CCMS Colloquium: Fractal Cohomology of the Ternary Cantor Set: An Invitation to the Imaginarium of Complex Dimensions — Shanna Dobson (Harvey Mudd College)

Abstract: We present our program of geometrizing cohomology on a totally disconnected space through the intrinsic Cantor function and the cohomological complex dimensions of the ternary Cantor set. The Cantor set is an uncountable, self-similar set of measure zero; yet beneath this apparent “dust” lies a remarkably rich geometry across scales. We show how cohomological information can be organized and transported through this multiscale geometry, giving rise to nontrivial local and global fractal cohomology groups.

We describe how this multiscale geometry gives rise to a hierarchy of cohomological complex dimensions and indicate the resulting connections among geometry, analysis, and cohomology. A dust of points can carry a coherent geometry across every scale—and that geometry is reflected in a hierarchy of complex dimensions. This talk is based on forthcoming joint work with Michel L. Lapidus.

Bio: Dr. Shanna Dobson is a Visiting Assistant Professor of Mathematics at Harvey Mudd College. She received her PhD in Mathematics (summa cum laude) from the University of California, Riverside, where she studied fractal and arithmetic geometry, and category theory. Dr. Dobson’s graduate research was supported by a National Science Foundation Mathematical Sciences Graduate Research Fellowship, and she participated in the Institute for Advanced Study Women and Mathematics Program and Ambassadorship. She also teaches mathematics at Cal State LA and ArtCenter College of Design in Pasadena. Her teaching emphasizes creativity, inquiry-based learning, and helping students build confidence and curiosity in mathematics. Dr. Dobson is the founder of the AWM research network Women, Art, and Mathematics (WAAM), and currently serves as Chair of the Southern California–Nevada Section of the Mathematical Association of America, and Associate Editor of the Anneli Lax New Mathematical Library.

For questions about the event, please contact your CCMS Colloquium chairs:

Prof. Robert Cass (Robert.Cass@ClaremontMcKenna.edu) or Prof. Qidi Peng (Qidi.Peng@cgu.edu)

For logistical questions, contact Maxwell Kooiker (Maxwell.Kooiker@cgu.edu), the CCMS Colloquium Assistant.