Abstract: The signature is a classical invariant of manifolds, which is a shadow of their intersection form. More refined topological indices of manifolds can be constructed by remembering the full intersection form.
In this talk, I will explain how the signature has an enhancement to a multiplicative map from geometric cobordism spectra of manifolds to algebraic cobordism spectra of Hermitian forms (known as Grothendieck-Witt or L-theory). This crucially depends on the construction of the signature on the level of cobordism categories. This approach generalises and unifies various existing constructions in the literature. I will conclude by explaining some interesting applications to geometric topology, such as a simple proof of the Weiss-Williams index theorem on characteristic classes of manifold bundles. Based on work in progress with Andrea Bianchi, Fabian Hebestreit, Kaif Hilman, Christian Kremer, Markus Land, Thomas Nikolaus and Wolfgang Steimle.

Vortrag 2: Logarithmische deRham-Witt-Garben I – Nils Witt (Universität Heidelberg)

Talk 2: Prismatic cohomology relative to δ-rings - Marvin Schneider/Rustam Steingart (Universität Heidelberg)

Vortrag 3: Logarithmische deRham-Witt-Garben II - tba

Talk 3: The prismatic crystal - Marvin Schneider / Rustam Steingart (Universität Heidelberg)

Talk 1: Leonie Nienhaus (Goethe Universität Frankfurt): Overview of seminar
Talk 2: tba

Vortrag 4: Logarithmische deRham-Witt-Garben III - tba

Talk 4: The Nygaard filtration, syntomic cohomology and the prismatic package - Marvin Schneider / Rustam Steingart (Universität Heidelberg)