Schedule, Notes/Handouts & Videos

Directed spaces, i.e., topological spaces equipped with a “sense of direction” of some sort, arise naturally in applications of topology, in particular in computer science and in neuroscience, where they play an increasingly important role.  It is natural to want to develop appropriate directed versions of familiar homotopy invariants, which turns out to be significantly harder than one might expect. 
For example, it is not clear how to formulate a good definition of “directed” homology, even when restricting to directed spaces built from simplices or cubes. To be of theoretical interest, directed homology should be an invariant of an acceptable notion of weak equivalence of directed spaces and should distinguish between at least some 
 directed spaces with the same underlying undirected space. To be of practical interest for applications, it should also be reasonably computable.

• L. Fajstrup, E. Goubault, E. Haucourt, S. Mimram, M. Raussen, Directed Algebraic Topology and Concurrency, Springer Verlag, 2016. 

• M. Grandis, Directed Algebraic Topology. Models of Non-reversible worlds. Cam- bridge University Press, 2010.
Available from Grandis’ website
http://www.dima.unige.it/∼ grandis/BkDAT page.html 

• M. J. Dubut, “Directed homotopy and homology theories of geometric models of true concurrency” (2017 PhD thesis): http://www.lsv.fr/∼dubut/manuscript.pdf 


A major reason the Euler characteristic is a fantastic topological invariant is because it is additive on subcomplexes. It is also a fixed point invariant - it is a signed count of the number of fixed points of the identity map. (Replace the map bya homotopic map with isolated fixed points.) These two observations raise questions about how to think about fixed point invariants. We'll describe an approach to fixed point invariants that, while initially motivated by classical invariants, is reaching toward prioritizing additivity.
Reading List:
 R. Geoghegan, \Nielsen Fixed Point Theory" pp. 499 - 521 in R.J. Daverman
and R.B. Sher (eds) Handbook of Geometric Topology, (2001), Elsevier B.V.
 K. Ponto and M. Shulman, \Traces in symmetric monoidal categories", Expo-
sitiones Mathematicae, Vol. 32, Issue 3, 2014. pp. 248 { 273.

Work of Joyal, Lurie and many other contributors can be summarized by saying that ordinary 1-category theory extends to (∞,1)-category theory: that is, there exist homotopical/derived analogs of 1-categorical results. As “brave new algebra” grows in influence, many areas of mathematics now require homotopical/derived analogs of 2-categorical results and this work largely remains to be done in a rigorous fashion.  In my talks, I will give an overview of the development of (∞, 1)-category theory in the quasi-categorical model and describe the main idea behind the proof that this theory is “model independent.” I’ll then suggest some models of (∞, 2)- categories that might prove fertile for studying extensions of 2-category theory and sketch a possible strategy to demonstrate model independence.

Reading List:

• J. Lurie “(∞, 2)-categories and the Goodwillie Calculus I”, October 8, 2009. 
Available from http://www.math.harvard.edu/∼lurie/papers/GoodwillieI.pdf. 

• D. Gaitsgory and N. Rozenblyum , Appendix A of A study in derived algebraic geometry, Mathematical Surveys and Monographs, Vol. 221 (2017), pp. 419 - 524.
Available from http://www.math.harvard.edu/∼gaitsgde/GL/. 

• G. M. Kelly “Elementary observations on 2-categorical limits”, Bull. Austral. Math. Soc., Vol. 39 (1989), pp. 301-317.


Potential participants should skim bits of the first two, but need not read either in full. 


I will give an overview of the development of symmetric spectra, orthogonal spectra, and other monoidal model categories of spectra as well as some generalizations and applications. I then expect small groups to each focus on one of the various references listed below.

Reading List:

Foundations:

• M. Hovey, B. Shipley, J. Smith, Symmetric spectra, J. Amer. Math. Soc., 13 (2000), 149–208. (Only the first three sections are essential.) 

• M. Mandell, J. P. May, S. Schwede, B. Shipley, Model categories of diagram spectra, Proc. London Math. Soc. 82 (2001), 441–512. 
Generalizations and applications: 


• M. Hovey, Spectra and symmetric spectra in general model categories, Journal of Pure and Applied Algebra 165 (2001) 63–127. 

• T. Geisser and L. Hesselholt, Topological cyclic homology of schemes. Algebraic K-Theory (Seattle, WA, 1997), 41–87, Proc. Sympos. Pure Math., 67, Amer. Math. Soc., Providence, RI, 1999. (In particular, section 6.1) Available at: http://www-math.mit.edu/∼larsh/papers/008/gh.pdf

• B. Shipley, HZ-algebra spectra are differential graded algebras, Amer. J. Math. 129 (2007) 351-379.

General reference:

• S. Schwede, Symmetric spectra, untitled book in progress. Available at: www.math.uni- bonn.de/people/schwede/SymSpec

Families of groups such as symmetric groups or general linear groups, and
families of spaces such as con guration spaces or certain moduli spaces of manifolds,
display a stability phenomenon in their homology. I'll describe this phenomenon and give an idea about how such theorems are proved.

Reading List:

First three sections of
 O. Randall-Willliams and N. Wahl, \Homological stability for autormorphism
groups", Advances in Mathematics, Vol. 318 (2017), pp. 534 - 626.