"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.

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, two-sided) 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 non-commutative, but the presence of the involution makes the usual left-right differences for non-commutative 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 Undergraduate-oriented 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 well-known 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 well-known) Mad Vet Scenarios have provided a source of thought-provoking 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 K-theory 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 K-theory has proven to be a very useful and powerful tool for classification. In this talk we will examine the algebraic K-theory 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 K-theory is notoriously difficult to compute in general, we will present cases in which the algebraic K-theory 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 K-theory and discuss what K-theoretic 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 K-12 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 Cuntz-Krieger 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 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.

 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 Link to abstract 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 Link to abstract 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 Link to abstract 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 AB-BA. 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 Link to abstract May 1 Pere Ara Universitat Autonoma de Barcelona Lamplighter groups and separated graphs Link to abstract

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.