**Emily Stone**

Email

# 2003 Colloquia

## Spring 2003

### May 8 - Mathematics in Stone and Bronze

**Mathematics in Stone and Bronze**Helaman Ferguson's mathematical sculptures in stone and bronze celebrate ancient and modern mathematical discovieries, melding the universal languages of sculpture and mathematics. Using slides and video, Helaman and Claire trace Helaman's creations from initial concept, mathematical design, computer graphics, diamond cutting and final form. Their lectures have fascinated audiences worldwide, bringing together multiple disciplines and stimulating dialogue among them.

Thursday, 8 May 2003

4:10 p.m. in North Underground Lecture Hall

Refreshments at 3:30 p.m. Underground Lecture Hall Lobby

### April 18 - Book-thickness of Graphs

**Book-thickness of Graphs**An *n*-book is a line in 3-space, called the spine, together with *n* half-planes, called pages, joined together at the spine. An *n*-book embedding of a graph *G* is an embedding of *G* in an *n*-book so that each vertex of *G* lies on the spine and each edge of *G* lies on a single page so that no two edges cross each other or the spine. The book-thickness of *G*, or *bt*(*G*) is the smallest *n* such that *G* admits an *n*-book embedding.

We will answer some questions about the relationship between genus and book-thickness originally posed by Bernhart and Kainen. We will look at bounds for the book-thickness of different families of graphs, giving an optimal bound for the complete graph. We will demonstrate a large class of planar graphs of book-thichness two, which includes *X*-trees, square grids, and planar bipartite graphs. We will then consider extensions to generalized books where the pages and spine are modified.

Friday, 18 April 2003

4:10 p.m. in Math 109

Coffee/treats at 3:30 p.m. Math 104 (Lounge)

### April 17 - New Results in Research on Undergraduate Mathematics Majors' Understanding of the Role of Definitions in Mathematics

**New Results in Research on Undergraduate Mathematics Majors' Understanding of the Role of Definitions in Mathematics**Undergraduate mathematics majors enrolled in their first Real Analysis course can have understandings about the role of mathematical definitions that affect their understandings of the content of the course and their abilities to construct and understand formal mathematical proofs. One might think that this is not so surprising since many of the concepts (and even their formal definitions) may have been encountered in the students' previous calculus courses, and since students' understandings of elementary calculus concepts can often be imperfect, these students might 'unconsciously' misuse the related definitions. My recent research (with Michael Ward at Western Oregon University) involving students in an introductory Abstract Algebra course has led to similar conclusions, however, even though many of the concepts in introductory abstract algebra may be completely new to the students. (Thus, theoretically, there should be no interference by previous incomplete understandings of the concepts.) I have also discovered that misconceptions about the role of mathematical definition are much harder to uproot than previously thought. I will also discuss a revised and refined theoretical framework for studying student's use of definitions in mathematics.

Thursday, 17 April 2003

4:10 p.m. in Math 109

Coffee/treats at 3:30 p.m. Math 104 (Lounge)

### April 7 - Radial Basis Function Methods for Partial Differential Equations

**Radial Basis Function Methods for Partial Differential Equations**Radial basis function (RBF) methods are a fairly new method for the numerical solution of PDEs. When compared to existing numerical PDE schemes, RBF methods are very accurate, easy to implement, and extremely flexible. The methods are grid free which allow them to be easily implemented in very complex geometries.

Despite showing extreme promise, several issues must be resolved before the RBF methods reach their potential. The issues include ill-conditioning, the presence of relatively large boundary region errors, and efficient implementation. In this talk, we discuss strategies for coping with the ill-conditioning problem and a method to reduce boundary region errors in PDE problems. Successfully dealing with the ill-conditioning problem and suppressing boundary region errors has a dramatic positive effect on the accuracy of RBF methods for derivative approximation. Numerical results are given for time dependent PDE problems.

Monday, 7 April 2003

4:10 p.m. in Math 109

Coffee/treats at 3:30 p.m. Math 104 (Lounge)

### April 4 - Two-Sided Vector Spaces

**Two-Sided Vector Spaces**A two-sided vector space over a field K is a set V and a pair of actions of K on V, *(x,v)* ι→ xv and *(v,x)* ι→ *vx*, such that each action individually makes V into a vector space over K, satisfying an associativity condition .

Every ordinary vector space is a two-sided vector space, but there are interesting examples where the left and right actions do not agree. Motivated by some problems in "noncommutative algebraic geometry," we look at the structure of two-sided vector spaces in some detail. The results end up depending on the arithmetic of the field K. The problem can be formulated purely in terms of linear algebra. (This work is joint with A. Nyman at the University of Montana.)

Friday, 4 April 2003

4:10 p.m. in Math 109

Coffee/treats at 3:30 p.m. Math 104 (Lounge)

### April 3 - An Inverse Problem in Atmospheric Imaging

**An Inverse Problem in Atmospheric Imaging**In this talk, I will discuss the solution of a class of inverse problems that originate in applications from image reconstruction. These applications include microscopy, medical imaging and, our particular application, astronomical imaging. We pose our inverse problem as a optimization problem. Incorporated into our choice of objective function is prior statistical knowledge about the noise in our data, while a regularization functional is incorporated for stability. Finally, the optimization problem we wish to solve is constrained. The constraints arise from the fact that the objects we are imaging (stars) are photon densities, or light intensities, and are therefore nonnegative. Since high resolution images are desired, the problem is large-scale. Consequently, efficient numerical techniques are required. I will discuss the problem of minimizing both quadratic and convex cost functions with nonnegativity constraints. An existing algorithm will be presented for the quadratic minimization problem, and ideas from this algorithm are extended to the problem of minimizing the convex function. Implementation of an efficient sparse preconditioner will also be discussed, and a numerical study of these algorithms will be presented.

Thursday, 3 April 2003

4:10 p.m. in Math 109

Coffee/treats at 3:30 p.m. Math 104 (Lounge)

### April 1 - Quantum Logics over Rationals

*nQuantum Logics over Rationals*We introduce the notion of quantum logics of idempotents in algebras of operators, and develop a measure theory for the set *P(H*) of all (not necessarily Hermitian) continuous linear projections on a Hilbert space *H*.

Any set can be considered as an idempotent in the algebra of multiplication operators, and any finitely additive measure µ can be considered as a function on these idernpotents.

Idempotents *P* and *Q* *(i.e., P ^{2} = p,Q^{2} = Q)* in an algebra of operators are said to be

*orthogonal*if

*PQ*=

*QP*= 0, in this case

*P*+

*Q*is also an idempotent. A function

*v*defined on a set ∏ of idempotents is called

*σ*-

*orthoadditive*if

*v(P+Q)*=

*v(P)*=

*v(Q)*whenever

*P*and Q are orthogonal.

*v*is called -

*orthoadditive*if in addition

*v*(Σ

*P*) = Σ

_{n}*v*(

*P*

_{n}) whenever

*P*are mutually orthogonal for

_{n}, P_{m}*n≠m*.

Let *X* be a real topological linear space and *P(X)* be the set of all continuous linear projections on *X*. For what *X* every relatively *σ*-additive function *μ:P(X)*→ ℝ admits an extension to a sequentially strongly continuous linear functional? Does there exist a non-Hilbert space X with this property?

**Theorem**: Let *P*(ℚ*n*) be the set of all linear projections on ℚ* ^{n}*,

*n*≥

*3*. Then every orthoadditive function

*μ:P(ℚn)*→

*ℚ*defines a unique linear operator

*T*on

*ℚ*.

^{n}Tuesday, 1 April 2003

4:10 p.m. in Math 109

Coffee/treats at 3:30 p.m. Math 104 (Lounge)

### February 20 - New Numerical Methods for PDE Models of Biological Populations

**New Numerical Methods for PDE Models of Biological Populations**We disccuss a class of numerical methods for partial differential equations that take into account age as well as space in modeling the dynamics of a population. The equations and their biological meaning will be presented. The new numerical methods and their utility will then be explored in the context of some examples, including the case of colony growth of the bacteria Proteus Mirabilis.

Thursday, 20 February 2003

4:10 p.m. in Math 109

Coffee/treats at 3:30 p.m. Math 104 (Lounge)

### February 18 - Nonlinear Models of Dynamics in Drilling

**Nonlinear Models of Dynamics in Drilling**In this talk I will discuss primarily my joint research with Abe Askari of The Boeing Company on the chatter instability in drilling. Chatter is a self-excited oscillation between the machine tool and the workpiece that limits productivity of machining operations, reduces the quality of the product and shortens machine tool life. Up until recently all models of chatter have been linear, with delay effects in the case of regenerative chatter. These models only partially explain the instabilities observed in the machining process.

In aircraft manufacture drilling is a critical machining process: over a million holes may be drilled in the creation of a commercial passenger jet. To address the problem of chatter in drilling, we are developing a suite of nonlinear models of metal cutting that can be merged with finite element studies of drill vibration modes and informed by large scale of simulations of metal cutting operations. Typically, engineering studies of chatter have restricted themselves to the question of linear stability of a steady cutting solution; in addition to that we are studying the effects of the nonlinear terms in the model on the resulting dynamics. Contact has been made with laboratory results from experiments conducted in Seattle and St. Louis, with the goal of directing tool design and allowing machine operators to avoid chatter regimes in drilling.

I will also take this opportunity to overview some of our student projects with the Mathematics and Engineering Analysis group at Boeing-Bellevue, and other areas of interest in our interdisciplinary mathematics program.

Tuesday, 18 February 2003

4:10 p.m. in Math 109

Coffee/treats at 3:30 p.m. Math 104 (Lounge)

### January 30 - Reversing the Design Cycle - Technology and Mathematics Instruction

**Reversing the Design Cycle - Technology and Mathematics Instruction**In education, we often adopt and adapt existing technologies that were originally intended for application in other endeavors. As a result, the creation and negotiation of meaning by learners mediated by such technologies sometimes takes surprising twists and turns and we often fail to reflect carefully on those opportunities. We will discuss examples based on the experiences of middle school teachers simulating the role of a Calculator Based Ranger (CBR) motion detector in making sense of graphs of displacement produced by a CBR. Then we will turn to a more fundamental question of whether or not it is possible to reverse the direction of the interplay between instructional and technologies design. Normally, we observe the effects technology has on learning and then adapt instruction to make more effective use of the technology. Instead, can we stipulate the effects we want instruction to have on learners and then design technology to achieve those goals? We will discuss some promising "first principles" as articulated by David Merrill and Catherine Fosnot that could guide efforts to realize that vision.

Thursday, 30 January 2003

4:10 p.m. in Math 109

Coffee/treats at 3:30 p.m. Math 104 (Lounge)

## Fall 2003

### December 12 - Intersection theory and the Jacobian Conjecture

**Intersection theory and the Jacobian Conjecture**I ntersection theory's goal is to "count" the number of ways geometric objects intersect each other. This can be applied to enumerative geometry questions (such as "How many conics go through 4 points and are tangent to a given line?"), or more geometric enterprises (calculating the Euler characteristic of a space).

In this talk I will give a brief introduction to the ideas of the subject, with a few examples (the intersection ring of projective n-space, the answer to the above conics question, and perhaps another reason why the Euler characteristic of the sphere is 2).

Friday, 12 December 2003

4:10 p.m. in Skaggs 117

### December 11 - Greedoid Invariant Theory and the Greedoid Tutte Polynomial

**Greedoid Invariant Theory and the Greedoid Tutte Polynomial**Thursday, 11 December 2003

4:10 p.m. in Math 109

Coffee/treats at 3:30 p.m. in Math 104

### November 21 - Using a Priori Information for Constructing Regularizing Algorithms

**Using a Priori Information for Constructing Regularizing Algorithms**Many problems of science, technology and engineering are posed in the form of operator equation of the first kind with operator and right part approximately known. Often such problems turn out to be ill-posed. It means that they may have no solutions, or may have non-unique solution, or/and these solutions may be unstable. Usually, non-existence and non-uniqueness can be overcome by searching some ''generalized'' solutions, the last is left to be unstable. So for solving such problems is necessary to use the special methods - regularizing algorithms.

The theory of solving linear and nonlinear ill-posed problems is advanced greatly today (see for example [1, 2]). A general scheme for constructing regularizing algorithms on the base of Tikhonov variational approach is considered in [2].

It is very well known that ill-posed problems have unpleasant properties even in the cases when there exist stable methods (regularizing algorithms) of their solution. So at first it is recommended to study all a priori information, to find all physical constraints, which may make it possible to construct a well-posed mathematical model of the physical phenomena.

Computational programs for linear ill-posed problems with a priori information (monotonicity, convexity, known number of extremes, sourcewise representation of unknown solution, etc.) could be found in [1] and other author's publications, and could be generalized for nonlinear problems also. If the constraints are not sufficient to make a problem well-posed, then it is necessary to use all these constraints but we must also know error level of experimental data. As examples of successful applications of these regularizing algorithms to practical problems we consider inverse problems of vibrational spectroscopy and electron microscopy.

References

- Tikhonov, A.N., Goncharsky, A.V., Stepanov, V.V. and Yagola, A.G., Numerical Methods for the Solution of Ill-Posed Problems. Kluwer Academic Publ., Dordrecht, (1995).
- Tikhonov, A.N., Leonov, A.S. and Yagola, A.G., Nonlinear Ill-Posed Problems. Chapman and Hall, London, (1995).

Friday, 21 November 2003

4:10 p.m. in Skaggs 117

### November 14 - Duality in Noncommutative Algebra and Geometry

**Duality in Noncommutative Algebra and Geometry**Duality is one of the fundamental concepts in mathematics. From the basic duality for finite dimensional vector spaces it extends in many directions: Banach spaces and topological groups in analysis, Poincare duality in topology, Serre duality in algebraic geometry, and so on. Grothendieck showed us that working with complexes in the derived category we get even more dualities.

Grothendieck's theory of dualizing complexes adapts well to noncommutative rings. It provides a powerful tool to study rings and their representations. It also makes sense on noncommutative algebraic spaces.

In the lecture I will sketch the basics of duality theory (with illuminating examples) and explain some applications in noncommutative ring theory. I will mention recent developments in noncommutative algebraic geometry, and some relations to (commutative) algebraic geometry and theoretical physics.

Friday, 14 November 2003

4:10 p.m. in Skaggs 117

Coffee/treats at 3:30 p.m. in Math 104

### November 6 - Technology and the Real Numbers: Studying irrational numbers at the high school level.

**Technology and the Real Numbers: Studying irrational numbers at the high school level.**We will use the TI Voyage 200 calculator to investigate rational approximations to the irrational numbers. We will investigate why 22/7 is used to approximate pi and why the golden ratio is the most unrational irrational number. If time permits, we might even discuss how technology can help students to build the understanding needed to prove classes of numbers are irrational.

Thursday, 6 November 2003

4:10 p.m. in Math 109

Coffee/treats at 3:30 p.m. in Math 104

### October 30 - Electronic Journals, ScienceDirect, and Dissertations Online - Oh My!

**Electronic Journals, ScienceDirect, and Dissertations Online - Oh My!**E lectronic publications are changing the nature of scholarly research and communication. The Mansfield Library crossed the threshold several years ago of providing access to more electronic journals than paper subscriptions. These ejournals are often purchased in packages which become databases and enable new searches, within article full text, never previously possible. A brief tour and discussion of our evolving infostructure will occur.

There are many universities worldwide that are publishing digital doctoral dissertations. The primary reason for digital dissertations is to increase access to the scholarly body of knowledge. The use of dissertations often dramatically increases when they are electronically available. A synopsis of opportunities available to the University of Montana will be covered.

Thursday, 30 October 2003

4:10 p.m. in Math 109

Coffee/treats at 3:30 p.m. in Math 104

### October 24 - A.J. Gibson's Women's Hall at the University of Montana, A Brief History

**A.J. Gibson's Women's Hall at the University of Montana, A Brief History**This year marmarks the 100th anniversary of the dedication of the Math Building at the University of Montana. Originally the Women's Hall, the third building on the campus was designed by noted architect A.J. Gibson to provide housing for the University's growing female population. The building was a significant departure from the earliest plans for the University in both its style and location, yet it added a new level of elegance to the emerging campus. More importantly, it revealed the University's singular dedication to the education of women at the turn of the last century. In a brief presentation, Professor Rafael Chacon from the University's Art Department will discuss the history of this venerable building and introduce the audience to his upcoming book on the life and work of A.J. Gibson, Missoula's most important architect.

Friday, 24 October 2003

4:10 p.m. in Math 109

Coffee/treats at 3:30 p.m. in Math 104

### October 23 - 58,000 AP Statistics Exams Later

**58,000 AP Statistics Exams Later**Last June 280 college/university and high school teachers of elementary statistics gathered in Lincoln, Nebraska to grade the six free-response questions on the AP Stat Exam. I was a first year reader, an "acorn." During this presentation, I will discuss the format of the exam; the roles of chief reader, table leaders, and readers; the organization and daily routine; the use of rubrics and training to ensure consistency and accuracy of grading. We will examine in detail at least one of the free-response questions on the 2003 exam and the implementation of the rubric for this question. I'll share some of the rewards I experienced as a reader and encourage others to apply.

Thursday, 23 October 2003

4:10 p.m. in Math 109

Coffee/treats at 3:30 p.m. in Math 104

### October 16 - Return map characterizations for a model of bursting with two slow variables

**Return map characterizations for a model of bursting with two slow variables**Many neurons and endocrine cells exhibit periodic bursting oscillations in their transmembrane electrical potential. The fast subsystems of the corresponding models exhibit bistability between stable equilibria and periodic orbits. Slow variables evolve in a manner which causes the solutions to switch between pseudo-stationary and oscillatory states resulting in a characteristic bursting pattern.

Most recent models involve two slow variables which tends to complicate their analyses. Using singular perturbation techniques we show that bursting solutions of such models correspond to fixed points of a one dimensional map constructed from the fast and slow subsystems. We further show that for some parameter values, bistability between bursting solutions and stable equilibria is possible.

Thursday, 16 October 2003

4:10 p.m. in Math 109

Coffee/treats at 3:30 p.m. in Math 104

### October 2 - Infectious Disease Ecology and Mathematical Research

**Infectious Disease Ecology and Mathematical Research**This presentation will focus on UM efforts to compete for National Science Foundation IGERT (Integrative Graduate Education and Research Traineeships) Program funds, and to solicit interest of mathematics faculty in participating in a new proposal submission. The goal of the NSF IGERT program is to develop new, interdisciplinary models in graduate education, by focusing the training of graduate students from different disciplines on a specific real-world research problem.

The IGERT proposal we are developing focuses on the ecology of infectious disease. The dynamics of many disease systems (West Nile virus, Hanta virus, SARS, plague, and others) are complex and poorly understood. They involve factors operating simultaneously at multiple spatial and temporal scales (differing by orders of magnitude) and include significantly nonlinear dynamics. Disease ecology systems present an array of computational, mathematical, biomedical, ecological, and conservation problems and opportunities. Research on these systems is best addressed by interdisciplinary teams spanning these disciplines. The goal of participation by mathematics faculty and graduate students would include advancing both basic and applied mathematics research (not simply fulfilling a service role in solving biological problems).

In this presentation, I will outline briefly the goals of the NSF IGERT program, recount the history of UM's proposal development including reviewer responses to the previous submission, and discuss needs and opportunities for mathematicians to participate.

Thursday, 2 October 2003

4:10 p.m. in Math 109

Coffee/treats at 3:30 p.m. in Math 104

### September 18 - MyMathLab, a Feasible, Useful Environment For Mathematics Distance Courses

*MyMathLab, a Feasible, Useful Environment For Mathematics Distance Courses*More and more post-secondary mathematics courses are being taught through Distance Learning departments of colleges. Many web environments exist to present these courses, from home-made teacher authored web sites to publisher produced sites. It seems that the home-made ones take a tremendous amount of time to author, and it also seems like many of the publisher produced sites are either too expensive to use, not very adaptable to mathematics course work, or too rigid/boring to be effective with.

We propose a site that is rich in mathematical "bells and whistles", yet flexible enough for unique contributions from the instructor-and neither instructor nor student need be a technology "wizz" to function adequately. This environment, free for instructor use, is called MyMathLab, and we will demonstrate some of its features and workings.

Thursday, 18 September 2003

4:10 p.m. in Math 109

Coffee/treats at 3:30 p.m. in Math 104

### September 12 - Coloring Graphs on Surfaces

**Coloring Graphs on Surfaces**Several central parts of Graph Theory have grown out of problems related to colorings of graphs embedded in the plane and in more general surfaces. Main results about the chromatic number of graphs embedded in surfaces will be surveyed. Different asymptotic behavior in terms of the genus will be outlined for usual colorings, for graphs of large girth, and for acyclic colorings. Additional emphasis will be given to coloring and flow duality in locally planar graphs.

Friday, 12 September 2003

4:10 p.m. in Skaggs 117

Refreshments at 3:30 p.m. in Lobby

This talk is part of The Big Sky Conference, and is sponsored in part by the National Science Foundation & the Department of Mathematical Sciences.

### September 11 - The Traveling Salesman Problem and Optimization on a Grand Scale

**The Traveling Salesman Problem and Optimization on a Grand Scale**Optimization problems of enormous size and complexity arise in areas from genome sequencing to VLSI design. In this talk we discuss the challenges in working with large-scale models by considering the well-known traveling salesman problem, or TSP for short, which asks for the cheapest way to visit a collection of cities and return to the starting point. For over fifty years the study of the TSP has led to improved solution methods for a wide range of practical problems. We will discuss the history and applications of the TSP and the methods used to attack very large instances. Along the way, we discuss the interplay of modern applied mathematics and increasingly more powerful computing platforms, including computational grids and distributed web-computing.

Thursday, 11 September 2003

8:00 p.m. in the Music Recital Hall

Refreshments following lecture in Lobby

This talk is part of The Big Sky Conference, and is sponsored in part by the National Science Foundation, The Presidents Office & the Department of Mathematical Sciences.

### September 11 - Planar Graphs, Parallel Computing, and the Traveling Salesman Problem

**Planar Graphs, Parallel Computing, and the Traveling Salesman Problem**The traveling salesman problem, or TSP for short, asks for the minimum-weight Hamiltonian cycle in a edge-weighted complete graph. The most successful approach known for solving instances of the TSP is the linear-programming based method of Dantzig, Fulkerson, and Johnson from the early 1950s. In this talk we discuss some recent work on extending this approach to larger TSP instances. Following the work of Fleischer and Tardos (1999) and Letchford (2000), we consider computational techniques for exploiting graph planarity in improving the linear-programming relaxations. We present computational results with a parallel implementation of this approach and discuss some open theoretical issues that could have impact on the success of the method in practice.

Thursday, 11 September 20034:10 p.m. in the UC Theatre

This talk is part of The Big Sky Conference, and is sponsored in part by the National Science Foundation & the Department of Mathematical Sciences.

### September 4 - Can the Mathematics Community Afford Not to Be Political?

**Can the Mathematics Community Afford Not to Be Political?**In the past year as President of NCTM, events have convinced me that the mathematics community of teachers, professors, and professionals cannot afford to take a back seat and allow themselves to be defined by others. From the No Child Left Behind Act in Washington to reports from research institutes to simple children's exercises that have been publicized in various ways, we must consider how departments such as this at The University of Montana can be more politically active and influence public perception about mathematics. Not to do so is to be derelict in our responsibility to the mathematical future of our children.

Thursday, 4 September 2003

4:10 p.m. in Math 109

Coffee/treats at 3:30 p.m. in Math 104