Topology seminar

Location

Mondays at 2:30 in Vincent Hall 311

Talks in Spring 2013

• Jan 28

  • Tyler Lawson (University of Minnesota)

  • Secondary operations

  Secondary operations featured prominently in Adams' proof of the Hopf invariant one conjecture, which implies there are no division algebra structures on $\mathbb{R}^n$ when $n \neq 1,2,4,8$. I'll discuss how secondary products and secondary operations appear in different contexts in algebra and topology.

• Feb 04

  • Alexander Voronov (University of Minnesota)

  • Secondary invariants from Dijkgraaf-Witten theory

  As Tyler Lawson explained to us in his last seminar talk, secondary operations arise when a primary operation vanishes or two primary operations are equal. If you have a bordism between two closed manifolds, their homology classes will be equal and, by definition, their cobordism classes will be equal. A Topological Field Theory (TFT) creates a secondary invariant in this case: a linear map between certain vector spaces associated to each manifold. Dijkgraaf-Witten (DW) theory is an example of a TFT associated to a finite group. Recently, there has been a push towards describing an Extended TFT, and that generated deeper understanding of DW theory. I will discuss new insights on DW theory as per Jacob Lurie and others.

• Feb 18

  • Tyler Lawson (University of Minnesota)

  • Forms of K-theory and elliptic cohomology

  I'll discuss K-theory, and how its theory of Chern classes differs from that for ordinary cohomology. I'll then discuss "forms" of K-theory, which provide some twist on the multiplicative rules of ordinary K-theory, and how real K-theory is universal among those. Elliptic cohomology theories will then be introduced, along with the analogous properties a universal object should have.

• Feb 25

  • Tyler Lawson (University of Minnesota)

  • Elliptic curves and elliptic cohomology theories

  I'll be continuing last week's talk by introducing elliptic cohomology theories, and universal objects among these.

• Mar 04

  • Erin Manlove (University of Minnesota)

  • Morava Stabilizer Groups

  Morava stabilizer groups arise in the study of homotopy groups of spheres as objects that encode topological information. Though Morava stabilizer groups themselves remain relatively mysterious, we may study them by looking at their maximal finite subgroups. In my talk, I will give two characterizations of Morava stabilizer groups, describe the type of subgroups that are of interest, and discuss how we connect these subgroups to geometric objects by identifying them with automorphism groups of points in an algebraic stack.

• Mar 11

  • Niles Johnson (Ohio State)

  • Obstruction theory for homotopical algebra maps

  We take an obstruction-theoretic approach to the question of algebraic structure in homotopical settings. At its heart, this is an application of the Bousfield-Kan spectral sequence adapted for the action of a monad $T$ on a topological model category. This yields an obstruction theory for lifting maps from $T$-algebras in a homotopy category to the homotopy category of $T$-algebras. Maps in this latter category are required to commute with a coherent system of higher homotopies encoding the $T$-action. We give general conditions under which the $E_2$ page of the resulting spectral sequence may be identified with André-Quillen cohomology.

  This talk will give an overview of the general theory, but focus mainly on the breadth of applications. These include categories of algebraic theories in the sense of Lawvere and categories of algebras over operads. We will describe specific examples in $G$-spaces, $G$-spectra, and rational $E_\infty$ algebras. Furthermore, we will outline calculations connected to rational unstable homotopy theory which distinguish between $E_\infty$ and $H_\infty$.

• Apr 01

  • Eric Peterson (University of California, Berkeley)

  • Cotangent spaces of certain spectra

  We outline a construction in derived algebraic geometry which produces elements of the $K(n)$-local Picard group. As an example, we produce a new model of a spectrum called the determinantal sphere, and as time permits we discuss newly indicated patterns in $K(n)$-local homotopy theory.

• Apr 08

  • Robert Hank (University of Minnesota)

  • Massey products in $A_\infty$ algebras

  During this talk, I'll develop a relationship between Massey products and multiplicative structure in such a way that the generalization of a Massey product from a strictly associative to an $A_\infty$ structure arises naturally.

• Apr 15

  • William Abram (University of Michigan)

  • The equivariant complex cobordism ring of a finite abelian group

  The calculation of the non-equivariant cobordism ring due to Milnor and Quillen was one of the great successes of algebraic topology. Equivariantly, Kriz described tom Dieck's stable equivariant complex cobordism ring $(MU_G)_*$, in the case $G=\mathbb Z/p$, as the pullback of a diagram of rings arising from the Tate diagram for $MU_{\mathbb Z/p}$. We extend this work to the case of $G$ a finite abelian group, where we describe $(MU_G)_*$ as the inverse limit over certain $G$-spectra $F(S)_*$ indexed over chains of subgroups of $G$. In the case $G=\mathbb Z/p^n$, we get a simple description of $(MU_G)_*$ as the n-fold pullback of a diagram of rings. In the general case, we are still able to compute the algebraic structure of $F(S)_*$ explicitly. I will discuss this computation in some detail.

• Apr 22

  • Yifei Zhu (University of Minnesota)

  • Power operations in an elliptic cohomology theory

  As a result of the geometry of vector bundles underlying K-theory, power operations in K-theory correspond to linear representations of finite symmetric groups. For elliptic cohomology theories, lacking analogous geometric interpretations, we can study their power operations via a correspondence to cyclic isogenies of elliptic curves, in hope of understanding their conjectural geometry. This approach is valid for Morava E-theories in general, including versions of K-theory and elliptic cohomology as special cases where computations can be carried out explicitly.

• Apr 29

  • Donald Kahn (University of Minnesota)

  • Schemes and homotopy after Fabian Morel

  I will attempt to survey the works of Morel on his construction of the homotopy category of schemes and the relation to motivic homotopy with applications to to the Grothendieck-witt ring of quadratic forms and questions about when relations in classic homotopy are actually algebraic. This will have to be a sketch.

• May 06

  • Nathaniel Stapleton (M.I.T.)

  • The E-theory of centralizers in symmetric groups

  Using cohomology theories closely related to Morava E-theory, we will provide an algebro-geometric description of the cohomology of particular centralizers in symmetric groups in terms of the scheme classifying subgroups of a certain p-divisible group.