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Mark Mahowald | |
---|---|

Born | |

Died | July 20, 2013 81) Illinois, United States | (aged

Nationality | United States |

Alma mater | University of Minnesota |

Known for | Homotopy groups of spheres |

Scientific career | |

Fields | Mathematics |

Institutions | Syracuse University Northwestern University |

Doctoral advisor | Bernard Russell Gelbaum |

Doctoral students | Michael J. Hopkins |

**Mark Edward Mahowald** (December 1, 1931 – July 20, 2013) was an American mathematician known for work in algebraic topology.^{ [1] }

Mahowald was born in Albany, Minnesota in 1931.^{ [2] } He received his Ph.D. from the University of Minnesota in 1955 under the direction of Bernard Russell Gelbaum with a thesis on *Measure in Groups*. In the sixties, he became professor at Syracuse University and around 1963 he went to Northwestern University in Evanston, Illinois.

Much of Mahowald's most important works concerns the homotopy groups of spheres, especially using the Adams spectral sequence at the prime 2. He is known for constructing one of the first known infinite families of elements in the stable homotopy groups of spheres by showing that the classes survive the Adams spectral sequence for . In addition, he made extensive computations of the structure of the Adams spectral sequence and the 2-primary stable homotopy groups of spheres up to dimension 64 together with Michael Barratt, Martin Tangora, and Stanley Kochman. Using these computations, he could show that a manifold of Kervaire invariant 1 exists in dimension 62.

In addition, he contributed to the chromatic picture of the homotopy groups of spheres: His earlier work contains much on the image of the J-homomorphism and recent work together with Paul Goerss, Hans-Werner Henn, Nasko Karamanov, and Charles Rezk does computations in stable homotopy localized at the Morava K-theory .

Besides the work on the homotopy groups of spheres and related spaces, he did important work on Thom spectra. This work was used heavily in the proof of the nilpotence theorem by Ethan Devinatz, Michael J. Hopkins, and Jeffrey Smith.

In 1998 he was an Invited Speaker of the International Congress of Mathematicians in Berlin.^{ [3] } In 2012 he became a fellow of the American Mathematical Society.^{ [4] }

- Mark E. Mahowald and Martin C. Tangora,
*Some differentials in the Adams spectral sequence*, Topology 6 (1967) 349–369. doi : 10.1016/0040-9383(67)90023-7 MR 0214072 - Michael G. Barratt, Mark E. Mahowald, and Martin C. Tangora,
*Some differentials in the Adams spectral sequence II*, Topology 9 (1970) 309–316. doi : 10.1016/0040-9383(70)90055-8 MR 0266215 - Stanley O. Kochman and Mark E. Mahowald,
*On the computation of stable stems*in*The Čech centennial: a Conference on Homotopy Theory, June 22–26, 1993*, pp. 299–316. MR 1320997 - Mark E. Mahowald,
*A new infinite family in*, Topology 16 (1977) 249–256. doi : 10.1016/0040-9383(77)90005-2 - Paul Goerss, Hans-Werner Henn, Mark E. Mahowald, and Charles Rezk,
*A resolution of the K(2)-local sphere at the prime 3*, Annals of Mathematics 162 (2005), 777–822. JSTOR 20159929

In mathematics, the **Hurewicz theorem** is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the **Hurewicz homomorphism**. The theorem is named after Witold Hurewicz, and generalizes earlier results of Henri Poincaré.

In the mathematical field of algebraic topology, the **homotopy groups of spheres** describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting about their precise geometry. Unlike homology groups, which are also topological invariants, the homotopy groups are surprisingly complex and difficult to compute.

**John Frank Adams** FRS was a British mathematician, one of the major contributors to homotopy theory.

In mathematics, **stable homotopy theory** is that part of homotopy theory concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor. A founding result was the Freudenthal suspension theorem, which states that given any pointed space , the homotopy groups stabilize for sufficiently large. In particular, the homotopy groups of spheres stabilize for . For example,

In mathematics, the ** J-homomorphism** is a mapping from the homotopy groups of the special orthogonal groups to the homotopy groups of spheres. It was defined by George W. Whitehead (1942), extending a construction of Heinz Hopf (1935).

In mathematics, the **Adams spectral sequence** is a spectral sequence introduced by J. Frank Adams (1958) which computes the stable homotopy groups of topological spaces. Like all spectral sequences, it is a computational tool; it relates homology theory to what is now called stable homotopy theory. It is a reformulation using homological algebra, and an extension, of a technique called 'killing homotopy groups' applied by the French school of Henri Cartan and Jean-Pierre Serre.

In mathematics, **topological modular forms (tmf)** is the name of a spectrum that describes a generalized cohomology theory. In concrete terms, for any integer *n* there is a topological space , and these spaces are equipped with certain maps between them, so that for any topological space *X*, one obtains an abelian group structure on the set of homotopy classes of continuous maps from *X* to . One feature that distinguishes tmf is the fact that its coefficient ring, (point), is almost the same as the graded ring of holomorphic modular forms with integral cusp expansions. Indeed, these two rings become isomorphic after inverting the primes 2 and 3, but this inversion erases a lot of torsion information in the coefficient ring.

In mathematics, specifically algebraic topology, there is a distinguished class of spectra called Eilenberg–Maclane spectra for any Abelian group ^{pg 134}. Note, this construction can be generalized to commutative rings as well from its underlying Abelian group. These are an important class of spectra because they model ordinary integral cohomology and cohomology with coefficients in an abelian group. In addition, they are a lift of the homological structure in the derived category of abelian groups in the homotopy category of spectra. In addition, these spectra can be used to construct resolutions of spectra, called Adams resolutions, which are used in the construction of the Adams spectral sequence.

In mathematics, **Sullivan conjecture** or **Sullivan's conjecture on maps from classifying spaces** can refer to any of several results and conjectures prompted by homotopy theory work of Dennis Sullivan. A basic theme and motivation concerns the fixed point set in group actions of a finite group . The most elementary formulation, however, is in terms of the classifying space of such a group. Roughly speaking, it is difficult to map such a space continuously into a finite CW complex in a non-trivial manner. Such a version of the Sullivan conjecture was first proved by Haynes Miller. Specifically, in 1984, Miller proved that the function space, carrying the compact-open topology, of base point-preserving mappings from to is weakly contractible.

In mathematics, the **Toda bracket** is an operation on homotopy classes of maps, in particular on homotopy groups of spheres, named after Hiroshi Toda, who defined them and used them to compute homotopy groups of spheres in.

In mathematics, the **EHP spectral sequence** is a spectral sequence used for inductively calculating the homotopy groups of spheres localized at some prime *p*. It is described in more detail in Ravenel and Mahowald (2001). It is related to the EHP long exact sequence of Whitehead (1953); the name "EHP" comes from the fact that George W. Whitehead named 3 of the maps of his sequence "E", "H", and "P".

In mathematics, **Brown–Peterson cohomology** is a generalized cohomology theory introduced by Edgar H. Brown and Franklin P. Peterson (1966), depending on a choice of prime *p*. It is described in detail by Douglas Ravenel . Its representing spectrum is denoted by BP.

In mathematics, the **Kervaire invariant** is an invariant of a framed -dimensional manifold that measures whether the manifold could be surgically converted into a sphere. This invariant evaluates to 0 if the manifold can be converted to a sphere, and 1 otherwise. This invariant was named after Michel Kervaire who built on work of Cahit Arf.

In homotopy theory, a branch of algebraic topology, a **Postnikov system** is a way of decomposing a topological space's homotopy groups using an inverse system of topological spaces whose homotopy type at degree agrees with the truncated homotopy type of the original space . Postnikov systems were introduced by, and are named after, Mikhail Postnikov.

**Michael Jerome Hopkins** is an American mathematician known for work in algebraic topology.

In algebraic topology, the **doomsday conjecture** was a conjecture about Ext groups over the Steenrod algebra made by Joel Cohen, named by Michael Barratt, published by Milgram and disproved by Mahowald (1977). Minami (1995) stated a modified version called the **new doomsday conjecture**.

**Douglas Conner Ravenel** is an American mathematician known for work in algebraic topology.

In mathematics, **chromatic homotopy theory** is a subfield of stable homotopy theory that studies complex-oriented cohomology theories from the "chromatic" point of view, which is based on Quillen's work relating cohomology theories to formal groups. In this picture, theories are classified in terms of their "chromatic levels"; i.e., the heights of the formal groups that define the theories via the Landweber exact functor theorem. Typical theories it studies include: complex K-theory, elliptic cohomology, Morava K-theory and tmf.

In topology, a discipline within mathematics, the **Brown–Gitler spectrum** is a spectrum whose cohomology is a certain cyclic module over the Steenrod algebra.

**Charles Waldo Rezk** is an American mathematician, specializing in algebraic topology, category theory, and spectral algebraic geometry.

- ↑ "MARK EDWARD MAHOWALD Obituary: View MARK MAHOWALD's Obituary by Chicago Tribune". Legacy.com. 2013-07-20. Retrieved 2013-07-24.
- ↑ R.R. Bowker Company. Database Publishing Group (2009).
*American Men & Women of Science*.**5**. Thomson/Gale. ISBN 9781414433059 . Retrieved 2014-12-14. - ↑ Mahowald, Mark (1998). "Toward a global understanding of π
_{*}()".*Doc. Math. (Bielefeld) Extra Vol. ICM Berlin, 1998, vol. II*. pp. 465–472. - ↑ List of Fellows of the American Mathematical Society, retrieved 2013-02-02.

- Mark Mahowald at the Mathematics Genealogy Project
- Homepage at Northwestern University
- "Miller, Ravenel about Mahowald's work on the homotopy groups of spheres" (PDF). Archived from the original (PDF) on 2012-03-08. (294 kB)

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