Nov 30, 2017
Thursday

09:00 AM  09:10 AM


Introduction and Welcome

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
 
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09:15 AM  09:45 AM


Direct proofs of properties and structures of model structures for (∞, 1)categories
Julie Bergner (University of Virginia)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
There are many different models for (∞,1)categories, each of which has an associated model category. Given these model structures, we’d like to know what additional properties they possess, for example, which are simplicial, cartesian, or left or right proper? We know several of these results, but not all; and in particular there are not always counterexamples in the literature for when a model structure does not have a desired property. Furthermore, some of these properties are known from very general results, and it would be nice to have a more concrete proof for a given model category. In this project, we’ll seek to fill in some of these gaps in our knowledge.
Reading List:
• J. Bergner, “A survey of (∞, 1)categories” In: Baez, J., May, J.P. (eds) Towards Higher Categories, Vol. 152. Springer, NY.
• P. Hirschhorn, Model Categories and their Localizations, for background on sim plicial, monoidal, proper model categories.
 Supplements


09:45 AM  10:15 AM


Directed homotopy and homology theory, with an eye towards applications
Lisbeth Fajstrup (Aalborg University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
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.
Reading List:
 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 Nonreversible 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
 Supplements


10:15 AM  10:45 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
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10:45 AM  11:15 AM


Connections between Ktheory and fixed point invariants
Cary Malkiewich (Binghamton University (SUNY)), Kathleen Ponto (University of Kentucky)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
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.
 Supplements


11:15 AM  11:45 AM


Foundations of (∞, 2) category theory
Emily Riehl (Johns Hopkins University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Work of Joyal, Lurie and many other contributors can be summarized by saying that ordinary 1category theory extends to (∞,1)category theory: that is, there exist homotopical/derived analogs of 1categorical results. As “brave new algebra” grows in influence, many areas of mathematics now require homotopical/derived analogs of 2categorical 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 quasicategorical 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 2category 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 2categorical limits”, Bull. Austral. Math. Soc., Vol. 39 (1989), pp. 301317.
Potential participants should skim bits of the first two, but need not read either in full.
 Supplements


11:45 AM  01:45 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
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01:45 PM  02:15 PM


Applications of monoidal model categories of spectra
Brooke Shipley (University of Illinois at Chicago)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
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 KTheory (Seattle, WA, 1997), 41–87, Proc. Sympos. Pure Math., 67, Amer. Math. Soc., Providence, RI, 1999. (In particular, section 6.1) Available at: http://wwwmath.mit.edu/∼larsh/papers/008/gh.pdf
• B. Shipley, HZalgebra spectra are differential graded algebras, Amer. J. Math. 129 (2007) 351379.
General reference:
• S. Schwede, Symmetric spectra, untitled book in progress. Available at: www.math.uni bonn.de/people/schwede/SymSpec
 Supplements


02:15 PM  02:45 PM


Homological Stability
Nathalie Wahl (University of Copenhagen)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
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. RandallWillliams and N. Wahl, \Homological stability for autormorphism
groups", Advances in Mathematics, Vol. 318 (2017), pp. 534  626.
 Supplements


03:00 PM  03:30 PM


Tea

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
 
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03:30 PM  04:00 PM


Team Introductions

 Location
 Simons Auditorium, Commons Room, Atrium, Space Science Lab
 Video


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04:00 PM  05:30 PM


First working session

 Location
 Simons Auditorium, Commons Room, Atrium, Space Science Lab
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05:30 PM  07:30 PM


Dinner on your own in downtown Berkeley

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08:00 PM  08:30 PM


Informal discussion

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