Ryan Blair, Ph.D.

Professor

Courses Taught at ºÃÉ«ÏÈÉú

  • MATH 550 - Elements in Topology
  • MATH 555 - Topics in Topology
  • MATH 361A - Introduction to Mathematical Analysis I
  • MATH 233 - Fundamental Concepts for Advanced Mathematics
  • MATH 122 - Calculus I
  • MATH 123 - Calculus II
  • MATH 224 - Calculus III

Research Interests

  • Low-dimensional Topology
  • Knot Theory
  • Graph Theory 
  • Material Science

Research Projects

Knot theory is the mathematical study of loops in 3D space. I study classical knot invariants like bridge number, meridional rank and Gabai width using tools like Coxeter groups and distance in the complex of curves.

I am interested in developing models for self-replication that naturally emerge from the study of low-dimensional topology. I make questions like, "Which shapes can self-replicate?" mathematically rigorous. Known models for self-replication include idempotents in topological categories and rep-tiles.

In collaboration with Dr. Klotz in the Department of Physics and Astronomy, I am interested in developing applications of knot theory and low-dimensional topology to material science.

I have recently begun pursuing problems related to zero-forcing of graphs and the inverse eigenvalue problem. These problems are at the intersection of graph theory and linear algebra. I hope to develop connections between these ideas and low-dimensional topology.

Key Publications

  • Three-dimensional Rep-tiles (with Z. Marley, I. Richards) Accepted to Proceedings of the American Mathematical Society
  • Percolation and Dissolution of Borromean Networks (with D. Ferschweiler and A. Klotz) Physical Review E 107, 2023
  • Coxeter groups and meridional rank of links (with S. Baader and A. Kjuchukova) Mathematische Annalen 379: 1533–1551, 2021
  • Distortion and the bridge distance of knots (with M. Campisi, S. Taylor and M. Tomova) Journal of Topology 13: 669-682, 2020
  • Exceptional and Cosmetic Surgeries on Knots (with M. Campisi, J. Johnson, S. Taylor and M. Tomova) Mathematische Annalen 367: 581-622, 2017
  • Width is not Additive (with M. Tomova) Geometry & Topology, 17: 93-156, 2013.

Education History

  • Hans Rademacher Instructor of Mathematics, University of Pennsylvania, 2010-2013
  • Ph.D., Mathematics, University of California Santa Barbara, 2010
  • B.S., Mathematics, University of California San Diego, 2004
  • UC Transfer Agreement Guarantee, Palomar Community College, 2002