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This page was the home of the Xtraordinary cohomology theory & K-theory Collective Discussion group* for the spring 2011 semester.

All talks at Berkeley were on Berkeley time (i.e. they started 10 minutes late).

Saturday, Feb. 19 -- 147 Jordon Hall (Stanford)

11:00-12:00 -- Surgery, part 1
The study of cobordism is in some sense the study of surgery. I'll define what it means to do surgery on a manifold, wave my hands about why this is equivalent to studying cobordisms, and then proceed to describe the technique of "elementary surgery," or surgery below the middle dimension. The result is that we can take a framed manifold and construct a cobordant manifold all of whose homotopy groups vanish below the middle dimension. I'll sketch some of the difficulty that occurs in the middle dimension and how it is related to the index or the Arf-Kervaire invariant. Finally, I'll state the main surgery theorem that appears in Browder's book.
Cary Malkiewich
references: Browder, Surgery on Simply-Connected Manifolds.

1:00-2:00 -- Introduction to the Goodwillie Calculus of Functors
Through the 90s, Goodwillie introduced a new organizing framework for homotopical algebra called the calculus of functors, which has since been used by Kuhn, McCarthy, and countless others to reinterpret classical results in stable homotopy theory from a unified perspective. We'll construct the calculus, note some of its cuter formal properties, and work whatever examples time permits. (Audience beware: this talk may well get split into two installments.)
Eric Peterson
references: Goodwillie, Calculus I, Calculus II, Calculus III.

Saturday, Feb. 26 -- 939 Evans

11:00-12:00 -- Surgery, part 2
Following Browder more closely, I'll show how elementary surgery can be carried over to the setting of normal cobordism. Then I'll discuss how to actually do surgery in the middle dimension, and outline the peculiar problems that arise when the dimension is congruent to 0, 1, 2, or 3 mod 4. If there's time, or next time, I'll discuss some theorems in differential topology that can be proven using Browder's results.
Cary Malkiewich

1:00-2:00 -- From Morse theory to Bott periodicity, part 1
In this series of talks I'll survey Bott's original proof of his celebrated periodicity theorem. I'll begin by building up the requisite machinery. In this first talk I'll give a rapid overview of Morse theory, and I'll also cover a bit of Riemannian geometry that we'll need in order to analyze the path space of a manifold.
Aaron Mazel-Gee
references: Milnor, Morse Theory. Lectures notes for the entire series. These give a quick introduction to basic Morse theory and Riemannian geometry and then proceed more carefully thereafter: sections 3 and 4 give a self-contained (and essentially complete) account the proof.

No meeting Saturday, Mar. 5.
Saturday, Mar. 12 -- 380D Sloan Hall (Stanford)

11:00-12:00 -- String topology
In 1999 Chas and Sullivan described a product on the homology of LM, the space of unbased maps of a loop S1 into a closed oriented manifold M. This yields a graded-commutative algebra that is a smooth invariant of M. Informally, the product can be thought of as using pairs of pants in M to join pairs of loops together to form a single loop. I'll introduce this product and discuss a few different ways to look at it, from the vague and informal to the totally rigorous. We'll also see what sort of structure it gives to the homology of LM and, if there's time, calculate this structure when M is an odd dimensional sphere.
Cary Malkiewich

1:00-2:00 -- From Morse theory to Bott periodicity, part 2
In this talk I'll illustrate how we can generalize the framework of Morse theory to obtain a CW-complex which is homotopy equivalent to the path space of a given manifold. This can be made quite explicit for symmetric spaces, which include compact Lie groups as special cases.
Aaron Mazel-Gee
references: Jump to the first lecture in this series.

Monday, Apr. 4 -- 891 Evans

2:00-3:00 -- The homology of the connective covers of BU, part 1
This is the first in a sequence of talks whose goal is to explore and understand the family of characteristic classes associated to the connective covers of BU. This talk will mostly be an overview: we'll begin by using some small bits of geometry to motivate the study of these spaces in the first place, and then we'll produce the comparison map from their homologies to a family of classical objects in arithmetic geometry. This map will turn out to be an isomorphism; proof of this fact and detailed analysis of these objects will occupy the talks following this one.
Eric Peterson
references: (As-yet incomplete) lecture notes for the entire series.

Wednesday, Apr. 13 -- varying locations

11:00-12:00 -- 740 Evans -- Leading to the perverse: an introduction to intersection (co)homology
In the 1970s Mark Goresky and Robert MacPherson discovered new topological invariants attached to any 'reasonably nice' singular space, the intersection homology groups of that space. After a 'fortuitous encounter' with Deligne at a Halloween party in France, Goresky and MacPherson developed their new theory in the language of sheaves, ultimately leading to the notion of perverse sheaves. These mysterious objects provide a link between geometry and representation theory and have had a profound impact in both areas.
     This talk will be the first of several aiming to provide a down to earth introduction to intersection homology. There will be many examples and pictures and the only prerequisite is a general understanding of your favourite homology theory.
George Melvin
references: Goresky & MacPherson, Intersection Homology Theory and Intersection Homology II. Kleiman, The Development of Intersection Homology Theory.

2:00-3:00 -- 961 Evans -- The homology of the connective covers of BU, part 2
Last time, we laid out the problem of studying the spaces BU<2k> and gave an unstructured computation of their mod p homology. This time, we will proceed to develop our algebraic model for these rings and reduce the study of these models in two ways: first, we need only study a small collection of them, and second, we'll study the structure of their tangent spaces as a simple approximation.
Eric Peterson
references: Jump to the first lecture in this series.

Monday, Apr. 25 -- 891 Evans

11:00-12:00 -- Symmetric spectra
Spectra are the objects we work with when we want to study topological spaces, but we want to trash anything that is killed by suspension. In particular, they are important in studying stable homotopy theory and cohomology theories. I'll review some basics about what spectra are, and what works or doesn't work when we try to use them. Then I'll describe symmetric spectra, one of the two "right" definitions of spectra. I'll also give some familiar examples and how they appear in symmetric land.
Cary Malkiewich
reference: Hovey, Shipley & Smith, Symmetric spectra.

2:00-3:00 -- The homology of the connective covers of BU, part 3
Eric Peterson
references: Jump to the first lecture in this series.

Monday, May 2 -- 891 Evans

11:00-12:00 -- Attempting to motivate the decomposition theorem POSTPONED
The Decomposition Theorem of Beilinson-Bernstein-Deligne-Gabber has been described as one of the deepest theorems in topology. It relates the topology of two complex algebraic varieties connected via a proper morphism f:X→Y and its expression requires the introduction of perverse sheaves.
     We will try to motivate why one would be led to such a theorem and look at some examples to keep us honest. Along the way we'll meet Lefschetz theorems and Leray spectral sequences. We may also be able to say how Ngo's recent work fits in with this... But probably not.
George Melvin

2:00-3:00 -- The homology of the connective covers of BU, part 4
Eric Peterson
references: Jump to the first lecture in this series.

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