.
The seminar is concerned with trace maps out of algebraic K-theory.
The ultimate goal of the seminar is to understand a proof of the
following result of Bökstedt, Hsiang and Madsen
which discovers a big portion inside
.
Theorem ([BHM])
Let 
be a group whose homology in each degree is a finitely generated abelian
group. Then the rationalized assembly map in algebraic K-theory
Here the left hand side admits a more explicit description in terms of group homology with rational coefficients. The result fits into the general philosophy of assembly maps. Among similar statements it has an astonishingly weak assumption on the group.
Its proof uses the cyclotomic trace map from K-theory to the so called topological cyclic homology TC, which has proven to be a powerful tool also in other K-theory computations. Topological cyclic homology will hence be the main object of study in our seminar. It is at least philosophically related to Connes' cyclic homology. But its construction works not on the level of homological algebra but on the level of spaces (or spectra). The definition involves some unexpected structure which is not visible anymore after passage to homological algebra.
Recently the definition of TC and the cyclotomic trace has been extended to schemes. Therefore the seminar could be appealing to algebraic geometers with an interest in algebraic K-theory.
To get a first impression one should read the introduction of Ib Madsen's survey article [M]. Moreover we highly recommend the notes [R1] of John Rognes, where the construction of topological cyclic homology and the cyclotomic trace is sketched on 14 pages without hiding any technical difficulties.