*
*Algebra
& Discrete Mathematics Seminar** **

**
Spring 2014 Schedule**

**Location:
**Minard 210**
Time: **Tuesday, 10:00-10:50 a.m.

**22 April 2014**

**15 April 2014**

**25 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.

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.

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

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

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.

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

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
**

Time:

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.

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.

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

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.

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

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)

**28 February 2012**

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

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)`

Azer Akhmedov, NDSU: Hamiltonian cycles in some homogeneous graphs

Abstract: 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 2012

Cătălin Ciupercă, NDSU: Kronecker (general) extensions II

24 January 2012

Cătălin Ciupercă, NDSU: Kronecker (general) extensions

**Fall
2011 Schedule
**

22 November 2011

Rocío Blanco, Universidad de Castilla-La Mancha: Combinatorial resolution of binomial ideals

Abstract: 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.

Kristen Beck, University of Arizona:Asymmetric linear complete resolutions over a short local ring

Abstract: 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 directions

Abstract: 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:

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

Abstract: 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.

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

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

**12
April 2011**

**Tom Dunn, NDSU:** Multiplicities
in Local Rings

**1
March 2011**

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

**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**

16 November 2010

Sean Sather-Wagstaff, NDSU:

Sean Sather-Wagstaff, NDSU:

Sean Sather-Wagstaff, NDSU:

12 October 2010

Cătălin Ciupercă, NDSU:

5 October 2010

Azer Akhmedov, NDSU:

Azer Akhmedov, NDSU:

**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:

**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)

Sean Sather-Wagstaff, NDSU:

Abstract:

**Fall
2009 Schedule**

**Location:
**Minard 304A (Seminar Room)

**Time: **Tuesday, 10:00-10:50 a.m.

**Organizer: **Cătălin Ciupercă

Azer Akhmedov, NDSU:

Abstract:

**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:

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
**

Abstract

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.

Title:

Abstract:

2 December 2008

**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

Cătălin Ciupercă, NDSU:

Cătălin Ciupercă, NDSU: