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Goal: We will study the topology of configuration spaces and E_n algebras. We will start with papers from the 1970's by people such as May, McDuff, and Segal. We will work our way towards modern generalizations such as Lurie's and Salvatore's models of topological chiral homology. We may also discuss applications to gauge theory (Atiyah-Jones), holomorphic function spaces (Segal), moduli spaces of surfaces (Madsen-Weiss) or group homology of automorphisms of the free group (Galatius). We may also discuss homology operations Ă la Cohen.
Past talks:
10/26/2012
Speaker: Jeremy Miller
Topic: Configuration spaces of points in a manifold part 1
Paper: McDuff "Configuration spaces of positive and negative particles"
Abstract: McDuff proves that as the number of particles tends to infinity, the topology of the space of distinct unordered particles on a manifold has the same homology as a certain space of sections. In part 1, we will focus on definitions.
11/02/2012
Speaker: Jeremy Miller
Topic: The group completion theorem part 1
Paper: McDuff-Segal "Homology fibrations and the group completion theorem"
Abstract: For A, a topological monoid satisfying mild hypothesis, they prove that a localization of the homology of H_*(A) is the homology of H_*(Omega B A). Here Omega is the based loop space and B is the classifying space functor. In this talk we make sense of the above terms and use it to define the K-theory of a ring.
11/09/2012
Speaker: Jeremy Miller
Topic: Configuration spaces of points in a manifold part 2
Paper: McDuff "Configuration spaces of positive and negative particles"
Abstract: In part 2, we focus on proofs.
11/16/2012
Speaker: Elizabeth Viadurre
Topic: Serre spectral sequence
Primary source: Serre's thesis
Expository source: Hatcher's Spectral Sequences in Algebraic Topology
Abstract: I will describe the construction of a spectral sequence using exact couples and then state the Serre spectral sequence, which can then be used to compute the cohomology ring of iterated loops on a sphere over the rationals..
11/30/2012
Speaker: Aron Fischer
Topic: Operads and E_n algebras
Paper: May "Geometry of iterated loop spaces"
Expository paper: Cohen-Voronov "Notes on string topology, chapter 2"
Abstract: We will define operads and their algebas and introduce important examples.
12/07/2012
Speaker: Jeremy Miller
Topic: The recognition principle
Paper: May "Geometry of iterated loop spaces"
Expository paper: Sections 1.1, 1.2 and 2.1 of Nathalie Wahl's thesis
Abstract: We will prove that every connected algebra over the little n-cubes operad is homotopy equivalent to an n-fold loop space. The primary tool is the monadic two-sided bar construction.
Warning: the notes have more mistakes than usual
2/01/2013
Speaker: Jeremy Miller
Topic: The Dold-Thom theorem and Poincare duality part 1
Paper: Kallel "Spaces of particles on manifolds and generalized poincare dualities" and something in German.
Abstract: I am going to give a proof of Dold and Thom's theorem that the homotopy groups of the symmetric product of a space are the homology groups of that space. Then, I will give Kallel's proof that the homotopy groups of the symmetric of a manifold are the cohomology groups of that manifold. This gives a proof of Poincare duality. This is the sense in which Lurie's non-abelian Poincare duality is a generalization of Poincare duality.
2/13/2013
Speaker: Jeremy Miller
Topic: The Dold-Thom theorem and Poincare duality part 2
Paper: Kallel "Spaces of particles on manifolds and generalized poincare dualities" and something in German.
Abstract: In part 1, we talked about the Dold-Thom theorem. In part 2, we use it to prove Poincare duality.
2/26/2013
Speaker: Jeremy Miller
Topic: Homological stability for symmetric groups
Paper: Nakaoka, Decomposition theorem for homology groups of symmetric groups.
Expository paper: Randal-Williams, Homological stability for unordered configuration spaces
Abstract: I will give Randal-WIlliam's proof that the homology of the symmetric groups stabilizes.
3/05/2013
Speaker: Martin Bendersky
Topic: Homological operations for E_n-algebras part 1
Papers: Browder, Homology operations and loop spaces. Cohen, Homology of iterated loop spaces.
Abstract: Martin will introduce the Dyer-Lashof operations and the Browder operation and describe some relations.
3/12/2013
Speaker: Martin Bendersky
Topic: Homological operations for E_n-algebras part 2
Papers: Browder, Homology operations and loop spaces. Cohen, Homology of iterated loop spaces.
Abstract: Martin will describe properties of the homology of iterated loop spaces of spheres and configuration spaces. He will then use this to deduce homological stability of the symmetric group.
3/19/2013
Speaker: Aron Fischer
Topic: Atiyah Duality part 1
Expository paper: Chapter 2 of Kuper's notes to Malkiewich's talk in Awesome Joint Berkeley-Stanford String Topology Seminar.
Abstract: Aron will introduce (pre-) specra and describe a refinement of Poincare duality.
4/09/2013
Speaker: Aron Fischer
Topic: Atiyah Duality part 2
Expository paper: Chapter 2 of Kuper's notes to Malkiewich's talk in Awesome Joint Berkeley-Stanford String Topology Seminar.
Abstract: Aron will introduce (pre-) specra and describe a refinement of Poincare duality.
4/23/2013
Speaker: Jeremy Miller
Topic: Intro to topological chiral homology
Papers: http://arxiv.org/abs/math/9907073, http://arxiv.org/pdf/1209.2773v3.pdf,
http://www.math.northwestern.edu/~jnkf/writ/facthomology.pdf, or DAG VI.
Abstract: Recall my talk "The Dold-Thom theorem and Poincare duality." If you replace the integers with an arbitrary (group-like) commutative monoid, everything works exactly the same. What happens if you want to drop the word commutative?
4/30/2013
Speaker: Jeremy Miller
Topic: The group completion theorem.
Abstract: I will prove the group completion theorem.
5/07/2013
Speaker: Manuel Rivera
Topic: Bott periodicity
Papers: Bott periodicity via simplicial spaces by Bruno Harris
Abstract: A wise man who rides a motorcycle once told me, if you want to prove something is homotopic to a loop space of something else, instead prove that something else is homotopic to the clasifying space of something. Bott periodicity is the statement that something is the twofold loop space of itself. You get the idea.
5/14/2013
Speaker: Joey Hirsh
Topic: Deligne's conjecture
Abstract: Deligne asked the hochschild cochains be given the structure of an algebra over the chains of the little disks operad. Joey will tell us something about this.
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