**What:**

**When:** Saturday and Sunday, February 6-7.

**Where:** Northwestern University, Evanston, IL, Swift 107

**Who:**

- Soren Galatius

**Some applications of cellular \(E_2\) algebras****Abstract:**

A cell structure on an \(E_2\) (little disks) algebra is a weak equivalence from an \(E_2\) algebra built by iterated cell attachments. I will recall this notion and discuss how to find cell structures on specific \(E_2\) algebras. Then I will explain some applications to the homology of mapping class groups and general linear groups. This is joint work with A. Kupers and O. Randal-Williams. - Eric Peterson

**Cocycle schemes and \(MU[2k,\infty)\)-orientations****Abstract:**

We recall the study of \(MU[2k, \infty)\)-orientations as elucidated by Ando, Hopkins, and Strickland. Their work prompts us to investigate a particular algebraic moduli which, after 2-localization, we fully describe for all values of k. It gives a strikingly good (but imperfect) approximation of our topological motivator. - Inna Zakharevich

**A topological proof of a theorem of Larsen and Lunts****Abstract:**

The Grothendieck ring of varieties \(K_0(\mathrm{V})\) is defined to be the free abelian group generated by varieties, modulo the relation that for any variety \(X\) and any subvariety \(Y\), \([X] = [Y] + [X \backslash Y]\). Multiplication is defined by \([X] [Y] = [X \times Y]\). This is the universal additive invariant, in the sense that if any function \(\chi\) on varieties satisfies the formula \(\chi(X) = \chi(Y) + \chi(X \backslash Y)\) (such as point counting or Euler characteristic) factors through this ring. When Kapranov introduced his motivic zeta function he conjectured that it would always be rational. However, in their 2002 paper Larsen and Lunts showed that this is in fact generally not the case. The main step in their proof was a computation of the quotient ring \(K_0(V)/([\mathbb{A}^1])\). In this talk we give an alternate proof of this computation by replacing the ring \(K_0(V)\) by an \(E_\infty\) ring spectrum \(K(V)\) and computing \(\pi_0\) of the cofiber of the map \(K(V) \to K(V)\) induced by multiplication by \(\mathbb{A}^1\). - Kirsten Wickelgren

**\(\mathbb{A}^1\)-Milnor number: counting zeros arithmetically****Abstract:**

The Milnor fibration is a fibration associated to a point p of a hypersurface f = 0 over the complex numbers. Its fiber is homotopy equivalent to a wedge of spheres and the number of these spheres has an interpretation in singularity theory: if p is a singularity, it is possible to deform p into nodes, which are the simplest type of singularities, and the number of these nodes is the same as the number of spheres, or the Milnor number. Nodes over fields other than C have arithmetic information associated with them. We use \(\mathbb{A}^1\)-homotopy theory to give an enrichment of Milnor?s theorem equating the number of nodes with the number of spheres to an equality in the Grothendieck-Witt group of a field of characteristic not 2. To do this, we prove that the local \(\mathbb{A}^1\)-Brouwer degree equals the quadratic form of Eisenbud-Khimshiashvili-Levine, answering a question posed by David Eisenbud in 1978. This is joint work with Jesse Kass. - Carolyn Yarnall

**Slices and Suspensions****Abstract:**

The equivariant slice filtration is an analogue of the Postnikov tower for G-spectra. However, unlike the Postnikov tower, the slice tower does not commute with taking ordinary suspensions and, in fact, what results when suspending slice towers is not understood in general. In this talk, after recalling the construction of the slice tower, we will look at the slice towers for integer-graded suspensions of \(H\mathbb{Z}\) and compare them to complementary results of Hill, Hopkins, and Ravenel concerning \(\lambda\)-suspensions. We will conclude with a brief look at future directions regarding the interplay between suspension and the slice filtration for general G-spectra. - Craig Westerland

**Iterated algebraic K-theory and T-duality for sphere bundles****Abstract:**

This talk will focus on two main topics. The first is the construction of a periodic form \(A_n\) of the (n-1)-fold iterated algebraic K-theory of ku, the connective K-theory spectrum. Following Baas-Dundas-Rognes-Richter, the spectrum \(A_n\) may be regarded as a non-connective form of a K-theory of higher categorical analogues of vector bundles. We develop some basic machinery for studying this cohomology theory, in particular a form of the Chern character. In the second part of the talk, we will prove an analogue of Bouwknegt-Evslin-Mathai's topological T-duality for higher dimensional sphere bundles using \(A_n\).

**Schedule:**

All talks are in Swift 107.

Saturday

- 9:00-10:00 Registration and coffee
- 10:00-11:00 Soren Galatius
- 11:30-12:30 Eric Peterson
- 12:30-3:00 Lunch
- 3:00-4:00 Inna Zakharevich
- 4:30-5:30 Kirsten Wickelgren
- 6:30 Reception (Harris 108)

Sunday

- 9:30-10:00 Coffee
- 10:00-11:00 Carolyn Yarnall
- 11:30-12:30 Craig Westerland

**How:** The deadline for applying for funding has passed. You can still register here.

**Hotel:** We have a special rate at the Homestead. Mention the Midwest Topology Seminar when you book your rooms, or use the promo code "topo16" if you register online. The rooms are not being held, so reserve them as soon as you can!

**Contact:**