Sergei Petrovich Novikov is a Soviet and Russian mathematician, noted for work in both algebraic topology and soliton theory. In 1970, he won the Fields Medal.
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Morava K-theory is one of a collection of cohomology theories introduced in algebraic topology by Jack Morava in unpublished preprints in the early 1970s.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 suppressed in the notation), it consists of theories K(n) for each nonnegative integer n, each a ring spectrum in the sense of homotopy theory. Johnson & Wilson (1975) published the first account of the theories.
In mathematics, complex cobordism is a generalized cohomology theory related to cobordism of manifolds. Its spectrum is denoted by MU. It is an exceptionally powerful cohomology theory, but can be quite hard to compute, so often instead of using it directly one uses some slightly weaker theories derived from it, such as Brown–Peterson cohomology or Morava K-theory, that are easier to compute.
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 was a British mathematician, one of the major contributors to homotopy theory.
Jeffrey Henderson Smith is a former professor of mathematics at Purdue University in Lafayette, Indiana. He received his Ph.D. from the Massachusetts Institute of Technology in 1981, under the supervision of Daniel Kan, and was promoted to full professor at Purdue in 1999. His primary research interest is algebraic topology; his best-cited work consists of two papers in the Annals of Mathematics on "nilpotence and stable homotopy".
In mathematics, stable homotopy theory is the 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 Adams spectral sequence is a spectral sequence introduced by J. Frank Adams 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.
Jack Johnson Morava is an American homotopy theorist at Johns Hopkins University.
In mathematics, Brown–Peterson cohomology is a generalized cohomology theory introduced by Edgar H. Brown and Franklin P. Peterson, 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 mathematics, the chromatic spectral sequence is a spectral sequence, introduced by Ravenel (1978), used for calculating the initial term of the Adams spectral sequence for Brown–Peterson cohomology, which is in turn used for calculating the stable homotopy groups of spheres.
In mathematics, the May spectral sequence is a spectral sequence, introduced by J. Peter May. It is used for calculating the initial term of the Adams spectral sequence, which is in turn used for calculating the stable homotopy groups of spheres. The May spectral sequence is described in detail in.
Michael Jerome Hopkins is an American mathematician known for work in algebraic topology.
Mark Edward Mahowald was an American mathematician known for work in algebraic topology.
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 . More precisely, it states that for any ring spectrum , the kernel of the map consists of nilpotent elements. It was conjectured by Douglas Ravenel and proved by Ethan S. Devinatz, Michael J. Hopkins, and Jeffrey H. Smith.
In mathematics, the Ravenel conjectures are a set of mathematical conjectures in the field of stable homotopy theory posed by Douglas Ravenel at the end of a paper published in 1984. It was earlier circulated in preprint. The problems involved have largely been resolved, with all but the "telescope conjecture" being proved in later papers by others. Ravenel's conjectures exerted influence on the field through the founding of the approach of chromatic homotopy theory.
Peter Steven Landweber is an American mathematician working 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.
