Jan 27, 2014
Monday

09:15 AM  09:30 AM


Welcome

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
 
 Supplements



09:30 AM  10:30 AM


Introduction to operads
William Dwyer (University of Notre Dame)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
The talk will explain what an operad is, give some examples, and illustrate how operads come up in various mathematical settings.
 Supplements


10:30 AM  11:00 AM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Goodwillie's calculus of functors
Michael Ching (University of Massachusetts, Amherst)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
The calculus of homotopy functors provides a systematic way to approximate a given functor (say from based spaces to spectra) by socalled `polynomial' functors. Each functor F that preserves weak equivalences has a `Taylor tower' (analogous to the Taylor series of ordinary calculus) which in turn is built from homogeneous pieces that are classified by certain `derivatives' for F. I will review this material and consider the problem of how the Taylor tower of F can be reconstructed from its derivatives. We will discuss some important examples built from mapping spaces. Then. if time permits, I will us this approach to give a classification of analytic functors from based spaces to spectra and try to describe some connections to the GoodwillieWeiss manifold calculus
 Supplements


12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Morita theory in stable homotopy
Brooke Shipley (University of Illinois at Chicago)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
: I will give an introduction to Morita theory in stable homotopy theory. This will include the context of differential graded algebra and of spectral algebra. I will also survey some recent related results.
 Supplements


03:00 PM  03:20 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



04:10 PM  05:00 PM


MSRI/Evans Lecture: Topological cyclic homology (Evans Hall, UC Berkeley)
Lars Hesselholt (Nagoya University)

 Location
 UC Berkeley, 60 Evans Hall
 Video


 Abstract
Topological cyclic homology is a topological refinement of Connes' cyclic homology. It was introduced twentyfive years ago by BökstedtHsiangMadsen who used it to prove the Ktheoretic Novikov conjecture for discrete groups all of whose integral homology groups are finitely generated. In this talk, I will give an introduction to topological cyclic homology and explain how results obtained in the intervening years lead to a short proof of this result in which the necessity of the finite generation hypothesis becomes transparent. In the end I will explain how one may hope to remove this restriction and discuss number theoretic consequences that would ensue.
 Supplements




Jan 28, 2014
Tuesday

09:30 AM  10:30 AM


Chromatic redshift
John Rognes (University of Oslo)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
 
 Supplements


10:30 AM  11:00 AM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Higher categories and algebraic Ktheory
Andrew Blumberg (University of Texas)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
In this talk, I will given an overview of recent work on formulating the structural properties of algebraic Ktheory using the framework for studying homotopical categories provided by the development of higher category theory
 Supplements


12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Towards explicit models for higher Ktheories
Christopher Douglas (University of Oxford)

 Location
 
 Video

 Abstract
 
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03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Models for homotopical higher categories
Julie Bergner (University of Virginia)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Models for homotopical categories, known as $(\infty,1)$categories, are now fairly welldeveloped. Comparisons are in place between them, and there is a wide variety of applications ranging from topology to algebra to algebraic geometry. Generalizing them to models for homotopical higher categories, or $(\infty,n)$categories, requires substantially greater technicality. Much work has been done in this direction, due to recent work of many authors, but the full picture is still very much work in progress. In this talk we will review different models for $(\infty,1)$categories and discuss the number of ways they are being generalized to models for $(\infty,n)$categories, as well as the known and conjectured comparisons between them
 Supplements


04:30 PM  06:20 PM


Reception

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements




Jan 29, 2014
Wednesday

09:00 AM  10:00 AM


Computations in the stable homotopy groups of spheres
Mark Behrens (Massachusetts Institute of Technology)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
I will give an overview of the rich structure in the stable homotopy groups of spheres, both giving a summary of what we do know, and what we don't know.
 Supplements


10:00 AM  10:30 AM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



10:30 AM  11:30 AM


Computations in motivic homotopy theory
Daniel Isaksen (Wayne State University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We will begin with an introduction to stable motivic homotopy theory. Then we will use the motivic version of the Adams spectral sequence to compute stable motivic homotopy groups. We will discuss a number of open questions concerning these computations. Along the way, we will encounter some new results about classical stable homotopy groups and equivariant stable homotopy groups.
 Supplements


11:30 AM  12:30 PM


Views on the Jhomomorphism
Craig Westerland (University of Minnesota Twin Cities)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
This will be a survey talk on subjects related to the Jhomomorphism. In addition to the original geometric construction which assigns to a vector bundle its associated spherical fibration, we will touch on more algebraic constructions derived form it, in particular via Picard and unit spectra. We will examine the map from a computational point of view, studying its plocalisation and interpretation in terms of K(1)local homotopy theory. We will almost certainly fail to do justice to any of these ideas in the time available. At the end, we will briefly extend some of these results to a higher chromatic setting.
 Supplements



Jan 30, 2014
Thursday

09:30 AM  10:30 AM


Local structure of groups and of their classifying spaces
Bob Oliver (Université de Paris XIII (ParisNord))

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
This will be a survey talk on the close relationship between the local structure of a nite group or compact Lie group and that of its classifying space. By the plocal structure of a group G, for a prime p, is meant the structure of a Sylow psubgroup S G (a maximal ptoral subgroup if G is compact Lie), together with all Gconjugacy relations between elements and subgroups of S. By the plocal structure of the classifying space BG is meant the structure (homotopy properties) of its pcompletion BG^p . For example, by a conjecture of Martino and Priddy, now a theorem, two nite groups G and H have equivalent plocal structures if and only if BG^p ' BH^p . This was used, in joint work with Broto and Møller, to prove a general theorem about local equivalences between nite Lie groups a result for which no purely algebraic proof is known. As another example, these ideas have allowed us to extend the family of pcompleted classifying spaces of (nite or compact Lie) groups to a much larger family of spaces which have many of the same very nice homotopy theoretic properties.
 Supplements


10:30 AM  11:00 AM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Equivariant homotopy and localization
Michael Hopkins (Harvard University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
This will be a general talk on equivariant homotopy theory
 Supplements


12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Homotopy theory of KacMoody groups
Nitya Kitchloo (Johns Hopkins University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
The homotopy theory of compact Lie groups is very well understood by now. The rich structure of these groups (for example: existence and uniqueness of maximal tori, corresponding Weyl groups etc.) may be exploited to classify these groups. This classification even extends to homotopical versions of these groups known as pcompact groups. In the last few decades a beautiful new class of (noncompact) topological groups has been constructed. These are known as KacMoody groups and they share most of the structure that compact Lie groups admit. KacMoody groups have been shown to be relevant in mathematical physics and further investigation by several mathematicians (including the speaker) seems to suggest that KacMoody groups are surprisingly amenable to homotopical techniques. This makes these groups prime candidates for study from the standpoint of homotopy theory
 Supplements


03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Topological automorphic forms
Tyler Lawson (University of Minnesota Twin Cities)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
: Topological modular forms and its generalizations are objects in stable homotopy that realize a connection to 1dimensional formal group laws. In this talk I'll describe how this perspective on stable homotopy has emerged, and how using algebraic geometry is providing us with a library of new objects in homotopy theory
 Supplements


04:30 PM  06:00 PM


Problem Session

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
 
 Supplements




Jan 31, 2014
Friday

09:30 AM  10:30 AM


Representation stability and applications to homological stability
Thomas Church (Stanford University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
I will give an introduction to the theory of representation stability, through the lens of its applications in homological stability. I'll focus on three applications: homological stability for configuration spaces of manifolds; understanding the stable (and unstable) homology of arithmetic lattices; and stability for twisted homology such as H_i( GL_n(R); R^n ), where the coefficients change along with the groups.
 Supplements


10:30 AM  11:00 AM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Stability of moduli spaces of manifolds
Oscar RandalWilliams (University of Cambridge)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
I will first discuss some fairly classical homological stability phenomena: spaces of 0manifolds (e.g. configuration spaces), and spaces of 2manifolds (e.g. Riemann's moduli space). I will explain the general method, introduced by Quillen, for proving these stability theorems, and then explain some recent work with Søren Galatius which proves such a stability theorem for moduli spaces of 2ndimensional manifolds (for n > 2).
 Supplements


12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Stable homology of moduli spaces of manifolds
Soren Galatius (Stanford University)

 Location
 MSRI:
 Video

 Abstract
I will discuss recent joint work with Oscar RandalWilliams, aimed at calculating the cohomology of BDiff(W) and related spaces, where W is a smooth 2ndimensional manifold, Diff(W) is the topological group of diffeomorphisms of W, and BDiff(W) is its classifying space. Surprisingly, the cohomology ring turns out to be partially independent of W through a range of degrees (homological stability). In this talk, I will discuss how infinite loop spaces can be used to describe the cohomology in this stable range.
 Supplements


03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Loop Groups, TQFTs and algebraic geometry
Constantin Teleman (UC Berkeley Math Faculty)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
In this survey talk, I will shall the role that loop groups have played in the construction of topological gauge theories in dimensions 2 and possibly 3.
 Supplements


