DATE

SPEAKER

TITLE

ABSTRACT

March 18

Alan Loper

Ohio State University

Intersections of ultrapowers

An ultrafilter can be used to build an ultraproduct of an infinite collection of rings.  The

ultraproduct is a quotient of the direct product.  It shares with the direct product the feature

that it is very large.  In this work we consider each component of the product to be the

same ring and refer to the ultraproduct as an ultrapower.  Let $D$ be a commutative

integral domain and fix a nonprincipal ultrafilter to use building ultrapowers.  If $D$ is local

then an ultrapower of $D$ is also local.  If $D$ has an infinite number of maximal ideals

then an ultrapower of $D$ has a lot of maximal ideals - perhaps more than can reasonably

be studied.  One way to ameliorate this situation a little is to go from $D$ to the localizations

$D_M$ for each maximal ideal $M$, construct an ultrapower of each localization, and then

intersect the resulting rings.  We consider this construction, and then move to a more general

setting.  Choose a collection of overrings of $D$ (rings that lie between $D$ and its quotient field)

that intersect to $D$,  construct the ultrapower of each overring, and then intersect the resulting

rings.  Of particular interest is the case where each overring is a valuation domain.  We

demonstrate some surprising properties of these intersections.

THURSDAY,

March 19

UCCS Math Department

Colloquium

Alan Loper

Ohio State University

Unions of chains of quadratic extensions of regular local rings

We start with a localization of the ring of polynomials in two or more variables over the complex numbers.  We build local extension rings by "blowing up" and then localizing.  The resulting regular local rings are known as a quadratic transforms.  The process of building quadratic transforms can be iterated, so we build a quadratic transform tree of successively larger and larger rings - in lots of different directions.  Abhyankar showed in 1956 that in the two-variable case the union of the rings comprising any one infinite branch is a valuation domain.  This is not true in dimensions three or higher.  In this talk we look at what does happen in dimension three.  The question of which branches do lead to valuation domains in three dimensions has surprising connections to matrix theory, number theory (continued fractions, farey numbers) and a fractal.  This is a wide open problem.  The presentation will be a combination of theorems and conjectures.  This is joint work with Nick Werner.

April 8

Luke Harmon

UCCS

Lower Finite Modules over Commutative Rings

A partially ordered set (P,<) is lower finite provided P is infinite and for each x in P, there are but finitely many elements y in P such that y<x. We will call a module M lower finite if the set of proper submodules of M, partially ordered by set-theoretic containment, is lower finite. We will use the (well-studied) class of Jonsson modules (along with other classical results) to classify the lower finite modules over a commutative ring with identity.

April 22

David Milovich

Welkin Sciences, Colorado Springs

Projective Boolean algebras in locally finite terms

A Boolean algebra is projective iff it is a retract

of a free Boolean algebra. This property is nontrivial

only for uncountable Boolean algebras.

Yet, I have obtained a locally finite characterization:

a Boolean algebra is projective iff it is the union of

a family F of finite subalgebras such that

(1) every A and B in F are contained in some C in F, and

(2) if n is a natural number, A_0,...,A_n are in F,

and A_0,...,A_{n-1} are subalgebras of A_n,

then A_n contains the Boolean algebra freely generated

by the partial algebra A_0 union A_1 union ... A_{n-1}.

Restricting the above n to be at most some fixed d defines

the d-ary strong Freese-Nation property (SFN_d), which

I have shown characterizes projectivity below size aleph_d.

Based on 1990s results of Heindorf and Shapiro,

SFN_1 does not imply SFN_2 does not imply SFN_3.

My most recent result is that SFN_d does not imply

SFN_{d+2} in general. I conjecture that SFN_d

does not imply SFN_{d+1} in general.

May 6

P. N. Ánh

Renyi Institute of Mathematics, Budapest

t.b.a.

DATE

SPEAKER

TITLE

ABSTRACT

February 13

Zak Mesyan

UCCS

Transformation Semigroups and Their Topologies

Endomorphism rings, permutation groups, and transformation semigroups play fundamental roles in the theories of rings, groups, and
semigroups, respectively. (E.g., consider Cayley's theorem.) In the case where these objects are infinite, it is often convenient to put topologies on them, when trying to understand their structure. (E.g., consider the Jacobson density theorem.) Consequently, topologies on
endomorphism rings and permutations groups have received considerable attention over the years. Transformation semigroups, on the other hand, have not been nearly as fortunate. I will describe my recent attempt (with James Mitchell and Yann Peresse) to rectify this state of affairs.
It turns out, for example, that the transformation semigroup of a countably infinite set has a unique Polish semigroup topology. During the talk I will review the necessary topology, and discuss connections between the aforementioned rings, groups, and semigroups.

February 20

[could be you!]

February 27

Shishir Agrawal

Colorado College

How local systems arise

We'll begin by meandering through a bit of algebraic topology and complex analysis, discussing what local systems are and how differential equations give rise to local systems via the Riemann-Hilbert correspondence. This will motivate a transition to algebra and a discussion of positive characteristic analogs of this correspondence. Due initially to Katz and then expanded upon by Emerton and Kisin, these analogs show that étale local systems on smooth algebraic varieties in positive characteristic arise via difference equations (which are encoded by objects known as F-crystals). We'll conclude with a discussion of how Bhatt and Scholze's pro-étale topology should streamline the proof of this positive characteristic version of the Riemann-Hilbert correspondence. 

March 6

[could be you!]

THURSDAY March 7

UCCS Math Dept Colloquium

Jason Boynton

North Dakota State U

Factorization in rings of polynomials of the form D+M (and generalizations)

https://www.uccs.edu/math/sites/math/files/inline-files/Colloquium_Boynton.pdf

March 13

CANCELLED due to blizzard, to be rescheduled

Derek Wise

Fractals and Monads

Fractals in geometry are deeply related to monads in algebra. In particular, fractals and other self-similar structures are often generated by "iterating" functions, but the relevant type of iteration is best understood in terms of certain monads. Moreover, a fractal can often be understood algebraically as an algebra for the corresponding monad. This talk will be a gentle introduction to monads, and some of my recent explorations in using them to understand fractals. I will take a practical approach, working mostly with examples, and also showing how monads can be used concretely in methods of actually drawing pictures of fractals. This will also also include a glimpse of the role of monads in computer science, though no particular knowledge of computer science will be assumed. 

March 20

CANCELLED, hopefully to be rescheduled for AY 2019/2020

Keith Kearnes

CU Boulder

Commutator theory for group-like algebras 

I will speak about a common generalization of the group commutator and ideal multiplication for rings. I will also discuss its generalization to higher arity commutator operations. 

THURSDAY

March 21 

UCCS Math Dept Colloquium 

Daniel Herden
Baylor University

Local automorphisms and incidence algebras 

https://www.uccs.edu/math/sites/math/files/inline-files/Colloquium_Herden.pdf 

[[March 27]]

[[ Spring Break ]]

April 3

Luke Harmon

UCCS

Lower-finite partially ordered sets

We will introduce the order-theoretic notion of a lower-finite partially ordered set and focus, in particular, on the partially ordered set formed by the submodules (ordered by containment) of a module over a commutative, unital ring. Our goal is to classify all of the lower-finite modules over a commutative, unital ring. The results we have obtained to this end will be presented in a highly accessible manner.

April 10

April 17

THURSDAY April 18

UCCS Math Dept Colloquium

John Lorch

Ball State University

April 24

May 1

Derek Wise

Fractals and Monads

Fractals in geometry are deeply related to monads in algebra. In particular, fractals and other self-similar structures are often generated by iterated application of some function, but the relevant function may be multi-valued, and "iterating" such a function is best understood algebraically in terms of monads. This talk will be a gentle introduction to monads, and some of my recent explorations in using them to understand fractals. I will take a practical approach, working mostly with examples, and also showing how monads can be used concretely in methods of actually drawing pictures of fractals. This will also also include a glimpse of the role of monads in computer science, though no particular knowledge of computer science will be assumed.

May 8

DATE

SPEAKER

TITLE

ABSTRACT

October 10

Greg Oman,

UCCS

Divisible multiplicative groups of fields

Some time ago, Laszlo Fuchs asked the following question: which abelian groups can be realized as the multiplicative group of (nonzero elements of) a field? This question and several variants have been studied by Fuchs, Contessa, Mott, Guralnick, and Wiegand, among others. In this talk, we outline a solution of the problem within the subclass of divisible abelian groups, that is, we characterize which divisible abelian groups can be realized as the multiplicative group of a field. Some simple applications as well as more recent progress on the question will be presented as well, time-permitting.

Ocober 24

Damiano Fulghesu,

Minnesota St. University, Moorhead

Tiling of finitely generated groups

Let G be a finitely generated group. We say that a subset K of G is a (right) tile if there exists a subset C of G such that the family of sets cK, as c varies in C, is a partition of the group G.
In this talk we give a quick overview of what it is known in the literature about tiles in finitely generated groups. We also present some open problems, like the Coven-Meyerowitz conjecture on the complete classification of tiles in Z, the group of integers. 

November 7

David Milovich, 

Welkin Sciences (Colo Spgs)

Higher-order amalgamation of algebraic structures

Call two algebraic structures A, B
overlapping if the algebraic operators of A and
the algebraic operators of B agree when restricted
to their common domain A intersect B. Say that
n pairwise overlapping algebraic structures
A_1,...,A_n amalgamate in class C
if there is a structure B in C simultaneously
extending each A_i.

For example, any two overlapping Boolean rings
amalgamate in the class of Boolean rings.
But there are three overlapping finite Boolean rings that do not amalgamate, not even in the class of commutative rings. On the other hand, any number of overlapping vector spaces (over a fixed field) always amalgamate in the class of vector spaces.

I will survey what I know about n-ary amalgamation for various familiar algebraic structures and then introduce a nontrivial sufficient condition for n-ary amalgamation stated in terms of category theory.

November 28

Beth Malmskog,

Colorado College

Locally Recoverable Codes with Many Recovery Sets from Fiber Products of Curves

Error correcting codes are systems for incorporating redundancy into stored or transmitted data, so that errors can be identified and even corrected. A good error correcting code is efficient and can correct many errors relative to its efficiency. These codes are ubiquitous in the digital age, and many excellent codes arise from algebraic constructions. The increasing importance of cloud computing and storage has created a need for codes that protect against server failure in large computing facilities. One way of approaching this problem is to ask for local recovery. An error correcting code is said to be locally recoverable if any symbol in a code word can be recovered by accessing a subset of the other symbols. This subset is known as the helper or recovery set for the given symbol. It may be desirable to have many disjoint recovery sets for each symbol, in case of multiple server failures or to provide many options for recovery. Barg, Tamo, and Vladut recently constructed LRCs with one and two disjoint recovery sets from algebraic curves. This talk presents a generalization of this construction to three or more recovery sets, using fiber products curves over finite fields. This is joint work with Kathryn Haymaker and Gretchen Matthews.

THURSDAY

November 29

12:30 - 1:30

Osborne Science Bldg

Room A327

(part of the UCCS math department colloquium series) 

Kulumani M. Rangaswamy

The multiplicative ideal theory of Leavitt path algebras: Are Leavitt path algebras really commutative algebras in non-commutative clothing? 

https://www.uccs.edu/math/sites/math/files/inline-files/Colloquium_Ranga.pdf

DATE

SPEAKER

TITLE

ABSTRACT

February 7

Greg Oman,

UCCS

Commutative (unital) Artinian rings are Noetherian: a brief history and an elegant proof

Beamer slides

The study of so-called "chain conditions" for ideals of a ring began to take root in work of Emmy Noether and Emil Artin dating back to the early 20th century. Recall that a commutative ring R is Noetherian if R does not have an infinite, strictly ascending chain of ideals; R is Artinian if R has no infinite, strictly decreasing chain of ideals (there are noncommutative analogs, of course). A curious fact is that if R is a commutative ring with identity (yes, 1 is required for the following result), then R being Artinian is sufficient to guarantee that R is Noetherian, but the converse does not hold. In this talk, I will outline the standard proof of this result and conclude with a (more recent) short and elegant proof due to Karamzadeh.

Feb 14

Feb 21

Zak Mesyan,

UCCS

Infinite-Dimensional Triangularization 

Beamer slides

One of the most commonly used tools in linear algebra, both theoretically and computationally, is the fact that every matrix with entries from an algebraically closed field is similar to an
upper-triangular matrix (with possible additional structure, such as the Jordan canonical form). I will describe my recent attempts to generalize
the notion of "upper-triangular" to linear transformations of an arbitrary vector space over any field, and to generalize some of the usual facts about triangular matrices to that context. Among other topics, I will discuss the Jordan canonical form, minimal polynomials,
nilpotent transformations, simultaneous triangularization of commuting
transformations, and triangularizable algebras.

Feb 28

[[ analysis seminar ]] 

Mar 7

Luke Harmon,

UCCS

Rings Between Z and Q

Beamer slides

In this expository talk, we will use fundamental techniques in commutative ring theory related to prime ideals, multiplicative sets, and localization to categorize all of the subrings of the rationals that contain the integers as a subring. 

Mar 14

Ethan Berkove,

Lafayette College

Short Paths and Long Titles: Travels through the Sierpinski carpet, Menger sponge, and beyond. 

(Joint work with Derek Smith)

The Sierpinski carpet and Menger sponge are fractals which can be thought of as two and three dimensional versions of the Cantor set. Like the Cantor set, each is formed by starting with a shape (a square for the carpet, a cube for the sponge) and then recursively removing certain subsets of it. Unlike the Cantor set, what remains is connected in the following sense: given any two points s and f in the carpet or sponge, there is a path from s to f that stays in the carpet or sponge. In this talk, we’ll discuss what we know about the shortest path from s to f in the carpet, sponge, and even higher dimensional versions of these fractals. The proofs required a surprising (at least to us) breadth of techniques, from combinatorics, geometry, and even linear programming. 

Mar 21

John McHugh,

UCCS and U. Vermont

An Introduction to Character Theory

Beamer slides

This presentation will begin with a discussion of the complex representation theory of finite groups including the theorems of Maschke and Wedderburn. After studying the basic properties of representations, we will move on to discuss the character theory of finite groups. This is a powerful subfield of representation theory which allows one to dissect the structure of a finite group via examinations of associated numerical invariants. For example, if one knows the "character table" of a given finite group G (which turns out to be a square matrix of finite dimension) then one can tell right away what the normal subgroups of G are and even whether or not G is abelian, nilpotent, solvable, or simple. This presentation will be aimed at those who have never seen the theory before or have not seen it in a long time. To understand everything in the presentation, one should have a decent grasp on finite group theory, ring theory (including algebras over fields), and modules. 

[Mar 28]

[[[ Spring Break ]]]

Apr 4

Kulumani M. Rangaswamy,

UCCS

The history and the evolution of the concept of primitive rings

This expository talk will trace the origin and the development of the concept of primitive rings and outline their properties, with a leisurely review of the earlier works of Lie, Wedderburn, Artin, Jacobson, Kaplansky and others. This talk will mostly be at a level that our graduate students will be able to follow.

Apr 11

[[ analysis seminar ]]

Apr 18

Gene Abrams,

UCCS

Leavitt path algebras are Bezout

We show that for any graph $E$ and any field $K$, any finitely generated left ideal of the Leavitt path algebra $L_K(E)$ is principal (i.e., generated by a single element). In other words, $L_K(E)$ is Bezout. We'll give some background on the subject, and present some of the more-intriguing aspects of the proof. 

Apr 25

Matt Welz,

Fort Lewis College

The Finite Simple Groups and an Introduction to Fusion Systems

The Classification of Finite Simple Groups is one of the greatest achievements of 20th century mathematics. An immense undertaking,
filling over 10,000 journal pages across hundreds of articles, the Classification provides a "periodic
table" of the
finite simple groups. The sometimes mysterious ramifications of this massive project extend not only to mathematics beyond group theory, but to theoretical physics and chemistry. In this talk, I shall discuss various aspects of the
finite simple groups, including the"Monster", the largest of the "sporadic" simple groups, and how the Classification motivates my research
on fusion systems.

May 2

[[ analysis seminar ]] 

DATE

SPEAKER

TITLE

ABSTRACT

Wednesday Sept 13

Greg Oman, UCCS

A gentle introduction to model theory (and what it can do for you algebraically)

Link to: BEAMER SLIDES

In this talk, I'll present a rigorous (but somewhat gentle) introduction to the syntax and semantics of first-order logic. Time-permitting, I will discuss the classical theorems, including the Compactness Theorem, the Lowenheim-Skolem Theorems, Ultraproducts, and Los' Theorem (read "Washes Theorem"). I'll provide applications to ring theory, Given recent applications of model theory to C*-algebras, my goal is to argue that the basic tools of model theory can be useful to the algebraist in a variety of contexts.

Wednesday Sept 27

K. M. Rangaswamy, UCCS

Cotorsioness of modules and a problem of George Bergman

Link to: BEAMER SLIDES

Link to abstract

Wednesday Oct 25

Dan Bossaller, Ohio U.

Leavitt Path Algebras having Bases Consisting Solely of Strongly Regular Elements

Link to: Beamer Slides

Leavitt path algebras were introduced independently by Abrams and Aranda-Pino in 2005 and Ara, Moreno, and Pardo in 2007 as purely algebraic analogues of the graph $C^*$ algebras introduced by Kumjian, Pask, Raeburn, and Renault in 1997 and 1998. Various recent papers deal with the family of so called ``invertible algebras," those algebras over arbitrary (not necessarily commutative) unital rings which have bases that consist solely of invertible elements. Many familiar algebras satisfy this property, including all finite dimension algebras over fields other than $\mathbb{F}_2$ and all $n \times n$ matrix algebras over unital rings. L\'opez-Permouth and Pilewski gave a complete characterization of precisely which Leavitt path algebras of finite graphs are invertible. In this talk I will introduce the concept of a locally invertible algebra, that is, an algebra $A$ having basis $\mathcal{B}$ such that for every $b \in \mathcal{B}$, there exists some idempotent $e$ for which $b$ is invertible in the corner algebra $eAe$. We will show that this definition is equivalent to the algebra having a basis consisting solely of strongly regular elements, and then we will give various examples and non-examples of locally invertible algebras. Afterwards we will examine local invertibility in the context of Leavitt path algebras, in particular giving a complete characterization of strongly regular monomials and, as a corollary, showing that all von Neumann regular and all directly finite Leavitt path algebras are locally invertible. If time permits, we will show that the algebraic analogues of many algebras from operator theory are locally invertible.

THURSDAY, October 26 

12:30-1:30

UCCS Mathematics Department Colloquium Series

Dan Bossaller, Ohio U.

Associativity and Infinite Matrices

Link to: Beamer Slides

In a typical first course in linear algebra, finding a solution to a set of $m$ linear equations in $n$ unknowns guides much of the course. The representation of this system as the equation $Ax = b$ where $A$ is an $m \times n$ matrix (with $m$ and $n$ finite), $x \in K^n$, and $b \in K^m$ for some field $K$ is a central feature throughout. In order to find a solution, one would find a solution to this matrix equation $Ax = b$ by multiplying both sides by some matrix of row operations $U$ to write the matrix $A$ into some easily solvable form such as upper upper triangular form, row echelon form, etc. From there one solves the transformed equation, $(UA)x = Ub$, to get a solution to the original equation. One assures that this is indeed a solution by multiplying on the left by the inverse matrix $V$ for $U$. In the case of an infinite number of equations in infinitely many unknowns, the relevant matrices are infinite so one must take care that $V$, $U$, and $A$ are all multipliable matrices and that their product is associative. In this talk we will explore conditions under which the product of three matrices is associative. This talk will be accessible to a wide mathematical audience, requiring no more than an understanding of basic linear algebra.

Wednesday Nov 1

Gerard Koffi, U. Nebraska Kearney

Classification of distributive/thin representations via incidence algebras

Link to: Beamer slides

This talk will overview our recent work on application of incidence algebras of a finite poset P to classify various type of representations. In particular, we provide a unified approach to classify distributive, thin and finitely many orbits representations. The main tool is to introduce deformations of the incidence algebras which turns out to be the locally heriditary semisdistributive algebras and are classified in terms of the cohomology of the simplicial realization of the poset P.

Wednesday Nov 15

Jonathan Poritz

CSU Pueblo

Algebraic Turing Machines, with Applications to Quantum Computation

Alan Turing created the concept of what we now call "Turing machines" in his work on the Entscheidungsproblem ["decision problem"], one of David Hilbert's great problems motivating 20th century mathematics. A key insight in Turing's work was the centrality of "universal Turing machines" [UTMs], which could simulate any other Turing machine. By now Turing machines are as fundamental to theoretical computer science as bugs are to entomology and in applications, we tend to think of real-world, physical computational devices as engineered (and sadly finite) realizations of UTMs. There are, however, other equivalent models of computation which tend to be used in specific situations, including [Turing's thesis advisor] Alonzo Church's "lambda-calculus" and the "circuit model."
In this talk, I will present yet another model of computation by means of a construction which relies heavily on algebraic structures. The advantage of this construction is that it admits mathematically natural generalizations, one of which is equivalent to the model of of "probabilistic Turing machines" which are used in crypto and complexity theory. Another generalization (not even particularly complex mathematically) yields the recent and revolutionary (and terrifying, if you do crypto) idea of quantum computation. An advantage of this approach (to mathematicians, at least) is that allows precise mathematical statements and conjectures about ideas like what is universality and why quantum computers seem to be exponentially faster than classical ones.

Wednesday Nov 29

John Beachy

Northern Illinois U.

Universal localization of piecewise Noetherian rings

Link to beamer slides

A commutative ring R is called piecewise Noetherian if it has a Noetherian spectrum and for each prime ideal P the set of P-primary
ideals satisfies the ACC. In the noncommutative case there is a comparable definition, equivalent to the statement that R has ACC on prime ideals and for each ideal I the factor ring R has finite
reduced rank. After introducing piecewise Noetherian rings, the talk will explore the conjecture that this class of rings is the right one in which to study universal localization at semiprime Goldie ideals.

DATE

SPEAKER

TITLE

ABSTRACT

*** Special Day ***

Friday April 14

Kevin O'Meara

U. Christchurch 

(New Zealand)

The Gerstenhaber Problem and a computer search for a counterexample

Fifty six years on, it remains an open question whether Gerstenhaber’s (most surprising) theorem (Annals Math 1961) for two commuting n×n matrices A,B 2 Mn(F) over a field F also holds for three commuting matrices A,B,C in Mn(F):
must the (unital) subalgebra F[A,B,C] generated by A,B,C have dimension at most n? Chuck Vinsonhaler and I got interested in this in 2003 in connection with perturbing a finite collection of matrices to simultaneously diagonalizable matrices,
and in turn this was of considerable interest to some folk working in phylogenetics. The attack on the Gerstenhaber problem has involved many different branches of mathematics over the years: canonical forms in linear algebra, commutative and
noncommutative rings, module theory, topology, and algebraic geometry. It is one of the most fun projects I have ever been involved with!

In this (informal) talk, I will describe a promising attack with John Holbrook (U of Guelph, Ontario) and Pace Nielsen (BYU, Provo) of constructing a computer- generated counterexample, working with commuting triples of up to 500 × 500 matrices, and an upcoming run of the programs on a supercomputer. The critical tool in all of this is theWeyr form, a matrix canonical form similar to the Jordan form, but far superior here! As background to all of these things, you may wish to consult the monograph “Advanced Topics in Linear Algebra: weaving matrix problems through the Weyr form” (Oxford University Press 2011) by O’Meara,
Clark, and Vinsonhaler.

Our programs are written in MATLAB and use exact integer arithmetic. The 500 × 500 matrix size is not a true indication of the very large matrices M we
work with, more like 100, 000 × 500, 000 when we compute the dimension of the subalgebra F[A,B,C] as the rank of a certain such matrix M. Here we do not use Matlab’s subroutine rank(M), because it is slow and unreliable. Instead, we use the special form of matrices that centralize a Weyr matrix and our own tailored rank function (based on integer row reduction). The Weyr form allows recursive
calculations involving smaller matrices — this is its great beauty.

Wednesday

April 26

Roozbeh Hazrat

University of Western Sydney

(Australia)

Weighted Leavitt path algebras

Weighted Leavitt path algebras are a generalisation of Leavitt path algebras (with graphs of weight 1) and cover the algebras $L_K(n,n+k)$ constructed by Leavitt. In this talk we explain the concept and prove some theorems on the structure of these algebras. This is a joint work with Raimund Preusser
(Bielefeld/Brasila).

DATE

SPEAKER

TITLE

ABSTRACT

August 24

Jason Bell

University of Waterloo

Iterative algebras

There has been a lot of recent work on graded nil algebras by Smoktunowicz, Greenfeld, and others. These are N-graded algebras in which every homogeneous element is nilpotent and their study was motivated in part by trying to gain insight into unsolved Kurosh/Koethe-type problems about rings by studying the graded counterparts of these problems. We give a combinatorial construction of a family of graded nil rings, which we call iterative algebras, and we use these to answer some questions of Greenfeld, Leroy, Smoktunowicz, and Ziembowski. We give a survey of some of the ring theoretic properties of these algebras and give some open questions about them. This is part of joint works with Blake Madill and Be'eri Greenfeld.

August 31

NO seminar

x

September 7

CANCELLED

rescheduled to Dec 7

Andrew Kelley

SUNY - Binghamton

Subgroup growth:  a brief survey

The phrase "growth of groups" may bring to mind Gromov's celebrated theorem on polynomial *word* growth. However, there is another area of growth of groups that has received much attention, namely subgroup growth:

How many subgroups of a given index does a finitely generated group have? As the index increases, this number may grow rapidly. How fast (asymptotically) is this so called "subgroup growth"? And how does it relate to the algebraic, structural properties of the group?

In this survey, I will mention a few important results and techniques. As it turns out, some questions in subgroup growth relate to rings and modules, and I plan to briefly mention this as well. 

September 14

September 21

Greg Oman

UCCS

How did you come up with that?

Over the years, I have had the good fortune to work with a variety of undergraduate and graduate students. Most of these students have asked me the following important question: "How did you come up with the problem you gave me?"
Giving a complete and fully satisfactory answer to this question is no easy task. In this talk, I will attempt to give *an* answer to this query. In particular, I will illustrate the problem-creating and problem-solving process of doing research mathematics via a microcosm: specifically, with a recreational problem I have created. My plan is to informally discuss the sorts of questions I have asked to come up with the problem, giving a "behind-the-scenes" peak into the process of inventing new mathematics.
The mathematics involves only the notion of "group" and "homomorphism"; having taken an introductory course in modern algebra is more than sufficient to guarantee that you will comprehend the mathematics presented in my talk. 

September 28

****

THURSDAY OCTOBER 6

12:15 - 1:30pm

(UCCS Math Department Colloquium)

***

Ben Steinberg

City College of New York  and

CUNY Graduate Center

Representation theory 

and random walks

We discuss how a number of very natural Markov chains, including card shuffling and the Tsetlin library, can be modeled as random walks on finite monoids and how the representation theory of these monoids can be used to compute the eigenvalues of the transition matrices of these walks. No prior knowledge about Markov chains or monoids will be assumed.

October 12

James Mitchell

St. Andrews University (Scotland)

Chains of subsemigroups

The length of a semigroup S is defined to be the largest size of a chain of subsemigroups of S. An exact formula for the length of the symmetric group on n points was found by Cameron, Solomon and Turull; the length is roughly 3n/2. In general, it follows by Lagrange's Theorem that the length of a group is at most the logarithm of the group order. Semigroups refuse to be as well-behaved. The only valid upper bound for the length of an arbitrary semigroup is its size. For example, any zero-semigroup has length equal to its size. Even for less degenerate and more natural examples of semigroups, the contrast to groups is noticable. We will see that the length of the full transformation semigroup on n points, the semigroup analogue to the symmetric group, is asymptotically at least a constant multiple of its size.

October 19

October 26

Indah Emilia Wijayanti

Universitas Gadjah Mada (Indonesia);

Fulbright Scholar, 

currently visiting Ohio University

On Nice Modules

A ring with an identity is called a clean ring if every element of the ring decomposes as a sum of an idempotent and a unit of the ring. We introduce and investigate the new concept of a nice module, that is a module whose every submodule has a clean endomorphism ring. We present some sufficient conditions under which a module is nice, give examples of modules that are nice and those are not nice, show how to construct nice modules and raise an interesting question in the meantime. 

November 2

November 9

November 16

November 23

NO seminar - Thanksgiving

x

November 30

THURSDAY

December 1

UCCS Math Dept Colloquium

12:30 - 1:30pm 

refreshments at 12:15

Osborne A204

Andrew Kelley

SUNY Binghamton

Maximal subgroup growth of some groups

Let m_n(G) denote the number of maximal subgroups of a finitely
generated group G of index n. How do the algebraic/structural properties of G
control the growth rate of m_n(G)? Others have researched the broad picture and
described what it means for m_n(G) to be bounded above by a polynomial in n.
However, there are only a few groups whose degree of growth is known. If we
restrict to particularly nice classes of groups however, then asymptotic formulas (or
bounds) can be given. We will focus on metabelian groups, especially those that
are abelian by cyclic. Beyond this, current progress on virtually abelian groups and
Baumslag-Solitar groups may also be mentioned.

December 7

Andrew Kelley

(rescheduled from Sept 7)

Subgroup growth: a brief survey

The phrase "growth of groups" may bring to mind Gromov's celebrated theorem on polynomial *word* growth. However, there is another area of growth of groups that has received much attention, namely subgroup growth:

How many subgroups of a given index does a finitely generated group have? As the index increases, this number may grow rapidly. How fast (asymptotically) is this so called "subgroup growth"? And how does it relate to the algebraic, structural properties of the group?

In this survey, I will mention a few important results and techniques. As it turns out, some questions in subgroup growth relate to rings and modules, and I plan to briefly mention this as well. 

DATE

SPEAKER

TITLE

ABSTRACT

February 3

[available]

February 10

Kulumani Rangaswamy

UCCS

On prime ideals in Leavitt path algebras

In this talk I’ll present some information about prime factorization of ideals in a Leavitt path algebra L. This may include some (preliminary) results showing how the behavior of graded ideals influence that of the non-graded ideals in L.

February 17

[available]

February 24

TALK TO BE HELD AT COLORADO COLLEGE,

TUTT SCIENCE BUILDING

ROOM 229

Michael Penn

Colorado College

Vertex Operator Algebras: Motivation, Definition, and Examples

Part I

Vertex operator algebras(VOA) are a relatively new class of algebraic objects which have found uses across many branches of mathematics and physics: representation theory, modular forms and q-series, the study of finite simple groups, string theory, and topological quantum field theory. In this talk we will explore three equivalent formulations of a vertex algebra and provide a few examples — the Heisenberg(VOA) and lattice VOAs. The Heisenberg VOA can be considered a “quantum” version of the classical polynomial algebra, while lattice VOAs will form the basis of a second talk where we will cover several recent results.

March 2

TALK TO BE HELD AT COLORADO COLLEGE,

TUTT SCIENCE BUILDING

ROOM 229

Michael Penn

Colorado College

Vertex Operator Algebras: Motivation, Definition, and Examples

Part II

                            (see above)

March 9

[available]

March 16

[talk rescheduled to April 6]

March 23

NONE - 

Spring Break

March 30

Daniel Goncalves

UFSC - SC - Brasil

Simplicity of partial skew group rings with applications to Leavitt path algebras

In this talk I will make an introduction to algebraic partial actions and their associated partial skew group rings, describing a simplicity criteria for certain partial skew group rings. As a motivating example I will show how to realize Leavitt path algebras as partial skew group rings and derive the simplicity criteria for Leavitt path algebras from partial skew ring theory.

April 6

Greg Oman

UCCS

Polynomial and power series rings 

with finite quotients

An old textbook problem posed by Kaplansky is to show that the group Z of integers is the unique infinite abelian group G with the property that G/H is finite for every nonzero subgroup H of G. In the literature, rings R with the property that R/I is finite for every nonzero two-sided ideal I are called rings with finite quotients (alternatively, residually finite rings). In this talk, we classify the rings R (assumed only to be associative, not necessarily with 1) for which the polynomial ring R[X] (respectively, power series ring R[[X]]) has finite quotients. An analogous problem for one-sided ideals will also be discussed. This is joint work with Adam Salminen of The University of Evansville.

THURSDAY

APRIL 7

12:30 - 1:30

UCCS COLLOQUIUM

Ikko Saito

UCCS

April 13

[available]

April 20

Be'eri Greenfeld

Bar Ilan Univ. - Israel

Growth, Geometry and Representations of some Non-commutative Graded Algebras 

The investigation of growth of finitely generated groups has been a fruitful, well studied area for mathematicians, since Gromov's remarkable theorem (that a finitely generated group with polynomially bounded growth is virtually nilpotent), through Grigorchuk's inspiring example of a group with intermediate (i.e. super-polynomial but subexponential) growth, and until recent results of Tao and others.

The parallel study for finitely generated (associative, not necessarily commutative) algebras turns out to be very much different. For example, we show how many super-polynomial functions are realized as the growth types of prime algebras, improving a recent result of Bartholdi and Smoktunowicz.

For algebras of polynomially bounded growth, the degree of the polynomial (the GK-dimension) turns out to be an important invariant of the algebra. It naturally appears in the study of holonomicity in the theory of D-modules, and also enabled Artin and Van den Bergh to investigate NC (non-commutative) projective schemes -- which are essentially graded domains with polynomial growth. While NC projective curves were classified, the conjectured classification of NC projective surfaces is still open.

We study the behavior of the classical (namely, Krull) dimension for surfaces, and show it is always bounded (even for non-noetherian surfaces).

From a representation-theoretic point of view, we finally show that prime, graded algebras with restricted growth either satisfy a polynomial identity or act faithfully on a simple module; this provides an affirmative answer to a graded version of a long-standing question

raised by Braun and Small.

This is partially based on joint work with Leroy, Smoktunowicz and Ziembowski and on separate sole research by the speaker.

FRIDAY

April 22

Sergio Lopez-Permouth

Ohio University

Modules over Infinite Dimensional Algebras

Let A be an infinite dimensional K- algebra, where K is a field and let B be a basis for A. In this talk we explore a property of the basis B that guarantees that K^B (the direct product of copies indexed by B of the field K) can be made into an A-module in a natural way. We call bases satisfying that property "amenable" and we show that not all amenable bases yield isomorphic A-modules. Then we consider a relation (which we name congeniality) that guarantees that two different bases yield isomorphic A-module structures on K^B. We will look at several examples in the familiar setting of the algebra K[x] of polynomials with coefficients in K. Finally, if time allows we will mention some results regarding these notions in the context of Leavitt Path Algebras (joint work with Lulwah A-Essa and Najat Muthana).

April 27

Cristobal Gil

Universidad de Malaga - Spain

The commutative core of a Leavitt path algebra.

Leavitt path algebras are the algebraic version of Cuntz-Krieger graph $C^*$-algebras. The relation between these two classes of graph algebras has been mutually beneficial. The algebraic and analytic theories share important similarities, but also present some remarkable differences. This is the case for the topic discussed in this talk: the analytic result was given by Nagy and Reznikoff for the $C^*$-algebras $C^*(E)$, and we give here the algebraic analogue for the Leavitt path algebras $L_R(E)$.
We will introduce the commutative core $M_R(E)$: a maximal commutative subalgebra inside $L_R(E)$, which in particular contains the diagonal subalgebra. Furthermore, we will characterize injectivity of representations which gives a generalization of the Cuntz-Krieger uniqueness theorem: the result says that a representation of $L_R(E)$ is injective if and only if it is injective on $M_R(E)$. On the other hand, we will generalize and simplify the result about commutative Leavitt path algebras over fields. This work was done jointly with Alireza Nasr-Isfahani.

May 4

[available]

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.

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

Universidad de Málaga

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).

Wednesday, April 1

Kulumani Rangaswamy

UCCS

Leavitt path algebras as 

graded algebras

(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

Universidad de Málaga

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

Let $n$ be a positive integer and for each $0\leq j\leq n-1$ 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

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 $0\leq j \leq n-1$ 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

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 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.

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

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 theory 

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).

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. 

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.

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

Link to abstract

** 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"

Link to abstract

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

Link to abstract

** 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

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?

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.