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Rational Homotopy Theory Ii. Morse Theory Smooth And Discrete. Imbeddings and Immersions Translations of Mathematical Monographs. Quantum Cohomology : Lectures Given at the C. Summer School He View Wishlist. Our Awards Booktopia's Charities. He works on the theoretical foundations of the subject, and also on the applied and computational aspects.
Recently, he has been especially interested in the design of practical software tools for TDA and in applications to biology. Michael completed his Ph. The main motivations come from studying and trying to classify manifolds and their symmetries.
For manifolds of sufficiently high dimension, such questions can be attacked using algebraic invariants, like for example the algebraic K-theory groups of group algebras. The structure of these groups is predicted by the far-reaching Farrell--Jones conjecture in algebraic K-theory.
My own work focuses on this conjecture and uses methods from stable and equivariant homotopy theory and geometric group theory. What kind of requirements I need? I have studied several books in this branch, Hatcher for first, Sirinivas, Atiyah, and many lectures and some papers. I have good background in Homology and Algebraic geometry too. I think that doing algebraic K-theory properly certainly requires a good background on stable homotopy theory, that is to say the homotopy theory of spectra. Unfortunately there are not many textbooks in the subject.
Let me mention two of them:. Stable homotopy and generalized cohomology by J. Frank Adams is an old classic. Its treatment of some topics is far from modern though, and in particular the development of localizations is flawed and should be complemented by reading Bousfield's original papers.
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He also doesn't talk about commutative i. Symmetric spectra by S.
A much more modern approach, covering a variety of topics you're going to need. Just don't get too hung on the model categorical subtleties of the model he chose I'm thinking mainly semistability here , 'cause they won't come up in practice.
Categories and cohomology theories by G. This is a short paper, but if you want to learn about group completion and its relation to spectra, reading this is probably the quickest thing you can do.
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Despite its age it is surprising modern in its approach. Moreover it is a pleasure to read. When you have a sufficiently good background in homotopy theory that the words spectra and group completion don't make you scream in terror, it's time to start with actual algebraic K-theory. Here are some useful starting points. The K-book by Charles Weibel has a lot of classical material and it is a useful bridge from the low dimensional, hand-defined groups to the more modern algebraic K-theory spectrum.
grupoavigase.com/includes/251/743-citas-en-apa.php It is a bit long though, and I'd treat it more as a reference than a book to be read from top to bottom. One of the fundamental papers on algebraic K-theory. It also has a decent introduction to Waldhausen's S-construction and it is worth reading in full.