do all groups have this property? The answer is negative; I will discuss some ingredients of the construction and related to that, we will see how it helps to construct tiles with arbitrarily big Heesch number in symmetric spaces of rank one simple Lie groups. Interestingly, the idea works in all symmetric spaces (including hyperbolic spaces of dimension greater than two) except for the hyperbolic plane.

Fall 2008 Schedule

Location:      Minard 304A (Seminar Room)
Time:             Tuesday, 11:00-11:50 a.m.
Organizer:    Cătălin Ciupercă

2 September 2008
Josh Lambert, NDSU: The Biplanar Crossing Number of C_k x C_l x C_{2m} x P_n
Abstract

9 September 2008
Azer Akhmedov, NDSU: Perturbations of Wreath Products and Quasi-Isometric Rigidity  I
Abstract:  Groups are often endowed with a left-invariant metric which allows them to be viewed as metric spaces along with the more traditional view of groups as isometries of metric spaces. Starting with the works of Cayley and Dehn, this approach to studying groups has proven to be very fruitful.
In the early 80's, M.Gromov initiated a broad program of classifying  groups up to quasi-isometry. Based on his deep insight, he conjectured that "algebraic properties of groups are geometric", i.e. groups with quasi-isometric Cayley graphs should share the same (or similar) algebraic properties. This phenomenon is called a quasi-isometric rigidity.
Some sporadic counterexamples to this conjecture were known. By introducing the notion of perturbation of wreath products of groups,I show that many-many algebraic properties fail to be invariants of quasi-isometry. In fact, one can initiate a counter-program to say that if a property does not satisfy certain finiteness condition then most likely it is not preserved under quasi-isometry.
For my constructions, I introduce a new class of groups which I call traveling salesman groups. These groups are interesting independently and have proven to be useful in other areas as well, e.g. in the theory of amenable groups.
The first talk is for a very general audience. In the second talk I will mainly discuss traveling salesman groups.

16 September 2008
Azer Akhmedov, NDSU: Perturbations of Wreath Products and Quasi-Isometric Rigidity   II

30 September 2008
Catalin Ciuperca, NDSU: Numerical criteria for integral dependence

7 October 2008
Catalin Ciuperca, NDSU: Numerical criteria for integral dependence II

14 October 2008
Sean Sather-Wagstaff, NDSU: Gorenstein presentations and semidualizing modules
Abstract: A famous result of Foxby, Reiten and Sharp says that a Cohen-Macaulay local ring admits a dualizing module if and only if it is a homomorphic image of a Gorenstein ring. We augment this result by showing that such a ring admits a nontrivial semidualizing module if and only if it admits a Gorenstein presentation Q/I such that the ideal I has a nontrivial decomposition. This is joint work with David Jorgensen and Graham Leuschke.

21 October 2008
Sean Sather-Wagstaff, NDSU: Gorenstein presentations and semidualizing modules II

28  October 2008
Title:
Contracting endomorphisms and Gorenstein modules
Abstract:
A finite module M over a noetherian local ring (R, m, k) is said to be Gorenstein if Ext_R^i(k,M)=0 for all i \ne dim R. An endomorphism f: R --> R of rings is called contracting if f^i(m) \subseteq m2 for some i \geq 1. Letting S denote the R-module R with action induced by f, we prove: A finite R-module M is Gorenstein if and only if Hom_R(S,M) \cong M and Ext_R^i(S,M) = 0 for 1 \leq i \leq \depth R.

4 November 2008
Yong Hou, NDSU
Title:  Geometry of Kleinian Group

18 November 2008
Yong Hou, NDSU
Title:  Fractal Dimensions and Geometric Dynamics

25 November 2008
Sean Sather-Wagstaff, NDSU
Title: Homological properties of modules
Abstract: In this talk, I will present some background information in preparation for David Jorgensen's seminar scheduled for 02 December. I will discuss Ext, depth, and some aspects of modules over Gorenstein rings.

2 December 2008
Dave Jorgensen, University of Texas at Arlington
Title:  Existence of totally reflexive modules
Abstract:  Totally reflexive modules over a commutative local ring behave much like maximal Cohen-Macaulay modules do over a Gorenstein ring.  The point of this talk is to investigate the existence of non-free totally reflexive modules over local (usually Cohen-Macaulay) non-Gorenstein rings.  We will briefly survey what is known, as well as discuss some recent results from joint work with Kristen Beck, and with Meri Hughes and Liana Sega.

Spring 2008 Schedule

Location:       Minard 304A (Seminar Room)
Time:             Thursday, 12:00 - 12:50 p.m
Organizer:     Cătălin Ciupercă

7 February 2008
Cătălin Ciupercă, NDSU:
Asymptotic properties of ideals

21 February 2008
Cătălin Ciupercă, NDSU:
Asymptotic properties of ideals II

28 February 2008
Cătălin Ciupercă, NDSU:
Asymptotic properties of ideals III

13 March 2008
Sean Sather-Wagstaff, NDSU: Duality in algebra
Abstract: I will present some examples, some theory and some applications of algebraic duality.

20 March 2008
Sean Sather-Wagstaff, NDSU: Duality in algebra II

27 March 2008
Sean Sather-Wagstaff, NDSU: Duality in algebra III

10 April 2008
Jim Coykendall, NDSU

17 April 2008
Jim Coykendall, NDSU

24 April 2008
Sandra Spiroff, Seattle University: A New Zero Divisor Graph
Abstract: A zero divisor graph of a ring R is a visual representation of the zero divisor activity in R. They have been studied by I. Beck, D. Anderson & P. Livingston, S. Mulay, and C. Wickham, to name just a few. Using a new zero divisor graph introduced by Mulay, one which is constructed from equivalence classes of zero divisors, we identify ring theoretic properties. We will compare and contrast these graphs with the original zero divisor graphs and discuss some results involving the associated primes of the ring. This is joint work with Cameron Wickham.