Jerry Lodder

Rigidity of secondary characteristic classes

The topic of this paper is the rigidity of secondary characteristic classes associated to a flat connection on a differentiable manifold M. Viewing the connection as a Lie-algebra valued one-form for a Lie algebra g, it is proven that if the Leibniz cohomology of g vanishes, then all secondary characteristic classes for g are rigid. Moreover, in the case when g is the Lie algebra of formal vector fields and M supports a family of codimension one foliations, the image of a characteristic map from HL4(g) to H∗dR(M) is computed, where HL∗ denotes Leibniz cohomology and H∗dR denotes de Rham cohomology.

Networks and Spanning Trees: The Juxtaposition of Prüfer and Boruvka.

Abstract This paper outlines a method for teaching topics in undergraduate mathematics or computer science via historical curricular modules. The contents of one module, “Networks and Spanning Trees,” are discussed from the original work of Arthur Cayley, Heinz Prüfer, and Otakar Borůvka that motivates the enumeration and application of trees in graph theory. Cayley correctly identifies a pattern for the number of (labeled) trees on n fixed vertices. Prüfer’s paper provides a rigorous verification of this pattern, whereas Borůvka’s paper offers one of the first algorithms for finding a minimal spanning tree over the domain of labeled trees. These latter two papers in juxtaposition offer a pleasing confluence of concepts and applications, written verbally before the modern terminology of graph theory had been formulated.

Teaching and Learning Mathematics from Primary Historical Sources.

Abstract Why would anyone think of teaching and learning mathematics directly from primary historical sources? We aim to answer this question while sharing our own experiences, and those of our students across several decades. We will first describe the evolution of our motivation for teaching with primary sources, and our current view of the advantages and challenges of a pedagogy based on teaching with primary sources. We then present three lower-division case studies based on our classroom experience of teaching discrete mathematics courses with student projects based on primary sources, and comment on how these could be adapted for use with other lower-division audiences.

A Comparison of Products in Hochschild Cohomology

In this paper we transport Steenrods cup-i products, i ≥ 0, from the singular cochains on the free loop space Maps(S1, BG) to Hochschilds original cochain complex Homk(k[G]⊗*, k[G]) defining Hochschild cohomology. Here G is a discrete group, k an arbitrary (commutative) coefficient ring, and BG the classifying space of G. This induces a natural action of the (mod 2) Steenrod algebra on the Hochschild cohomology of a group ring. For cochains supported on BG, we prove that Gerstenhabers cup product agrees with the simplicial cup product and Gerstenhabers pre-Lie product agrees with Steenrods cup-one product. As a consequence, for cocycles f and g supported on BG, the Gerstenhaber bracket [f, g] = 0 in HH*(k[G]; k[G]). This is interpreted in terms of the Batalin–Vilkovisky structure on HH*(k[G]; k[G]).

Historical sources as a teaching tool

The session will introduce participants to curricular modules (projects) based entirely on primary historical source material, developed by an interdisciplinary team of seven computer science and mathematical sciences faculty at New Mexico State University and Colorado State University Pueblo. More than twenty projects have been developed and are available on the Internet at: http://www.cs.nmsu.edu/historical-projects/ . The projects are intended for courses in discrete mathematics, algorithm design, automata, graph theory, and logic.

Primary Historical Sources in the Classroom

I study student response to learning from a specific historical curricular module and compare this to advantages of learning from historical sources cited in education literature. The curricular module is “Networks and Spanning Trees,” based on the original works of Arthur Cayley, Heinz Prufer and Otakar Borůvka. Cayley identifies a compelling pattern in the enumeration of (labeled) trees, although his counting argument is incomplete. Prufer provides an alternate proof of “Cayley’s formula” by counting all railway networks connecting n towns that contain the least number of segments. Borůvka develops one of the first algorithms for finding a minimal spanning tree by considering how best to connect n towns to an electrical network.

Teaching Discrete Mathematics Entirely From Primary Historical Sources

Abstract We describe teaching an introductory discrete mathematics course entirely from student projects based on primary historical sources. We present case studies of four projects that cover the content of a one-semester course, and mention various other courses that we have taught with primary source projects.

Resources for Teaching Discrete Mathematics: A Study of Logic and Programming via Turing Machines

During the International Congress of Mathematicians in Paris in 1900 David Hilbert (1862–1943), one of the leading mathematicians of the last century, proposed a list of problems for following generations to ponder [8, p. 290–329] [9]. On the list was whether the axioms of arithmetic are consistent, a question which would have profound consequences for the foundations of mathematics. Continuing in this direction, in 1928 Hilbert proposed the decision problem (das Entscheidungsproblem) [10, 11, 12], which asked whether there was a standard procedure that can be applied to decide whether a given mathematical statement is true. Both Alonzo Church (1903–1995) [2, 3] and Alan Turing (1912–1954) [13] published papers in 1936 demonstrating that the decision problem has no solution, although it is the algorithmic character of Turing’s paper “On Computable Numbers, with an Application to the Entscheidungsproblem” [13] that forms the basis for the modern programmable computer. Today his construction is known as a Turing machine. Let’s first study a few excerpts from Turing’s original paper [13, p. 231–234], and then design a few machines to perform certain tasks.