Return to Gene Abrams' home page
The Colorado Springs Algebra Seminar
"Rings and Wings"
The Colorado Springs Algebra Seminar typically meets every other Wednesday, from 3:45 until 5:00. 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 2016 Schedule (TENTATIVE)
All talks in ENGR 201 (Engineering Dean's Conference Room) on the UCCS campus, unless otherwise indicated.
(note: this is a different room than the one we've used in the past, but the same room as we used in Spring 2015...)
All "Rings and Wings" talks begin at 3:45pm, 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 
February 3 
[available] 

February 10  [available]  
February 17  [available]  
February 24  Michael Penn Colorado College 

March 2  Michael Penn Colorado College 

March 9  [available]  
March 16  Greg Oman UCCS 

March 23 
NONE  Spring Break


March 30  Daniel Goncalves UFSC  SC  Brasil 

April 6  Gautam Webb Colorado College 

April 13  [available] 

April 20  Be'eri Greenfeld Bar Ilan Univ.  Israel 

THURSDAY APRIL 21 12:30  1:30 UCCS COLLOQUIUM 
Ikko Saito UCCS 

April 27  [available]  
May 4  Cristobal Gil Universidad de Malaga  Spain 

Fall 2015 Schedule
All talks in ENGR 201 (Engineering Dean's Conference Room) on the UCCS campus, unless otherwise indicated.
(note: this is a different room than the one we've used in the past, but the same room as we used in Spring 2015...)
All "Rings and Wings" talks begin at 3:45pm, 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, September 9 
Bob Carlson UCCS 
Quantum Cayley Graphs for Free Groups (Talk 1 of 2) Talk #1 will aim for a broad audience, and will be “introductory”. The second followup talk will drill down deeper. We will schedule the second talk at the end of the first talk.

The spectral theory of selfadjoint operators provides an abstract framework for solving some of the main differential equations of mathematical physics: the heat equation, wave equation, and Schr{\"o}dinger equation. When the operators are invariant under a group action, a much more detailed analysis is often possible. This work on invariant differential operators on the metric Cayley graphs of free groups throws differential equations, graphs, group actions, functional analysis, algebraic topology, linear algebra, and a pinch of algebraic geometry into the blender. What emerges is a surprisingly satisfying concoction.
CLICK HERE FOR THE BEAMER SLIDES OF THE TALKS

Wednesday, September 30 
Bob Carlson UCCS 
Quantum Cayley Graphs for Free Groups (Talk 2 of 2) 
see above 
Wednesday, October 14 
Darren Funk Neubauer Colorado State University Pueblo 
Bidiagonal Pairs and Bidiagonal Triples  Roughly speaking, a bidiagonal pair is a pair of diagonalizable linear maps on a finitedimensional vector space, each of which acts in a bidiagonal fashion on the eigenspaces of the other. Bidiagonal pairs have been classified. The proof of this classification reveals that every bidiagonal pair can be extended to a triple of linear maps, in which each map acts bidiagonally on the eigenspaces of the other two. This leads to the notion of a bidiagonal triple. This talk will describe two different ways to define a bidiagonal triple, and then discuss the relationship between bidiagonal pairs and these two types of triples. There are a number of connections between bidiagonal pairs/triples and the representation theory of various well known algebras. The talk will describe these connections in detail. 
Wednesday, October 28 
Joe Timmer University of Colorado Boulder 
Hopf Algebras and Group Factorizations  With a group factorization $L=FG$ of a finite group and a field $k$ one may construct the bismash product Hopf algebra $H = k^G \# k F$. The connections between the representations of $L$ and $H$ have many similar qualities, especially when one looks at the FrobeniusSchur indicator.
In this talk, we will review (almost) all the background necessary for understanding the constructions of these Hopf algebras in question, the concept of indicators for Hopf algebras and also consider the case of factorizing the symmetric group $S_n$. 
Wednesday, November 11 
Gonzalo Aranda Pino Universidad de Málaga 
Decomposable Leavitt path algebras for arbitrary 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 natural generalizations of the aforementioned path and classical Leavitt algebras. In this talk we will characterize the Leavitt path algebras that are indecomposable (as a direct sum of twosided ideals) in terms of the underlying graph. When the algebra decomposes, it actually does so as a direct sum of Leavitt path algebras for some suitable graphs, and under certain finiteness conditions a unique indecomposable decomposition exists. This is a report on the joint paper: G. Aranda Pino, A. NasrIsfahani, "Decomposable Leavitt path algebras for arbitrary graphs", Forum Mathematicum, (to appear)" 
Wednesday, December 2 
John Beachy Northern Illinois University 
Universal Localization at Semiprime Ideals  P.M.Cohn defined the universal localization at a semiprime ideal S of a Noetherian ring to be the ring universal with respect to inverting all matrices regular modulo S. The universal localization coincides with the Ore localization, when that exists, and Goldie's localization is a homomorphic image of the universal localization. I will review the construction and some basic properties, and then focus on some of the difficulties and open questions. While wellbehaved modulo powers of its Jacobson radical, the "bottom" part of the universal localization is hard to calculate. It also fails to preserve chain conditions even for very wellbehaved rings. But I still believe that it may provide the language necessary to extend some commutative localization techniques to the noncommutative case. 
Spring 2015 Schedule
All talks in ENGR 201 (Engineering Dean's Conference Room) on the UCCS campus, unless otherwise indicated.
(note: this is a different room than the one we've used in the past ...)
All "Rings and Wings" talks begin at 3:45pm, 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, February 25

Zak Mesyan UCCS 
InfiniteDimensional Diagonalization 
Let V be an arbitrary vector space over a field K, and let End(V) be the ring of all Klinear transformations of V. We characterize the diagonalizable linear transformations in End(V), as well as the (simultaneously) diagonalizable subalgebras of End(V), generalizing results from classical finitedimensional linear algebra. These characterizations are formulated in terms of a natural topology on End(V), which reduces to the discrete topology when V is finitedimensional. This work was done jointly with Mio Iovanov and Manny Reyes. 
Wednesday, March 4 
Gonzalo Aranda Pino Universidad de Málaga

KumjianPask algebras of higher rank graphs. 
In this talk I will present the basic definitions and first results of the theory of the KumjianPask algebras. These algebras are both the higherrank generalizations of the Leavitt path algebras (a higherrank graph essentially being a multilayered or multidimensional graph) and the algebraic analogs of the graph C*algebras of higherrank graphs. After the main definitions and examples, we prove graded and CuntzKrieger uniqueness theorems for these algebras, establish the simplicity result and analyze their ideal structure. This is a report of the paper:G. Aranda Pino, J. Clark, A. an Huef and I. Raeburn, "KumjianPask algebras of higher rank graphs", Trans. Amer. Math. Soc. 365 (7), 36133641 (2013). 
Wednesday, April 1

Kulumani Rangaswamy UCCS 
Leavitt path algebras asgraded algebras(Preliminary report)


Wednesday, April 15**
**This talk will happen 4:30  5:45 
Ehsaan Hossain University of Waterloo 
Quillen and Suslin's Famous Theorem  In differential geometry and topology, it's well known that all continuous vector bundles on a contractible topological space, such as $\mathbf{R}^n$, are continously trivial. This can be extended to even show that all holomorphic vector bundles on $\mathbf{C}^n$ are \textit{holomorphically} trivial. Since the affine $n$space $\mathbf{A}^n$ is also contractible in the Zariski topology, it was conjectured by Serre in the 50's that $mathbf{A}^n$ admits no nontrivial \textit{algebraic} vector bundles. This became known as Serre's Problem. On the algebraic side of things, this question can be interpreted as asking whether all finite projective modules are free over a polynomial ring $k[x_1,\ldots,x_n]$. As a result of the work of D. Quillen in the '70s, and independently by A. Suslin, it was shown that all finite projective modules over $k[x_1,\ldots,x_n]$ are free, and consequently Quillen earned the Fields Medal in '78. L. Vaserstein gave an elementary proof of Quillen's result using the theory of unimodular rows. In this talk I hope to share the rich history of Serre's Problem and give an overview of Vaserstein's simplification. 
Wednesday, April 22**
**This talk will happen 4:30  5:45 This talk will happen in ENGR 247

Cristóbal Gil Universidad de Málaga 
Leavitt path algebras of Cayley graphs $C_n^3$. 
Let $n$ be a positive integer and for each $0\leq j\leq n1$ we let $C_n^j$ denote Cayley graph for the cyclic group $\mathbb{Z}_n$ with respect to the subset $\{1,j\}$. For each pair $(n,j)$, the size of the Grothendieck group of the Leavitt path algebra $L_K(C_n^j)$ is related to a collection of integer sequences described by Haselgrove. When $j = 0,1,2$ it is possible to analyze the Grothendieck group of the Leavitt path algebras $L_K(C_n^j)$ in order to explicitly realize them as the Leavitt path algebras of graphs having at most three vertices, thanks to a KirchbergPhillips type result. The case $j=2$ has some surprising connection to the classical Fibonacci sequence and also in case $j=3$, it is related to a sequence of Fibonaccitype, called Narayana's cow sequence. 
Thursday, April 30 UCCS Math Department Colloquium 12:15  1:30pm OSB A327 
Cristóbal Gil Universidad de Málaga 
Leavitt path algebras of Cayley graphs.  Leavitt path algebras are natural generalizations of path algebras (or algebras associated to graphs). On the other hand, they include the algebras without the Invariant Basis Number (IBN) property originally introduced by Leavitt, and many other interesting properties. Also Leavitt path algebras are the algebraic counterparts of C*algebras. Let $n$ be a positive integer and for each $0\leq j \leq n1$ we let $C_n^j$ denote Cayley graph for the cyclic group $\mathbb{Z}_n$ with respect to the subset $\{1,j\}$. When $j = 0,1,2$ it is possible to analyze the Grothendieck group of the Leavitt path algebras $L_K(C_n^j)$ in order to explicitly realize them as the Leavitt path algebras of graphs having at most three vertices. The case $j=2$ has some surprising connection to the classical Fibonacci sequence. In case $j=3$, it is related to a ``Fibonaccilike" sequence, called Narayana's cow sequence. I will discuss some known properties of the structure of the group in the $j=3$ case, and give some conjecture. 
Wednesday, May 6 3:45  5:XX ENGR 201 
Francesca Mantese Università degli Studi di Verona

Extensions of Chen simple modules over Leavitt path algebras

Let $E$ be a directed graph, K any field, and let $L_K(E)$ denote the Leavitt path algebra of $E$ with coefficients in $K$. 
Thursday, May 7 UCCS Math Department Colloquium 12:15  1:30pm OSB A327 
Alberto Tonolo Università degli Studi di Padova 
Equivalences between categories of modules.  Two equal problems have equal solutions: therefore it is not necessary to solve both of them. There are relationships weaker than equality which are useful in the same sense. For instance, two equivalent linear systems are not equal, but they have the same solutions. Another example that one meets very soon in abstract algebra is the concept of isomorphism: two isomorphic objects share the same algebraic properties. In this talk we will concentrate on Rings. Isomorphic rings are not equal, but they can be thought as essentially the same, only with different labels on the individual elements. In this talk I will present other notions of ``similarity'' between rings; in particular we will discuss Morita equivalent and Tilting equivalent rings. They are relationships defined between rings that preserves some ringtheoretic properties. Giving priority more to comprehensibility than to precision, I hope to be able to give you through examples the taste of an important contemporary field of research. 







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 GelfandKirillov 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

The center of rings associated to directed graphs 
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 
Game theoryand 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 socalled 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 ENGR 239 
Jeff Boersema Seattle University (visiting U. New Mexico) 
Real structures in graph C*algebras  Real C*algebras are the counterpart to C*algebras in which we replace the field of complex scalars by the reals. Any real C*algebra A can be embedded uniquely in a complex C*algebra isomorphic to A + iA (called the complexification); but two different real C*algebras can have the same complexification up to isomorphism. The general problem is to find, for a given complex C*algebra, all real corresponding C*algebras. We will introduce a graphbased construction for identifying real structures in graph algebras. 
Wednesday November 5 4:00 5:XX 
Greg Oman UCCS 
Strongly Jonsson binary relational structures 
Let X be a set, and let R be a binary relation on X. Say that a subset Y of X is an Rlower set provided whenever y is in Y and xRy, then also x is in Y. Say that the structure (X,R) is strongly Jonsson provided distinct Rlower subsets of X have distinct cardinalities. In this note, we consider the problem of classifying the strongly Jonsson binary relational structures. In particular, we relate the problem to the wellknown (and unsolved) "distinct subset sum problem" in combinatorics. 
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) 
An Elementary Introduction to Weak Hopf Algebras  Weak Hopf algebras combine two important generalizations of the notion of a group: Hopf algebras and groupoids. In a Hopf algebra, we replace the set of group elements with a vector space and systematically require all of the structure, such as multiplication and inversion, to be linear. This generalizes the "classical" notion of symmetry to "quantum" symmetry. In a groupoid, on the other hand, we replace the group multiplication by a partiallydefined operation: only specified pairs of elements can be multiplied. This generalizes the notion of symmetry to include not only transformations of a single object but also symmetries between different objects. The theory of weak Hopf algebras combines key aspects of these two generalizations, describing "multiobject quantum symmetries." 
POSTPONED UNTIL SPRING 2015

Gonzalo Aranda Pino U. Malaga (Spain) 
KumjianPask algebras of higher rank graphs.  In this talk I will present the basic definitions and first results of the theory of the KumjianPask algebras. These algebras are both the higherrank generalizations of the Leavitt path algebras (a higherrank graph essentially being a multilayered or multidimensional graph) and the algebraic analogs of the graph C*algebras of higherrank graphs. After the main definitions and examples, we prove graded and CuntzKrieger uniqueness theorems for these algebras, establish the simplicity result and analyze their ideal structure. This is a report of the paper: G. Aranda Pino, J. Clark, A. an Huef and I. Raeburn, "KumjianPask algebras of higher rank graphs", Trans. Amer. Math. Soc. 365 (7), 36133641 (2013). 
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. 