Schedule of Talks for the 2009-2010 Academic Year
Date: Wednesday, October 7, 2009
Speaker: Prof. Daniel E. Otero
Department of Mathematics
Xavier University
Email: oetro at xavier dot edu
Title: Determining the determinant
Abstract: Nearly every undergraduate student of mathematics learns how to solve linear systems with the help of determinants, so it may come as a surprise that the history of the development of the determinant is not better known than it is. In fact, there may be a good reason for this: befitting the complexity of the idea, its history is also quite complicated. The story of its genesis and evolution involves the interplay of a number of different problems, perspectives and approaches, and contributions were made by dozens of people over centuries. We plan to survey a key period of this history, from the time of Leibniz at the end of the 17th century, up to the watershed day of November 30, 1812, when Binet and Cauchy both presented papers on the determinant at the same meeting in Paris.
Date: Wednesday, November 18, 2009
Speaker: Prof. Ethan Coven
Department of Mathematics
Wesleyan University
Email: ecoven at wesleyan dot edu
Title: The origins of modern symbolic dynamics
Abstract: For the purposes of this talk, symbolic dynamics is the study of the iterates, under composition, of the shift transformation on compact spaces of sequences with entries from a finite alphabet.

It is generally believed that symbolic dynamics as defined above started with the seminal papers of Morse and Hedlund (Symbolic Dynamics and Symbolic Dynamics II, Amer. J. Math., 1938 and 1940). However, these papers show very little in common with present-day symbolic dynamics. On the other hand, Hedlund's 1944 Amer. J. Math. paper, Sturmian Minimal Sets, could have been written yesterday. I will trace the change to a 1941 letter from Hedlund to Morse.

Much of what I will say, including the Hedlund to Morse letter, appears in Coven and Nitecki, On the genesis of symbolic dynamics as we know it, Colloq. Math. 110 (2008), 227-242.

Date: Wednesday, February 10, 2010
Speaker: Prof. Karen Parshall
Departments of Mathematics and History
University of Virginia
Email: khp3k at virginia dot edu
Title: Algebra: Creating New Mathematical Entities in Victorian Britain
Abstract: Analytic geometry and mathematical physics may have interested a majority of mathematicians in Victorian Britain, but algebra also served to focus their mathematical attention. In the century's first half, algebraic work centered on the development of the so-called "symbolical algebra" and the creation of new algebras, while in its second, the theory of invariants dominated and the abstract theory of groups witnessed key developments. Underlying much of this research was the philosophical question of how free mathematicians were to create new mathematical entities. The Victorian British response was ultimately, "quite."
Date: Wednesday, March 3, 2010
Speaker: Prof. Erik Tou
Department of Mathematics
Carthage College
Email: etou at carthage dot edu
Title: Navigation in the Time of Euler
Abstract: Terrestrial navigation is one of the oldest human endeavors; once societies formed significant sea trade, the ability to avoid maritime disasters was of paramount importance. While some aspects of navigation (e.g., finding one's latitude) were easily resolved by rudimentary instruments and mathematics, the problem of finding longitude remained unresolved well into the 18th century. Ultimately, the chronometers of English clockmaker John Harrison proved to be the most practical and durable solution. However, there were several other competing methods that were equally capable of resolving the problem, and it was not clear for several decades which would be the most practical or accurate. We will trace some history of navigation from ancient times to the Enlightenment, and explore Leonhard Euler's work on this subject.
Date: Wednesday, April 7, 2010
Speaker: John Clagett, Architect
Director, Center for Ecumenical Research in the Arts and Sciences
Englewood, NJ
Email: clagett1 at verizon dot net
Title: A Case Study of Applied Projective Geometry
Abstract: The ecclesiastical architecture of the Baroque period [1600-1750] presents a conception of the world in continuous motion and change. This conception, or sensory impression, was produced with the aid of a design process that included of a set of mathematico-architectural transformative operations. These operations acted to reposition and/or reconfigure-often in unexpected ways-the parts of the building, including its domes, arches and vaults.

Of this set of transformative operations, the most advanced involved applied projective geometry. My talk will trace the origins of this geometry to the Renaissance architect Filippo Brunelleschi [1377-1446] and his interest in perspectiva artificialis-the drawing of an image as the eye perceives it. Perspectival methods and their implications continued to interest architects up to and throughout the Baroque.

My lecture builds on the research of the late Werner Müller as well as Baroque period treatises by Abraham Bosse, Girard Desargues, Amédée-François Frézier, Guarino Guarini and Johann Jacob Schubler.

Date: Wednesday, May 5, 2010
Speaker: Prof. Janet Barnett
Department of Mathematics
Colorado State University - Pueblo
Email: janet.barnett at colostate-pueblo dot edu
Title: Abstract awakenings in algebra: Teaching and learning group theory through the works of Lagrange, Cauchy, and Cayley
Abstract: The seeds of group theory can be recognized in several early nineteenth century mathematical developments. The common features of these apparently disparate developments were first explicitly recognized by Arthur Cayley (1812 - 1895). In his 1854 paper "On the theory of groups, as depending on the symbolic equation ϑn = 1," Cayley noted: "The idea of a group as applied to permutations or substitutions is due to Galois, and the introduction of it may be considered as marking an epoch in the progress of the theory of algebraical equations."

Cayley himself extended the idea of "a group" well beyond its application to permutations. In addition to defining a group as any (finite) system of symbols subject to certain algebraic laws, Cayley stated several important group theorems and proceeded to classify all groups up to order seven. Although focused on the general properties of arbitrary groups, he also did not neglect to motivate this abstraction through references to specific nineteenth century appearances of the group concept. As a result, Cayley's paper provides a powerful lens on the process and power of mathematical abstraction.

In this talk, we will examine this earliest paper on group theory through excerpts from a student module that uses this same powerful lens (together with several pre-Cayley original sources to provide historical and mathematical context for his paper) as a means to develop elementary group theory. In addition to an overview of the module and the historical developments on which it is based, we will consider how the module can be used in an introductory abstract algebra course. An overview of the rationale which guides the NSF-funded project Learning Discrete Mathematics and Computer Science via Primary Historical Sources that supported the development of this and 15+ other such student modules will also be provided.