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The Colorado Springs Algebra Seminar
"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 outoftown 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.
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.
DATE 
SPEAKER 
TITLE 
ABSTRACT 
Wednesday, March 12 OSB A342 (Small Conference Room)

Greg Oman UCCS 
Small and large ideals of an associative ring 
Abstract. Let R be an associative ring with identity, and let I be an (left, right, twosided) ideal of R. Say that I is small if I<R and large if R/I<R. In this talk, I will present results on small and large ideals. In particular, I will discuss their interdependence and how they influence the structure of R. Conversely, I will give examples to show how the ideal structure of R determines the existence of small and large ideals. 
Thursday, April 3 UCCS Math Dept. Colloquium OSB A324 (Daniels K12 Room) 12:30  1:30 
Kulumani Rangaswamy UCCS

The Leavitt path algebras of directed graphs  The Leavitt path algebra L of a directed graph E over a field K is endowed with nicely amalgamating different structures: L is an associative algebra over the field K, it is a graded ring, L possesses a compatible involution and all these structures are intertwined by the enveloping properties of the graph E. L is highly noncommutative, but the presence of the involution makes the usual leftright differences for noncommutative algebras disappear in L. This talk will describe some of the intrinsic properties and recent results on Leavitt path algebras and illustrate how Leavitt path algebras have become powerful tools in constructing examples of various types of algebras. 
Friday, April 25 CC Fearless Friday Series 12:00  1:00 Tutt Lecture Hall, Tutt Science Bldg. Colorado College Undergraduateoriented presentation

Gene Abrams UCCS 
Fibonacci’s Rabbits Visit the Mad Veterinarian 
Since its origin (more than eight centuries ago) as a puzzle about the number of rabbits in a (fantasmagorically expanding) colony, the Fibonacci Sequence 1,1,2,3,5,8,13,... has arguably become the most wellknown of numerical lists, due in part to its simple recursion formula, as well as to the numerous connections it enjoys with many branches of mathematics and science. Since their origins (less than two decades ago) as puzzles about the number of animals in a (fantasmagorically strange) veterinarian’s office, the (not so wellknown) Mad Vet Scenarios have provided a source of thoughtprovoking entertainment to internet gamers and math enthusiasts alike. In this talk we’ll show how Fibonacci’s puzzle about rabbits is naturally connected to the puzzles found in the Mad Vet’s office. Along the way, we’ll show how an investigation into Mad Vet Scenarios has led to the discovery of some heretofore unrecorded properties of the Fibonacci Sequence. This talk is rated G, meaning that it is intended for the most General of audiences. No prior familiarity with Fibonacci’s breeding rabbits, or with the Mad Veterinarian’s transmogrification machines, or with any other type of fantasmagorical animal population dynamics, will be assumed. 
Wednesday, April 30 OSB A342 (Small Conference Room) 
Mark Tomforde University of Houston 
Classification of Leavitt path algebras using algebraic Ktheory  Leavitt path algebras are algebraic counterparts of graph C*algebras that include the algebras without the Invariant Number Basis (IBN) property originally introduced by Leavitt, several ultramatricial algebras, and many other interesting examples. In the theory of C*algebras, operator algebra Ktheory has proven to be a very useful and powerful tool for classification. In this talk we will examine the algebraic Ktheory of the Leavitt path algebras and the extent to which it can be used to classify Leavitt path algebras up to Morita equivalence. While algebraic Ktheory is notoriously difficult to compute in general, we will present cases in which the algebraic Ktheory of Leavitt path algebras may be explicitly computed using some simple formulas. We will also describe various classes of Leavitt path algebras that can be classified up to Morita equivalence by algebraic Ktheory and discuss what Ktheoretic data is needed for each class. Surprisingly, the underlying field of the algebra plays an important role in this classification. 
Thursday, May 1 OSB A 324 (Daniels K12 Room) 
Mark Tomforde University of Houston 
Using results from dynamical systems to classify algebras and C*algebras 
In the subject of symbolic dynamics, the shift spaces of finite type arise as edge shifts of finite directed graphs. The classification of these shift spaces was used in the 1980’s to classify certain C*algebras constructed from directed graphs, known as CuntzKrieger C*algebras, and moreover, the dynamical systems methods were key ingredients in the proofs. More recently, similar techniques have been used to classify certain algebras constructed from directed graphs, which are known as Leavitt path algebras. In this talk I will give an overview of the dynamical systems results, describe how they have led to methods for classifying C*algebras and algebras, and discuss the current status of these classification programs and existing open problems. 
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 ABBA, or stated another way, that evaluating the polynomial
f(x,y)=xyyx 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 nonunital 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 ringtheoretic properties of the endomorphism ring A of L(E) and relate them to corresponding graphtheoretical 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 quasicyclic 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 $0\leq j \leq n1$ 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 KirchbergPhillipstype 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.
DATE 
SPEAKER 
TITLE 
ABSTRACT 
January 30

Stefan Erickson The Colorado College TALK TO BE HELD IN TUTT SCIENCE BUILDING ON THE COLORADO COLLEGE CAMPUS, ROOM 221 
Endomorphism Rings of Elliptic Curves 

February 13 
(cancelled) 

February 27 
P.N. Anh Mathematics Institute, Hungarian Academy of Sciences, Budapest TALK TO BE HELD IN TUTT SCIENCE BUILDING ON THE COLORADO COLLEGE CAMPUS, ROOM 229 
A generalization of Clifford's Theorem 

March 13 
Mike Siddoway The Colorado College TALK TO BE HELD IN TUTT SCIENCE BUILDING ON THE COLORADO COLLEGE CAMPUS, ROOM 229 
Ideals, Gauss' Lemma, Valuations, Eisenstein's Criterion 

** Thursday, March 21 12:30  1:30 UCCS Math Dept. Colloquium 
Murad Ozaydin University of Oklahoma 
"The linear Diophantine Frobenius problem: an elementary introduction to numerical methods" 

April 3 
Benjamin Schoonmaker MS Applied Math student, UCCS 
An examination of the K_0 groups of the Leavitt path algebras of some Cayley graphs 

April 17 
Zak Mesyan UCCS 
Generalizations of Shoda's Theorem 
Abstract: A celebrated theorem of Shoda from 1936 states that over any field (of characteristic 0), every matrix with trace 0 can be expressed as a commutator ABBA. I will describe various attempts to generalize this result over the years. 
** Friday April 19 2:00  3:00 UCCS, room tba 
Efren Ruiz University of Hawai'i Hilo 
Classification of graph algebras: The Invariant and Status Quo 

** Tuesday, April 31 12:30  1:30 UCCS Math Dept. Colloquium 
Mercedes Siles Molina Universidad de Malaga (Spain) 
Graph algebras: from analysis to algebra and back 

May 1 
Pere Ara Universitat Autonoma de Barcelona 
Lamplighter groups and separated graphs 

Previous semesters:
Fall 2012 Schedule
DATE 
SPEAKER 
TITLE 
ABSTRACT 
November 28 
Darren FunkNeubauer 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 groupflavoured Hopf algebras via their categories of [co]modules. I'll also mention some results from a recent paper by Cornelissen, expanding on BostConnes 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: *** 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 XiaoWu Chen 

September 12 
Greg Oman UCCS 
Rings whose multiplicative endomorphisms are power functions. 
Let F be a finite field of order p^n. It is wellknown 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 iprofile (resp. right pprofile) to be the collection of injectivity domains (resp. projectivity domains) of right Rmodules. We study the lattice theoretic properties of these profiles. We show that the iprofile 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 iprofile of a ring. We show through and example that the pprofile of a ring is not necessarily a set, and we characterize the pprofile of a right perfect ring. We then apply our results to the study of a special class of QFrings. 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, LopezPermouth and Sokmez, and by Holston, LopezPermouth and OrhanErtas. 
March 7 
Darren FunkNeubauer 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 FunkNeubauer CSU  Pueblo 
Introduction to the theory of tridiagonal and bidiagonal pairs: Part 2 

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 wellstudied 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 socalled defining characteristic. Little is known in general about these rings. It is not even known in which positive degree the first nontrivial cohomology classes occurs, a question that was first posed and partially answered by Quillen in the 1970s. Being far from wellunderstood, 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 longsettled 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 semihereditary Bezout semigroup if and only if it is isomorphic to the semigroup of principal ideals in a semihereditary 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. 