"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 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 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 follow-up talk will drill down deeper. We will schedule the second talk at the end of the first talk. The spectral theory of self-adjoint 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 finite-dimensional 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 Frobenius-Schur 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 two-sided 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. Nasr-Isfahani, "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 well-behaved 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 well-behaved rings. But I still believe that it may provide the language necessary to extend some commutative localization techniques to the noncommutative case.

Spring 2016 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

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

Infinite-Dimensional Diagonalization

Let V be an arbitrary vector space over a field K, and let End(V) be the ring of all K-linear 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 finite-dimensional linear algebra. These characterizations are formulated in terms of a natural topology on End(V), which reduces to the discrete topology when V is finite-dimensional. This work was done jointly with Mio Iovanov and Manny Reyes.

Wednesday, March 4

Gonzalo Aranda Pino

# In this talk I will present the basic definitions and first results of the theory of the Kumjian-Pask algebras. These algebras are both the higher-rank generalizations of the Leavitt path algebras (a higher-rank graph essentially being a multilayered or multidimensional graph) and the algebraic analogs of the graph C*-algebras of higher-rank graphs. After the main definitions and examples, we prove graded and Cuntz-Krieger 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, "Kumjian-Pask algebras of higher rank graphs", Trans. Amer. Math. Soc. 365 (7), 3613-3641 (2013).

Wednesday, April 1

Kulumani Rangaswamy

UCCS

###### Leavitt path algebras as

(Preliminary report)

This talk will describe some of the properties related to the graded structure of Leavitt path algebras over arbitrary graphs.

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

Leavitt path algebras of Cayley graphs $C_n^3$.

Let $n$ be a positive integer and for each $0\leq j\leq n-1$ 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 Kirchberg-Phillips 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 Fibonacci-type, called Narayana's cow sequence.

Thursday, April 30

UCCS Math Department Colloquium

12:15 - 1:30pm

OSB A327

Cristóbal Gil

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 $0\leq j \leq n-1$ 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 Fibonacci-like" 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 $K$.
In this talk we first give an explicit description of a projective resolutions of S, where S is a Chen simple module over $L_K(E)$. When E is a finite graph, this will allow us to analyze the Ext groups $\Ext^1_{L_K(E)}(S, T)$ for any two Chen simple modules S and T. As an application, when E is a finite graph containing at least one cycle, we show the existence of indecomposable left $L_K(E)$-modules of any prescribed finite length. The talk is based on a joint paper with Gene Abrams and Alberto Tonolo.

Thursday, May 7

UCCS Math Department Colloquium

12:15 - 1:30pm

OSB A327

Alberto Tonolo

Università degli

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 ring-theoretic 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 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

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 graph-based 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 R-lower 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 R-lower 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 well-known (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 partially-defined 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 "multi-object quantum symmetries."

POSTPONED UNTIL SPRING 2015

Wednesday

December 10

4:00 - 5:XX

Gonzalo Aranda Pino

U. Malaga (Spain)

Kumjian-Pask algebras of higher rank graphs.

In this talk I will present the basic definitions and first results of the theory of the Kumjian-Pask algebras. These algebras are both the higher-rank generalizations of the Leavitt path algebras (a higher-rank graph essentially being a multilayered or multidimensional graph) and the algebraic analogs of the graph C*-algebras of higher-rank graphs. After the main definitions and examples, we prove graded and Cuntz-Krieger 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, "Kumjian-Pask algebras of higher rank graphs", Trans. Amer. Math. Soc. 365 (7), 3613-3641 (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, 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.