"Rings and Wings"

The Colorado Springs Algebra Seminar typically meets every other Wednesday, from 4:00 until 5:XX (where 00 \leq XX \leq 30).   Often the meeting place is the campus of the University of Colorado at Colorado Springs, but other venues have been used as well.

We encourage talks from all areas of algebra.

Talks are typically attended by math faculty from throughout the Pikes Peak region, including the University of Colorado at Colorado Springs, The Colorado College, and Colorado State University - Pueblo.   Talks are also often attended by graduate students and advanced undergraduate students.

Talks are typically given by those who typically attend, as well as any out-of-town visiting algebraists who may happen along ...

It is now traditional that, on completion of the presentation, those who are interested head to a local eatery / watering hole (often Clyde's on the UCCS campus) for dinner or liquid refreshment or snacks (e.g., Wings?).

Contact the seminar organizer, Gene Abrams   abrams@math.uccs.edu   if you are interested in participating.

On this page we will also typically include other talks which will happen in the Pikes Peak region which may be of interest to algebraists.

Fall 2014 Schedule

All talks in ENGR 239 (Engineering Conference Rooms)  on the UCCS campus, unless otherwise indicated.

All talks begin at 4:00pm, unless otherwise indicated.

Contact Gene Abrams (preferably a few days in advance of the talk) if you need a UCCS parking permit.

DATE

SPEAKER

TITLE
ABSTRACT

Wednesday, Sept 10

K.M. Rangaswamy

UCCS

Leavitt path algebras satisfying a polynomial identity.

Leavitt path algebras L of arbitrary graphs E satisfying polynomial identities are characterized both graphically and algebraically. Connections to the Gelfand-Kirillov dimension of L are explored.

Thursday,

October 9

UCCS Math Dept. Colloquium

OSB A324

(Daniels K12 Room)

12:30 - 1:30

Jonathan Brown

Kansas St. University

# In 2005 Abrams and Aranda Pino began a program studying rings constructed from directed graphs. These rings, called Leavitt Path algebras, generalized the rings without invariant basis number introduced by Leavitt in the 1950's. Leavitt path algebras are the algebraic analogues of the graph C*-algebras and have provided a bridge for communication between ring theorists and operator algebraists. Many of the properties of Leavitt path algebras can be inferred from properties of the graph, and for this reason provide a convenient way to construct examples of algebras with a particular set of attributes. In this talk we will explore how central elements of the algebra can be read from the graph.

Thursday,

October 23

UCCS Math Dept. Annual Distinguished Lecture

UCCS Library,

2nd Floor Apse

12:30 - 1:30

(refereshments @12:15)

Jason Bell

University of Waterloo

###### and the mathematics of altruism

Game theory is a branch of mathematics that deals with strategy and decision making and is applied in economics, computer science, biology, and many other disciplines as well. We will discuss some of the basic points of game theory and discuss the so-called iterated prisoner dilemma, a game that is of central importance in the study of cooperation between individuals. We will then describe various strategies to this game and explain why altruism is something that can evolve naturally.

Friday October 31

2:00 - 3:00p

Room tba

Jeff Boersema

Seattle University

(visiting U. New Mexico)

to be announced

Wednesday

November 5

4:00 -5:XX

Greg Oman

UCCS

to be announced

Wednesday

November 19

4:00 - 5:XX

Zak Mesyan

UCCS

to be announced

Wednesday

December 3

4:00 - 5:XX

Derek Wise

U. Erlangen (Germany)

to be announced

Wednesday

December 10

4:00 - 5:XX

Gonzalo Aranda Pino

U. Malaga (Spain)

to be announced

Spring 2014 Schedule

All talks in OSB A342 (Small Conference Room) on the UCCS campus, unless otherwise indicated.

All talks begin at 4:00pm, unless otherwise indicated.

Contact Gene Abrams (preferably a few days in advance of the talk) if you need a UCCS parking permit.

Fall 2013 Schedule

All talks in ENGR 239 on the UCCS campus, unless otherwise indicated.

All talks begin at 4:00pm, unless otherwise indicated.

Contact Gene Abrams (preferably a few days in advance of the talk) if you need a UCCS parking permit.

 DATE SPEAKER TITLE ABSTRACT Thursday, August 29 *** UCCS Colloquium Series Zak Mesyan UCCS Evaluating polynomials on matrices A classical theorem of Shoda from 1936 says that over any field K (of characteristic 0), every matrix with trace 0 can be expressed as a commutator AB-BA, or stated another way, that evaluating the polynomial f(x,y)=xy-yx on matrices over K gives precisely all the matrices having trace 0. I will describe various attempts over the years to generalize this result. September 25 Kulumani Rangaswamy UCCS Endomorphism rings of Leavitt path algebras Leavitt path algebras L(E) of a graph E are in general non-unital rings unless the number of vertices in E is finite. Regarding a Leavitt path algebra L(E) as a right L(E)-module, I will talk about the various ring-theoretic properties of the endomorphism ring A of L(E) and relate them to corresponding graph-theoretical properties of E. Interestingly, these properties of the graph E are much stronger that those on E in order for L(E) to have the same property as A. October 16 Greg Oman UCCS Strongly Jonsson and strongly HS modules Borrowing from universal algebraic terminology, an infinite module M over a ring R is called a Jonsson module provided every proper submodule of M has smaller cardinality than M. Call M strongly Jonsson (and drop the requirement that M be infinite) provided distinct submodules of M have distinct cardinalities (the cyclic and quasi-cyclic groups have this property). Dually, M is said to be homomorphically smaller (HS for short) if M/N has smaller cardinality than M for every nonzero submodule N of M. Say that M is strongly HS provided M/N and M/K have distinct cardinalities for distinct subgroups N and K of M (the abelian group Z of integers has this property; again, we drop the requirement that M be infinite). In this talk, we discuss the above notions. In particular, we present classification theorems for modules over an arbitrary commutative ring R. October 30 Gonzalo Aranda Pino Universidad de Malaga (Spain) Leavitt path algebras of generalized Cayley graphs Path algebras are algebras associated to graphs whose bases as vector spaces consist of the sets of all paths in a graph. On the other hand, classical Leavitt algebras are universal examples of algebras without the Invariant Basis Number condition (that is, of algebras having bases with different cardinal). Leavitt path algebras are then natural generalizations of the aforementioned path and classical Leavitt algebras, as well as algebraic versions of graph C*-algebras.      Let $n$ be a positive integer. For each $0\leq j \leq n-1$ we let $C_n^j$ denote Cayley graph for the cyclic group $Z_n$ with respect to the subset $\{1, j\}$. For any such pair $(n,j)$ we compute the size of the Grothendieck group of the Leavitt path algebra $L_K(C_n^j)$; the analysis is related to a collection of integer sequences described by Haselgrove in the 1940's. When $j=0,1,$ or $2$, we are able to extract enough additional information about the structure of these Grothendieck groups so that we may apply a Kirchberg-Phillips-type result to explicitly realize the algebras $L_K(C_n^j)$ as the Leavitt path algebras of graphs having at most three vertices. The analysis in the $j=2$ case leads us to some perhaps surprising and apparently nontrivial connections to the classical Fibonacci sequence.

Spring 2013 Schedule

All talks in ENGR 239 on the UCCS campus, unless otherwise indicated.

All talks begin at 4:00pm, unless otherwise indicated.

Contact Gene Abrams (preferably a few days in advance of the talk) if you need a UCCS parking permit.

Previous semesters:

Fall 2012 Schedule

 DATE SPEAKER TITLE ABSTRACT November 28 Darren Funk-Neubauer Colorado State University - Pueblo An Introduction to Bidiagonal Pairs I will introduce a linear algebraic object called a bidiagonal pair and present a theorem which classifies these objects. Roughly speaking, a bidiagonal pair is an ordered pair of diagonalizable linear transformations on a finite dimensional vector space, each of which acts in a bidiagonal fashion on the eigenspaces of the other. Understanding the definition of a bidiagonal pair and the statement of the classification theorem only requires a basic knowledge of undergraduate linear algebra. However, the proof of the classification theorem makes use of the representation theory of Lie algebras and quantum groups. I will discuss the origin of bidiagonal pairs in Lie theory, but no Lie theory will be assumed in following the talk. November 14 Matthew Eric Bassett Queen's College, London A Tour of Hopf Algebras and Their Applications, plus some remarks about class field theory From their beginnings in algebraic topology, Hopf algebras - later quantum groups - have found uses ranged from number theory to noncommutative geometry. In this talk, we'll discuss their uses in studying Galois modules, to constructing noncommutative geometries, and, time permitting, say a few words about the structure of the quantum group-flavoured Hopf algebras via their categories of [co]modules.  I'll also mention some results from a recent paper by Cornelissen, expanding on Bost-Connes type system results. October 24 Kulumani M. Rangaswamy UCCS Centers of path algebras, Cohn algebras, and Leavitt path algebras This talk will attempt to describe the centers of path algebras, the Cohn algebras and the Leavitt path algebras of an arbitrary graph E over a field K. October 10 Muge Kanuni Er Boğaziçi Univesity Dept of Mathematics Visiting Fulbright Scholar to UCCS "An approach to calculating the global dimension of some Artinian algebras" In this talk, we will focus on two aspects: 1) To represent an Artinian algebra A over a field k as a directed graph by the uniquely determined set of orthogonal primitive idempotents of A that decompose the unity. 2) By using homological tools, solely on this directed graph, to develop a procedure for getting upper and lower bounds for the global dimension of a class of Artinian algebra. *** joint work with A. Kaygun September 26 Kulumani M. Rangaswamy UCCS A descriptinon of results in the article "Irreducible representations of Leavitt path algebras" by Xiao-Wu Chen Chen's article September 12 Greg Oman UCCS Rings whose multiplicative endomorphisms are power functions. Let F be a finite field of order p^n. It is well-known that there are exactly n field automorphisms of F. In particular, they are all power functions. In this note, we "throw away" addition and enlarge the class of rings to the class of commutative rings with identity. We then consider the following question: For which rings R is it the case that every multiplicative endomorphism of R (a map which preserves multiplication, sends 0 to 0, and sends 1 to 1) is equal to a power function?

Spring 2012 Schedule

 DATE SPEAKER TITLE ABSTRACT February 1 Zak Mesyan UCCS Simple Lie algebras arising from Leavitt path algebras February 15 Sergio Lopez Permouth Ohio University Characterizing rings in terms of the extent of the injectivity and projectivity of their modules. Given a ring R, we define its right i-profile (resp. right p-profile) to be the collection of injectivity domains (resp. projectivity domains) of right R-modules. We study the lattice theoretic properties of these profiles. We show that the i-profile is equivalent to an interval of the lattice of linear filters of right ideals of R. This allows us to apply torsion theoretic techniques to study the i-profile of a ring. We show through and example that the p-profile of a ring is not necessarily a set, and we characterize the p-profile of a right perfect ring. We then apply our results to the study of a special class of QF-rings. The study of rings in terms of their (i or p-)profile generalizes the study of rings with no right (i or p-)middle class, initiated in recent papers by Er, Lopez-Permouth and Sokmez, and by Holston, Lopez-Permouth and Orhan-Ertas. March 7 Darren Funk-Neubauer CSU - Pueblo Introduction to the theory of tridiagonal and bidiagonal pairs: Part 1 Tridiagonal pairs and bidiagonal pairs are linear algebraic objects which originally arose in the context of algebraic graph theory, but now appear in many other areas of mathematics. Roughly speaking, a tridiagonal (resp. bidiagonal) pair is a pair of diagonalizable linear transformations on a finite dimensional vector space which act tridiagonally (resp. bidiagonally) on each others eigenspaces. In these two seminar talks I will do the following: give the formal defintions of tridiagonal and bidiagonal pair, give some concrete examples of them, discuss the history of how they arose, discuss how these objects arise in the representation theory of Lie algebras (my research area), and discuss attempts to classify these objects. These talks will not assume any specialized knowledge of graph theory, Lie theory, or representation theory. The only prerequiste for the talk is a solid understanding of linear algebra. March 21 Darren Funk-Neubauer CSU - Pueblo Introduction to the theory of tridiagonal and bidiagonal pairs: Part 2 (see above) April 4 Greg Oman UCCS Modules which are isomorphic to their factor modules. Call a module M homomorphically congruent (HC) provided M is infinite and M\cong M/N for every submodule N of M for which |M|=|M/N|. We will give some background showing how we arrived at this definition (i.e. how it is a natural outgrowth of notions well-studied in the literature) and present the principal results on HC modules. Friday, April 20 Cornelius Pillen University of South Alabama Cohomology of Finite Groups of Lie Type A longstanding open problem of major interest for  algebraists and topologists has been to determine the cohomology  rings of finite groups of Lie type in the so-called defining  characteristic. Little is known in general about these rings. It is  not even known in which positive degree the first non-trivial  cohomology classes occurs, a question that was first posed and partially answered by Quillen in the 1970s.  Being far from well-understood, the cohomologies of the corresponding Lie algebras and Frobenius kernels, nevertheless, are better understood theories that can be used as a tool for investigations into the cohomology rings of finite groups. Recently developed methods and techniques in linking these theories will be presented. These techniques will allow us to use Kostant's partition functions to provide answers to Quillen's question. May 2 *** at Colorado College Mike Siddoway Colorado College Divisibility Theory of Rings with Zero Divisors The "divisibility theory" of a commutative ring is the semigroup of finitely generated ideals partially ordered by inverse inclusion. For instance, for a Bezout ring this amounts to the semigroup of principal ideals. The divisibility theory for valuation domains is well known through the groundbreaking work of Krull in the early 20th century, though his approach did not directly consider the semigroup of finitely generated ideals. Rings with zero divisors present interesting complications. Aside from the long-settled case of valuation rings, the divisibility theory of rings with zero divisors has not added a new class of rings since the 1960s. One of our results states that a semigroup is a semi-hereditary Bezout semigroup if and only if it is isomorphic to the semigroup of principal ideals in a semi-hereditary Bezout ring partially ordered by reverse inclusion. This is joint work with Pham Ngoc Anh of the Hungarian Academy of Sciences and recalls the studies of Krull on valuation domains and Kaplansky, Jaffard, and Ohm on Bezout domains. Our results are the first major developments along these lines for rings with zero divisors.