College of Arts and Sciences

Department of Mathematical Sciences

Topology Group

Faculty: Elizabeth Gasparim (Algebraic geometry, Algebraic topology); Jerry Lodder (Algebraic topology, Algebraic K- theory, Cyclic homology, Leibniz homology); David Pengelley (Algebraic topology, History of mathematics, Mathematics education); Ross Staffeldt (Algebraic topology, Algebraic K-theory); Ted Stanford (Knot theory, Three dimensional manifolds).

Emeritus Faculty: Frank Williams (Algebraic topology)

Doctoral Students: Candelario Castaneda, Jay Cho, John Sanders, Mohammed Yasein.

Recent Ph.D. Graduates: Khaled Bataineh (2003), Hyunju Oh (2003), Eduardo Quinones-Rico (1997)

Visitors: In 2002 and 2003 the topology group hosted short term visits by Ralph Cohen (Stanford University), Jim Conant (Cornell University), Jim Milgram (Stanford University), Mara Neusel (Texas Tech University), Constantin Teleman (Cambridge University), and Charles Thomas (Cambridge University). Also, in 2003 the group hosted visits by Christophe Eyral (Tokyo Metropolitan University, Japan), Ruxandra Moraru (University of Toronto, Canada), Pedro Ontaneda (Federal University of Pernambuco, Brazil), and Morwen Thistlethwaite (University of Tennessee).

Group Activities: Our group maintains a weekly Topology Seminar. Not only are meetings used to discuss research and current results, but we also devote some sessions to talks by doctoral students on background material related to their research. Occasionally, students in the graduate topology courses speak on topics slightly outside standard course material, as preparation for the oral part of the comprehensive examination. Now and then we also devote a block of time to a group reading and discussion project revolving around a specific topic.

Future Plans: In the early 1990's there was a joint seminar on geometry and topology that involved mathematicians from the University of New Mexico, New Mexico State University, and University of El Paso. We hope to find a funding source that will permit us to bring this seminar back to life in the near future.

Faculty Interests

Elizabeth Gasparim:
Areas of Interest: Algebraic Geometry, Algebraic Topology,and Mathematical Physics.

Research Topics: Topology of moduli spaces of vector bundles over surfaces and their topological changes under birational transformations of the base. Construction of vector bundles and sheaves over complex surfaces and threefolds. Numerical invariants of vector bundles and sheaves. Curves and surface singularities. Instantons and their moduli spaces. Deformations of vector bundles. Computational algorithms for numerical invariants.

Jerry M. Lodder:

My scholarly activities are characterized by both breadth and depth, ranging from traditional research topics in algebraic topology, to applications in differential geometry to ground breaking publications in mathematics pedagogy. My work has found external funding in three separate grants from the National Science Foundation, additional funding as a research fellow at the Institut de Recherche Mathématique Avancée in Strasbourg, the Institut des Hautes Etudes Scientifiques near Paris, as well as a further research grant from the Centre National de la Recherche Scientifique (CNRS). Outside New Mexico State University, I have delivered numerous seminar lectures, conference addresses, colloquia, workshops, and special session presentations, in a variety of venues, ranging from regional meetings of the American Mathematical Society to national meetings to international conferences, including the quadrennial International Congress of Mathematicians.

As an area of specialization, I have pioneered Leibniz cohomology as a tool for the study of algebraic structure and geometric space, capturing classical ideas of curvature as cochains in the Leibniz complex, while detecting modern invariants of K-theory and characteristic classes, both primary and exotic, as non-trivial Leibniz cohomology classes. When applied to differentiable manifolds, Leibniz cohomology offers new geometric information with the potential to be a complete diffeomorphism invariant. Born as a collaboration with Distinguished Research Professor Jean-Louis Loday of Strasbourg, my work significantly enlargens the scope of the Leibniz theory, with students of the Loday school (Frabetti and Wagemann) writing papers that directly build on my results. The subject will endure for the foreseeable future.

David Pengelley:

My research in topology is primarily on the structure of classifying spaces for various types of vector bundles and related spaces. I am currently especially interested in the structure of their homology and cohomology as unstable algebras over the Steenrod algebra and over related algebras of operations like the Kudo-Araki-May algebra, and in the connections to the Dickson algebras and other invariant subalgebras of polynomial algebras under action by subgroups of the general linear
groups. Most recent published papers are coauthored with Frank Williams, also of our department.

In mathematics education I am working primarily on the pedagogy of teaching with primary historical sources at all levels, in particular bringing collections of primary historical sources together in translation and publication for use in various curricular settings. I am completing a second coauthored book of annotated primary sources in five branches of mathematics. I am part of a group of five faculty in mathematics and computer science developing course materials of student projects based on
primary sources for discrete mathematics, with support form the National Science Foundation. Elaine Cohen is working towards a doctoral dissertation in using history in teaching mathematics with me.

In the history of mathematics I am particularly involved in translation, study, and exposition of eighteenth and nineteenth century mathematics, with some emphasis towards number theory. I am currently coauthoring work on Sophie Germain's handwritten manuscripts of her research on Fermat's Last Theorem in the early nineteenth century.

Ross Staffeldt: My areas of interest are algebraic topology, applications of topology, and geometry of other subjects, like physics or differential equations.

My own research is concerned with the algebraic K-theory of topological spaces and with the related topic of the algebraic K-theory of S-algebras. The algebraic K-theory of topological spaces was invented to study the space of h-cobordisms associated to a manifold, so it is a complicated algebraic tool for studying delicate geometric problems. My current research with Roland Schwänzl (Osnabrück, Germany) and Friedhelm Waldhausen (Bielefeld, Germany) is on the algebraic K-theory of S-algebras. Our results indicate that it is possible to analyze to a certain extent the space of h-cobordisms of a manifold written as a union of two manifolds glued together along a common boundary in terms of the spaces h-cobordisms of the presumably simpler pieces.

I am currently advising Hyunju Oh on her dissertation research, which involves the study of the knot invariant associated to "the universal weight system associated to the Lie superalgebra osp(1|2)." Her dissertation is nearly complete. Ted Stanford and I are working with John Sanders to try to develop for John a thesis topic related to one of the competing definitions of "knot energy." At the undergraduate level, I am interested in developing computers as tools to help faculty present material for second year calculus and to help students experiment with and learn the concepts developed in these courses. This year, after teaching a course in vector calculus, I have gotten interested in the motion of the top, or gyroscope, as a problem which one can use to motivate a lot of the material covered in vector analysis.

Ted Stanford: My areas of interest are "knot theory and topological dynamics" or maybe "low-dimensional topology and topological dynamics."

My research is mostly concerned with knots and links in S3. I have done some work with finite-type invariants (also called Vassiliev invariants), and some of their generalizations. I have worked on other topics related to knots and links, such as braids and knotted graphs. I am also doing some work on knot and link groups and some of their representations into a finite group or into finite-dimensional algebras. Current and recent collaborators in knot theory include Jacob Mostovoy, Jim Conant, and Swatee Naik. Along other lines, I have a research project with Erik Bollt and others to understand the symbol dynamics arising form a non-generating partition of a topological dynamical system.

Under my supervision, Khaled Bataineh has completed his Ph.D. dissertation, which is about finite-type invariants for knots in a solid torus. Ross Staffeldt and I are working with John Sanders to develop a dissertation research topic related to knot theory energy. I am also working with Candelario Castaneda and Mohammed Yasein on selecting a dissertation research topic.