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Michael Hopkins (mathematician)
Michael Jerome Hopkins (born April 18, 1958) is an American mathematician known for work in algebraic topology. Life He received his PhD from Northwestern University in 1984 under the direction of Mark Mahowald, with thesis ''Stable Decompositions of Certain Loop Spaces''. Also in 1984 he also received his D.Phil. from the University of Oxford under the supervision of Ioan James. He has been professor of mathematics at Harvard University since 2005, after fifteen years at the Massachusetts Institute of Technology, a few years of teaching at Princeton University, a one-year position with the University of Chicago, and a visiting lecturer position at Lehigh University. Work Hopkins' work concentrates on algebraic topology, especially stable homotopy theory. It can roughly be divided into four parts (while the list of topics below is by no means exhaustive): The Ravenel conjectures The Ravenel conjectures very roughly say: complex cobordism (and its variants) see more in the stabl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Harvard University
Harvard University is a Private university, private Ivy League research university in Cambridge, Massachusetts, United States. Founded in 1636 and named for its first benefactor, the History of the Puritans in North America, Puritan clergyman John Harvard (clergyman), John Harvard, it is the oldest institution of higher learning in the United States. Its influence, wealth, and rankings have made it one of the most prestigious universities in the world. Harvard was founded and authorized by the Massachusetts General Court, the governing legislature of Colonial history of the United States, colonial-era Massachusetts Bay Colony. While never formally affiliated with any Religious denomination, denomination, Harvard trained Congregationalism in the United States, Congregational clergy until its curriculum and student body were gradually secularized in the 18th century. By the 19th century, Harvard emerged as the most prominent academic and cultural institution among the Boston B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
University Of Oxford
The University of Oxford is a collegiate university, collegiate research university in Oxford, England. There is evidence of teaching as early as 1096, making it the oldest university in the English-speaking world and the List of oldest universities in continuous operation, second-oldest continuously operating university globally. It expanded rapidly from 1167, when Henry II of England, Henry II prohibited English students from attending the University of Paris. When disputes erupted between students and the Oxford townspeople, some Oxford academics fled northeast to Cambridge, where they established the University of Cambridge in 1209. The two English Ancient university, ancient universities share many common features and are jointly referred to as ''Oxbridge''. The University of Oxford comprises 43 constituent colleges, consisting of 36 Colleges of the University of Oxford, semi-autonomous colleges, four permanent private halls and three societies (colleges that are depar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Jeffrey H
Jeffrey may refer to: * Jeffrey (name), including a list of people with the name *Jeffrey's, Newfoundland and Labrador, Canada *Jeffrey City, Wyoming, United States *Jeffrey Street, Sydney, Australia *Jeffreys Bay, Western Cape, South Africa Art and entertainment * ''Jeffrey'' (play), a 1992 off-Broadway play by Paul Rudnick * ''Jeffrey'' (1995 film), a 1995 film by Paul Rudnick, based on Rudnick's play of the same name * ''Jeffrey'' (2016 film), a 2016 Dominican Republic documentary film * Jeffrey's sketch, a sketch on American TV show ''Saturday Night Live'' *'' Nurse Jeffrey'', a spin-off miniseries from the American medical drama series ''House, MD'' People with the surname * Alexander Jeffrey (1806–1874), Scottish solicitor and historian * Carol Jeffrey (1898–1998), English psychotherapist, writer *Charles Jeffrey (footballer) (died 1915), Scottish footballer *E. C. Jeffrey (1866–1952), Canadian-American botanist *Grant Jeffrey (1948–2012), Canadian writer * Hes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
CW Complex
In mathematics, and specifically in topology, a CW complex (also cellular complex or cell complex) is a topological space that is built by gluing together topological balls (so-called ''cells'') of different dimensions in specific ways. It generalizes both manifolds and simplicial complexes and has particular significance for algebraic topology. It was initially introduced by J. H. C. Whitehead to meet the needs of homotopy theory. (open access) CW complexes have better categorical properties than simplicial complexes, but still retain a combinatorial nature that allows for computation (often with a much smaller complex). The C in CW stands for "closure-finite", and the W for "weak" topology. Definition CW complex A CW complex is constructed by taking the union of a sequence of topological spaces \emptyset = X_ \subset X_0 \subset X_1 \subset \cdots such that each X_k is obtained from X_ by gluing copies of k-cells (e^k_\alpha)_\alpha, each homeomorphic to the open k- bal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Suspension (topology)
In topology, a branch of mathematics, the suspension of a topological space ''X'' is intuitively obtained by stretching ''X'' into a cylinder and then collapsing both end faces to points. One views ''X'' as "suspended" between these end points. The suspension of ''X'' is denoted by ''SX'' or susp(''X''). There is a variant of the suspension for a pointed space, which is called the reduced suspension and denoted by Σ''X''. The "usual" suspension ''SX'' is sometimes called the unreduced suspension, unbased suspension, or free suspension of ''X'', to distinguish it from Σ''X.'' Free suspension The (free) suspension SX of a topological space X can be defined in several ways. 1. SX is the quotient space (X \times ,1/(X\times \)\big/ ( X\times \). In other words, it can be constructed as follows: * Construct the cylinder X \times ,1/math>. * Consider the entire set X\times \ as a single point ("glue" all its points together). * Consider the entire set X\times \ as a single p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Nilpotence Conjecture
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, the kernel of the map \pi_\ast R \to \mathrm_\ast(R) consists of nilpotent elements. It was conjectured by and proved by . Nishida's theorem showed that elements of positive degree of the homotopy groups of spheres are nilpotent. This is a special case of the nilpotence theorem. See also * Ravenel's conjectures 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 ... References * * . Open online version.* Further reading Connection of ''X(n)'' spectra to formal group laws Homotopy theory Theorems in algebraic topology {{Topology ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Stable Homotopy Category
A stable is a building in which working animals are kept, especially horses or oxen. The building is usually divided into stalls, and may include storage for equipment and feed. Styles There are many different types of stables in use today; the American-style stable called a barn, for instance, is a large barn with a door at each end and individual stalls inside or free-standing stables with top and bottom-opening doors. The term "stable" is additionally utilised to denote a business or a collection of animals under the care of a single owner, irrespective of their housing or whereabouts. A building with tie stalls is also known as stanchion or stall barn, where animals are tethered by the head or neck to their stall. It is mostly used in the dairy cow industry, but traditionally horses were also tied up. The exterior design of a stable can vary widely based on climate, building materials, historical period and cultural styles of architecture. A wide range of building m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Complex Cobordism
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. The generalized homology and cohomology complex cobordism theories were introduced by using the Thom spectrum. Spectrum of complex cobordism The complex bordism MU^*(X) of a space X is roughly the group of bordism classes of manifolds over X with a complex linear structure on the stable normal bundle. Complex bordism is a generalized homology theory, corresponding to a spectrum MU that can be described explicitly in terms of Thom spaces as follows. The space MU(n) is the Thom space of the universal n-plane bundle over the classifying space BU(n) of the unitary group U(n). The natural i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Ravenel Conjectures
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. The first of the seven conjectures, then the ''nilpotence conjecture'', was proved in 1988 and is now known as the nilpotence theorem. The telescope conjecture, which was fourth on the original list, remains of substantial interest because of its connection with the convergence of an Adams–Novikov spectral sequence. While opinion has been generally against the truth of the original statement, investigations of associated phenomena (for a triangulated category in general) have become a research area in its own right ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Stable Homotopy Theory
In mathematics, stable homotopy theory is the part of homotopy theory (and thus algebraic topology) 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 X, the homotopy groups \pi_(\Sigma^n X) stabilize for n sufficiently large. In particular, the homotopy groups of spheres \pi_(S^n) stabilize for n\ge k + 2. For example, :\langle \text_\rangle = \Z = \pi_1(S^1)\cong \pi_2(S^2)\cong \pi_3(S^3)\cong\cdots :\langle \eta \rangle = \Z = \pi_3(S^2)\to \pi_4(S^3)\cong \pi_5(S^4)\cong\cdots In the two examples above all the maps between homotopy groups are applications of the suspension functor. The first example is a standard corollary of the Hurewicz theorem, that \pi_n(S^n)\cong \Z. In the second example the Hopf map, \eta, is mapped to its suspension \Sigma\eta, which generates \pi_4(S^3)\cong \Z/2. One of the mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Lehigh University
Lehigh University (LU), in Bethlehem, Pennsylvania, United States, is a private university, private research university. The university was established in 1865 by businessman Asa Packer. Lehigh University's undergraduate programs have been mixed-sex education, coeducational since the 1971–72 academic year. , the university had 5,911 undergraduate students and 1,781 graduate students. Lehigh has five colleges: the P.C. Rossin College of Engineering and Applied Science, the College of Arts and Sciences, the College of Business, the College of Education, and the College of Health. The College of Arts and Sciences is the largest, with 38% of the university's students. The university offers Interdisciplinarity, Interdisciplinary Studies, Bachelor of Arts, Bachelor of Science, Master of Arts, Master of Science, Master of Business Administration, Master of Engineering, Master of Education, Doctor of Philosophy, and Doctor of Education degrees. The university is Carnegie Classific ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
