Mark W. Johnson's Research

Professor of Mathematics

Penn State Altoona

Homepage

Conference Information Link

I am an organizer of a conference to be held at the University of Illinois (at Urbana-Champaign) July 17 to 21, 2017.

Old Conference Information

• I was an organizer of an American Mathematical Society special session at the Jan. 2014 Joint Mathematics meetings in Baltimore, along with a satellite conference at Johns Hopkins
• I was an organizer of an American Mathematical Society special session at the Jan. 2012 Joint Mathematics meetings in Boston, along with a satellite conference at MIT.
• I was corresponding organizer of a special session at the Fall 2009 AMS Eastern Section Meeting in State College, PA, October 24 and 25, as well as an associated mini-conference at Penn State Altoona on October 23. Session details
• I was corresponding organizer of an American Mathematical Society special session at the Jan. 2009 Joint Mathematics meetings in Washington, D. C.
• I was corresponding organizer of a special session at the Fall 2007 AMS Southeastern Section Meeting in Murfreesboro, TN, November 3 and 4.

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).

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

• "On Minimal Higher Homotopy Operations", Topology Seminar, Johns Hopkins University, October 2011. Also at Union College Mathematics Conference, April 2011.
• "Deconstructing Waldhausen's S_\bullet construction", American Mathematical Society Eastern Sectional Meeting, Ithaca NY, September 2011.
• "Quillen's Language of Model Categories", Mathematics Colloquium, University of Haifa, Israel, May 2011.
• "Stable Homotopy and the Language of Sheaves", Topology Student Seminar, Hebrew University of Jerusalem, Israel, May 2011.
• "Minimal Higher Homotopy Operations and Realizing Resolutions", Deformation Theory Seminar, University of Pennsylvania, February 2011.
• "On Homotopy Invariance for Algebras over Colored PROPs", Geometry and Physics Seminar, Penn State, November 2010.
• "Thoughts on higher Segal maps and internal categories", Topology/Geometry Seminar, Penn State, February 2010.
• "Spec of a category and a Tot spectral sequence", Topology/Geometry Seminar, Penn State, October 2009.
• "On homotopy invariance for algebras over colored PROPs", Mathematics Colloquium, University of Haifa, June 2009.
• "Pi-algebras and higher homotopy operations", Topology Seminar, City University of New York, May 2009.
• "On homotopy invariance for algebras over colored PROPs", Topology/Geometry Seminar, Penn State, April 2009.
• "On modified Reedy structures", Topology Seminar, Wayne State University, November 2008.
• "Homotopy theory of colored PROPs", Homotopy Theory special session of the Fall 2008 AMS Central Section Meeting, organized by Michele Intermont (Kalamazoo), John Martino (Western Michigan), and Jeff Strom (Western Michigan).
• "Pi-algebras and higher homotopy operations," Topology Seminar, Ohio State University, May 2008.
• "On obstructions to realizing diagrams of Pi-algebras," Geometric Topology special session of the Spring 2008 AMS Eastern Section Meeting, organized by David Rosenthal (St. John's) and Marco Varisco (Binghamton).
• "A local-to-global approach to Andre-Quillen cohomology of diagrams," Topology Seminar, University of Chicago, May 2007.
• "A local-to-global approach to Andre-Quillen cohomology of diagrams," Deformation Theory Seminar, University of Pennsylvania, May 2007.
• "Factoring the Becker-Gottlieb transfer through the trace map," Topology Seminar, MIT, October 2006.
• "A different view of the algebraic K-theory of a model category," Topology/Geometry Seminar, Penn State, September 2006.
• "Factoring the Becker-Gottlieb transfer through the trace map," Topology Proseminar, University of Chicago, March 2006.
• "Factoring the Becker-Gottlieb transfer through the trace map," Topology Seminar, Northwestern University, March 2006.
• "Realizing diagrams of Pi-algebras," Union College Mathematics Conference, Contributed Talks in Algebraic Topology, December 2005.
• "Toward the Andre-Quillen cohomology of diagrams," Topology/Geometry Seminar, Penn State, September 2005.
• "Realizing generalized Pi-algebras," Special Session at AMS Eastern Section Meeting, April 2005.
• "Resolution Model Categories," Topology/Geometry Seminar, Penn State, March 2005.

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