Daniel C. Isaksen's Books

Stable Stems

By: Daniel C. Isaksen

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The author presents a detailed analysis of 2-complete stable homotopy groups, both in the classical context and in the motivic context over C. He uses the motivic May spectral sequence to compute the cohomology of the motivic Steenrod algebra over C through the 70-stem. He then uses the motivic Adams spectral sequence to obtain motivic stable homotopy groups through the 59-stem. He also describes the complete calculation to the 65-stem, but defers the proofs in this range to forthcoming publications. In addition to finding all Adams differentials, the author also resolves all hidden extensions by 2, η, and ν through the 59-stem, except for a few carefully enumerated exceptions that remain unknown. The analogous classical stable homotopy groups are easy consequences. The author also computes the motivic stable homotopy groups of the cofiber of the motivic element τ. This computation is essential for resolving hidden extensions in the Adams spectral sequence. He shows that the homotopy groups of the cofiber of τ are the same as the E2-page of the classical Adams-Novikov spectral sequence. This allows him to compute the classical Adams-Novikov spectral sequence, including differentials and hidden extensions, in a larger range than was previously known.

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Page Count

174

Publisher

American Mathematical Soc.

Published Date

ISBN 10

1470437880

ISBN 13

9781470437886

Homotopy Theory: Tools and Applications

By: Daniel G. Davis

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Homotopy Theory: Tools and Applications

By: Daniel G. Davis

This volume contains the proceedings of the conference Homotopy Theory: Tools and Applications, in honor of Paul Goerss's 60th birthday, held from July 17–21, 2017, at the University of Illinois at Urbana-Champaign, Urbana, IL. The articles cover a variety of topics spanning the current research frontier of homotopy theory. This includes articles concerning both computations and the formal theory of chromatic homotopy, different aspects of equivariant homotopy theory and K-theory, as well as articles concerned with structured ring spectra, cyclotomic spectra associated to perfectoid fields, and the theory of higher homotopy operations.

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282

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American Mathematical Soc.

Published Date

ISBN 10

1470442442

ISBN 13

9781470442446

An Elementary Recursive Bound for Effective Positivstellensatz and Hilbert’s 17th Problem

By: Henri Lombardi,Daniel Perrucci,Marie-Françoise Roy

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An Elementary Recursive Bound for Effective Positivstellensatz and Hilbert’s 17th Problem

By: Henri Lombardi,Daniel Perrucci,Marie-Françoise Roy

The authors prove an elementary recursive bound on the degrees for Hilbert's 17th problem. More precisely they express a nonnegative polynomial as a sum of squares of rational functions and obtain as degree estimates for the numerators and denominators the following tower of five exponentials 222d4k where d is the number of variables of the input polynomial. The authors' method is based on the proof of an elementary recursive bound on the degrees for Stengle's Positivstellensatz. More precisely the authors give an algebraic certificate of the emptyness of the realization of a system of sign conditions and obtain as degree bounds for this certificate a tower of five exponentials, namely 22(2max{2,d}4k+s2kmax{2,d}16kbit(d)) where d is a bound on the degrees, s is the number of polynomials and k is the number of variables of the input polynomials.

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Page Count

138

Publisher

American Mathematical Soc.

Published Date

ISBN 10

147044108X

ISBN 13

9781470441081

Book's by Daniel C. Isaksen

Stable Stems

Stable Stems

Homotopy Theory: Tools and Applications

Homotopy Theory: Tools and Applications

An Elementary Recursive Bound for Effective Positivstellensatz and Hilbert’s 17th Problem

An Elementary Recursive Bound for Effective Positivstellensatz and Hilbert’s 17th Problem

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