# Seminars in geometry/topology

## Spring 2016

## January 18, 14:15 - 15:15, room 734, Sentralbygg 2

**Eivind Dahl (UiO): The set of ring spectrum maps from \(\Sigma^\infty B^3\mathbb{Z}_+\) to \(K_2\)**

**Abstract:** We briefly introduce the Morava \(K\)-theories and computations of the \(K_2\) homology of \(B^3\mathbb{Z}=K(\mathbb{Z},3)\) following Ravenel-Wilson. We then compute the set of all ring spectrum maps from \(\Sigma^\infty B^3\mathbb{Z}_+\) to \(K_2\) as the set of group-like elements in \(K_2^0 B^3\mathbb{Z}\). This computation is a stepping-stone in a program to produce a ring spectrum map from \(K(\mathit{ku})\) to \(E_2\).

## January 25, 14:15 - 15:15, room 734, Sentralbygg 2

**Valentin Krasontovitsch (UiB): Iterated THH: the de Rham-Burnside-Witt complex**

**Abstract:** Computations in algebraic K-theory (which is an important invariant of (commutative) rings) lend themselves to so-called trace methods: One approximates K-theory with topological cyclic homology, or short TC, via the cyclotomic trace map trc: K \(\to\) TC, which (for some rings) is an equivalence after p-completion, on connective covers (here, p is a prime).
To determine TC, one must first determine its building blocks, topological Hochschild homology, or short THH. Now THH has a circle action, and its homotopy groups inherit a rich, rather rigid structure. This was used by Hesselholt and Madsen to for computations of THH.
We intend to do the same thing, but for iterated THH, based on a product of circles instead of one circle. This gives an even richer structure on the homotopy groups, which we investigate, culminating in the proof of the existence of an initial object in (loosely said) a certain algebraic category, which is inhabited in particular by the homotopy groups of iterated THH. We hope to compare the initial object to a topological example in order to decude some computational results.
In the talk, I would like to talk about these things in more detail and present some of my work.

## February 1, 14:15 - 15:15, room 734, Sentralbygg 2

**Alexander Schmeding (NTNU): Riemannian geometry on Lie group valued shape spaces**

**Abstract:** Shape analysis methods have in the past few years become very popular, both for theoretical exploration as well as from an application point of view. Originally developed for planar curves, these methods have been expanded to higher dimensional curves and surfaces. To solve problems arising in applications one endows the (infinite-dimensional) shape spaces with the structure of a Riemannian manifold and studies the associated geometry. We will explore one possible construction of such a geometry (using the so called "square root velocity transform") for shape spaces of curves with values in a Lie group. The main idea of our approach is to exploit the additional geometry imposed on the shape space by the Lie group geometry.

This research was motivated by problems from computer animations. Namely, our aim was to find cyclic approximations of non-cyclic character animations and interpolate between existing animations to generate new ones. See E. Celledoni, M. Eslitzbichler, A. Schmeding: Shape Analysis on Lie Groups with Applications in Computer Animation, arXiv:1506.00783.

## Wednesday, April 6, 09:15 - 10:15, room 734, Sentralbygg 2

**Drew Heard (MPIM Bonn): Picard groups of higher real \(K\)-theory spectra**

**Abstract:** For any finite subgroup \(G\) of the Morava stabilizer group of height \(n\) at a prime \(p\), there is an associated ring spectrum \(EO = (E_n)^{hG}\) of homotopy fixed points of Morava \(E\)-theory \(E_n\) under its \(G\)-action. I will discuss joint work with Akhil Mathew and Vesna Stojansoka , in which we show that, when \(n=p-1\), \(Pic(EO)\) is always cyclic, so that every invertible \(EO\) module is a suspension of \(EO\).

## April 25, 14:15 - 15:15, room 734, Sentralbygg 2

**Dustin Tate Clausen (Copenhagen): Number theory and the topology of tori**

**Abstract:** In the vast array of topological spaces, tori are among the simplest and easiest to understand. Yet, they admit close connections to number theory, and sometimes number-theoretic phenomena can be given fairly transparent topological explanations in terms of tori. I will explain abstractly why this should happen, then give several examples.