Mark Edward Mahowald (December 1, 1931 – July 20, 2013) was an American

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

known for work in

algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

Life

Mahowald was born in

Albany, Minnesota Albany is a city in Stearns County, Minnesota, United States. The population was 2,561 at the 2010 census. It is part of the St. Cloud Metropolitan Statistical Area. History File:Albany Minn Oct 8 1911.jpg, A real photo postcard captured ...

in 1931. He received his Ph.D. from the

University of Minnesota The University of Minnesota Twin Cities (historically known as University of Minnesota) is a public university, public Land-grant university, land-grant research university in the Minneapolis–Saint Paul, Twin Cities of Minneapolis and Saint ...

in 1955 under the direction of

Bernard Russell Gelbaum Bernard Russell Gelbaum (died March 22, 2005, Laguna Beach, California) was a mathematician and academic administrator having served as a professor at the University of Minnesota, University of California, Irvine (where he was the first chair of ...

with a thesis on ''Measure in Groups''. In the sixties, he became professor at

Syracuse University Syracuse University (informally 'Cuse or SU) is a Private university, private research university in Syracuse, New York, United States. It was established in 1870 with roots in the Methodist Episcopal Church but has been nonsectarian since 1920 ...

and around 1963 he went to

Northwestern University Northwestern University (NU) is a Private university, private research university in Evanston, Illinois, United States. Established in 1851 to serve the historic Northwest Territory, it is the oldest University charter, chartered university in ...

Evanston, Illinois Evanston is a city in Cook County, Illinois, United States, situated on the North Shore (Chicago), North Shore along Lake Michigan. A suburb of Chicago, Evanston is north of Chicago Loop, downtown Chicago, bordered by Chicago to the south, Skok ...

Work

Much of Mahowald's most important works concerns the

homotopy groups of spheres 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 ...

, especially using the

Adams spectral sequence In mathematics, the Adams spectral sequence is a spectral sequence introduced by 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 c ...

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

h_1h_j

survive the Adams spectral sequence for

j\geq 3

. 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 In mathematics, the Kervaire invariant is an invariant of a framed (4k+2)-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 sp ...

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 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 , extending a construction of . Definition Whitehead's original homomorphism is de ...

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 In stable homotopy theory, a branch of mathematics, Morava K-theory is one of a collection of cohomology theories introduced in algebraic topology by Jack Morava in unpublished preprints in the early 1970s. For every prime number ''p'' (which is s ...

K(2)

. 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 In algebraic topology, the nilpotence theorem gives a condition for an element in the homotopy groups of a ring spectrum to be nilpotent, in terms of the complex cobordism spectrum \mathrm. More precisely, it states that for any ring spectrum R, th ...

by Ethan Devinatz, Michael J. Hopkins, and Jeffrey Smith.

Awards and honors

In 1998 he was an Invited Speaker of the

International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the IMU Abacus Medal (known before ...

in Berlin. In 2012 he became a fellow of the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

.List of Fellows of the American Mathematical Society
retrieved 2013-02-02.

Selected publications

*Mark E. Mahowald and Martin C. Tangora, ''Some differentials in the Adams spectral sequence'',

Topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

6 (1967) 349–369. *Michael G. Barratt, Mark E. Mahowald, and Martin C. Tangora, ''Some differentials in the Adams spectral sequence II'', Topology 9 (1970) 309–316. *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. * Mark E. Mahowald,'' A new infinite family in

_2\pi_*^S

'', Topology 16 (1977) 249–256. *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 The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as t ...

162 (2005), 777–822. *Prasit Bhattacharya, Philip Egger and Mark E. Mahowald, ''On the periodic v2-self-map of A1'', Algebraic and Geometric Topology 17 (2017) 657–692. doi:10.2140/agt.2017.17.657

References

External links

*
Homepage at Northwestern University
* (294 kB) 1931 births 20th-century American mathematicians 21st-century American mathematicians University of Minnesota alumni Northwestern University faculty Fellows of the American Mathematical Society 2013 deaths People from Albany, Minnesota Mathematicians from Minnesota American topologists Syracuse University faculty {{Mathematician-stub