Talk 1 (18.10.2017 – Gesina Schwalbe): Overview of some classical results. Recall the classical theorems of Freudenthal and Hurewicz and some elementary calculations of homotopy groups of spheres. Recall the definition of the stable homotopy groups of spheres and explain the ring structure. Show that the Hopf maps are stably essential, i.e. not stably null-homotopic, using Steenrod operations. ...

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This paper studies Kontsevich’s characteristic classes of smooth bundles with fiber a ‘singularly’ framed odd-dimensional homology sphere which are defined through his graph complex and configuration space integral. We will give a systematic construction of smooth bundles parametrized by trivalent graphs and will show that our smooth bundles are nontrivially detected by Kontsevich’s characteris...

Large scale phenomena in homotopy theory (Jordan Hall 124) Paul Goerss (Northwestern University) In the late 1940s it wasn’t clear which was the harder problem: the cohomology of Eilenberg-MacLane spaces or the stable homotopy groups of spheres. After the Cartan-Serre work on Eilenberg-MacLane spaces in the early ’50s, people then optimistically set about computing the homotopy groups of sphere...

Let N and P be smooth closed manifolds of dimensions n and p respectively. Given a Thom-Boardman symbol I, a smooth map f : N → P is called an Ω -regular map if and only if the Thom-Boardman symbol of each singular point of f is not greater than I in the lexicographic order. We will represent the group of all cobordism classes of Ω -regular maps of n-dimensional closed manifolds into P in terms...

For groups of prime order, equivariant stable maps between equivariant representation spheres are investigated using the Borel cohomology Adams spectral sequence. Features of the equivariant stable homotopy category, such as stability and duality, are shown to lift to the category of modules over the associated Steenrod algebra. The dependence on the dimension functions of the representations i...