Mark W. Johnson's Research

Professor of Mathematics

Penn State Altoona

Homepage
Conference Information Link
I was an organizer of a conference to held at the University of Illinois (at Urbana-Champaign) July 17 to 21, 2017.
Old Conference Information
Topology/Geometry Seminar
The Penn State Topology/Geometry Seminar meets almost exclusively in Altoona these days, but used to have occassional meetings at University Park. If you are interested in giving a talk, feel free to contact me.
PROPs book
This is a research monograph, with Donald Yau, attempting to provide a unified treatment of all kinds of operational structure (operads, properads, dioperads, wheeled PROPs, etc.) and the things they naturally act upon (i.e. algebras and modules). Nearly final pdf file.
Publications in Print
An alternative, often more up-to-date for preprints, is to look through my submissions to the arXiv or with their atom feed. This arxiv list excludes only the last three items below.
  1. Higher homotopy operations and Andre-Quillen cohomology, Advances in Mathematics, 230 (2012) 777-817, with David Blanc of the University of Haifa and James M. Turner of Calvin College.
  2. Factoring the Becker-Gottlieb Transfer Through the Trace Map, Pure and Applied Mathematics Quarterly, 8 no. 1 (2012) (the Farrell-Jones proceedings), with Wojciech Dorabiala also of Penn State Altoona.
  3. On modified Reedy and modified projective model structures, Theory and Applications of Categories, 24 (2010) 179-208.
  4. Higher homotopy operations and cohomology, Journal of K-theory 5 (2010) 167-200, with David Blanc of the University of Haifa and James M. Turner of Calvin College.
  5. On homotopy invariance for algebras over colored PROPs, Journal of Homotopy and Related Structures 4 (2009) 275-315, with Donald Yau of the Ohio State University at Newark.
  6. Local-to-global spectral sequences for the cohomology of diagrams, Journal of Pure and Applied Algebra 213 (2009) 34-53, with David Blanc of the University of Haifa and James M. Turner of Calvin College.
  7. On Realizing Diagrams of Pi-algebras, Algebraic and Geometric Topology 6 (2006) 763-807, with David Blanc of the University of Haifa and James M. Turner of Calvin College.
  8. Realizing Kasparov's KK-theory Groups as the Homotopy Classes of Maps of a Quillen Model Category, in Contemporary Mathematics, Volume 399 (2006) An Alpine Anthology of Homotopy Theory, Dominique Arlettaz and Kathryn Hess (Editors), with Michael Joachim of Universitaet Muenster.
  9. The product theorem for parametrized homotopy Reidemeister torsion, Journal of Pure and Applied Algebra, 196 (2005) 53-90 with Wojciech Dorabiala also of Penn State Altoona.
  10. Model structures on the category of ex-spaces, Topology and its Applications, 119 (2002) 325-353, with Michele Intermont of Kalamazoo College.
  11. A sheaf-theoretic view of loop spaces, Theory and Applications of Categories, 8 (2001) 490-508.
Preprints in progress
Once again, you can see my arXiv listing or the atom feed to see if any have been posted.
  1. On higher Segal maps and internal categories.
  2. On describing Waldhausen's S. construction using adjoints.
  3. Ideals in categories and Cech cohomology for diagrams, with David Blanc and James Turner.
  4. Generalized metrics, diameters and one-sided enrichments, with Hossein Movahedi-Lanakarani and Bob Wells.
  5. Foundations of the Theory of PROPs, a research monograph in progress with Donald Yau.
Current Research Collaborations
  1. David Blanc, Jim Turner and I received a grant supporting our ongoing work with diagrams of Pi-algebras, Andre-Quillen cohomology, and higher homotopy operations in Quillen model categories. We have been gradually shifting our focus toward the last of these. Here are slides from a 20-minute talk I gave in March 2008 on this project. We hope to finish the ideals SS article this summer and hopefully use it to study higher homotopy operations.
  2. Wojciech Dorabiala and I are continuing to talk about things related to Walhausen's algebraic K-theory of spaces. Most recently, we have been discussing an approach to the Additivity Theorem which would generalize to an adjoint functors approach to A-theory.
  3. Tom Fiore is telling me about double categories (e.g. internal categories in Cat), and we're talking about generalizing various constructions to them. With luck, this may eventually lead to a version of Waldhausen's K-theory functor for double categories.
  4. Donald Yau has been discussing various model category constructions with me, mostly for extending our article on colored PROPs into a research monograph. The content of the monograph is to be a careful discussion of graph substitution, Markl's notion of pasting scheme, generalized PROPs which include things like properads and half-PROPs in the same theory, and the appropriate notions of algebra or module. As a related side project, we hope to work with Terry Bisson to adapt his recent model category of graphs techniques to our version of colored wheeled graphs.
  5. Hossein Movahedi-Lankarani, Bob Wells, and I are talking about various constructions for generalized metric spaces, particularly trying to build the machinery of Lipschitz topology in this context. The nominal goal is to understand Hausdorff-Gromov limits in a quite general context.
Research Collaborators (Past, Present, and Future)
  1. Terry Bisson -- Canisius College
  2. David Blanc -- U. Haifa
  3. Wojciech Dorabiala -- Penn State Altoona
  4. Tom Fiore -- U. Michigan-Dearborn
  5. Michele Intermont -- Kalamazoo College
  6. Michael Joachim -- U. Munster
  7. Hossein Movahedi-Lankarani -- Penn State Altoona
  8. Jonathan Scott -- Cleveland State University
  9. Jim Turner -- Calvin College
  10. Robert Wells -- Penn State (University Park) Emeritus
  11. Donald Yau -- Ohio State Newark
Recent Presentations

Comments and questions on this website should be addressed to Dr. Mark W. Johnson.