Algebra & Discrete Mathematics Seminar

Spring 2010 Schedule


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

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
Hamid Rahmati, University of Nebraska-Lincoln
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.