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Topic: Bott periodicity

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	Raoul Bott; top explorer of the math behind surfaces and spaces - The Boston Globe
		Tony Bott was 12 when he first ventured up to his father's third-floor study in Newton and burst through the door without knocking, only to find he was the one in for a surprise.
		Among the mathematics awards Dr. Bott received were the National Medal of Science in 1987, the Wolf Prize in Israel in 2000, and two from the American Mathematical Society -- the Oswald Veblen Prize in 1964 and the Steele Prize for lifetime achievement in 1990.
		Bott then taught the University of Michigan and accepted a professorship at Harvard in 1959, where he remained until retiring to emeritus status in 1999.
	www.boston.com /news/globe/obituaries/articles/2006/01/04/raoul_bott_top_explorer_of_the_math_behind_surfaces_and_spaces (1010 words)

	Bott periodicity theorem - Wikipedia, the free encyclopedia
		In mathematics, the Bott periodicity theorem is a result from homotopy theory discovered by Raoul Bott during the latter part of the 1950s, which proved to be of foundational significance for much further research, in particular in K-theory of stable complex vector bundles, as well as the stable homotopy groups of spheres.
		Bott periodicity can be formulated in numerous ways, with the periodicity in question always appearing as a period 2 phenomenon, with respect to dimension, for the theory associated to the unitary group.
		The context of Bott periodicity is that the homotopy groups of spheres, which would be expected to play the basic part in algebraic topology by analogy with homology theory, have proved elusive (and the theory is complicated).
	en.wikipedia.org /wiki/Bott_periodicity_theorem (734 words)

	Raoul Bott - Wikipedia, the free encyclopedia
		Raoul Bott, FRS (born September 24, 1923, died December 20, 2005) was a mathematician known for numerous basic contributions to geometry in its broad sense.
		He studied the homotopy theory of Lie groups, using methods from Morse theory, leading to the Bott periodicity theorem (1956).
		Bott made important contributions towards the index theorem, especially in formulating related fixed-point theorems, in particular the so-called 'Woods Hole fixed-point theorem', a combination of the Riemann-Roch theorem and Lefschetz fixed-point theorem (it is named after Woods Hole, Massachusetts, the site of a conference at which collective discussion formulated it [1]).
	en.wikipedia.org /wiki/Raoul_Bott (441 words)

	Raoul Bott; theorems advanced topology, geometry \| The San Diego Union-Tribune
		Tony Bott was 12 when he first ventured up to his father's third-floor study in Newton, Mass., and burst through the door without knocking, only to find he was the one in for a surprise.
		Beginning in the 1960s, Dr. Bott's collaboration with Atiyah to find topological ways of investigating solutions to differential equations yielded the Atiyah-Bott fixed-point theorem, which in part shows that a mathematical map has a fixed point and also provides a means to count the number of fixed points on a given map.
		Among the mathematics awards Dr. Bott received were the National Medal of Science in 1987, the Wolf Prize in Israel in 2000 and two from the American Mathematical Society – the Oswald Veblen Prize in 1964 and the Steele Prize for lifetime achievement in 1990.
	www.signonsandiego.com /uniontrib/20060115/news_lz1j15bott.html (799 words)

	OP1 and Bott Periodicity
		This is but one of many related `period-8' phenomena that go by the name of Bott periodicity.
		It also means that to prove Bott periodicity for these homotopy groups:
		There is much more to say about this fact and how it relates to Bott periodicity for Clifford algebras, but alas, this would take us too far afield.
	math.ucr.edu /home/baez/octonions/node10.html (529 words)

	Vineyard Gazette - Obituaries
		Raoul Bott, a groundbreaking mathematician in the field of topology and geometry and longtime professor at Harvard University, died of cancer Dec. 20 at home in Carlsbad, Calif. He was 82.
		Raoul Bott was born in Budapest, Hungary, on Sept. 24, 1923.
		Bott is survived by his wife of 58 years, Phyllis; a son, Anthony, of Harwich; three daughters, Candace Bott of Cambridge, Jocelyn Scott of Rancho Sante Fe, Calif., and Renee Bott of Berkeley, Calif.; and nine grandchildren.
	www.mvgazette.com /features/obituaries/?name=raoul_bott&edition_date=2006/01/20 (486 words)

	Raoul Bott, an Innovator in Mathematics, Dies at 82 - New York Times
		Raoul Bott, a mathematician who made innovative contributions to differential geometry and topology, a field that examines the properties of spaces, died on Dec. 20 in Carlsbad, Calif. He was 82.
		Although he began his career studying engineering, Dr. Bott, who was born in Hungary, turned to mathematics as a graduate student in the 1940's and later defined a mathematician as "someone who likes to get at the root of things." He taught at Harvard from 1959 until 1999.
		Bott was also widely known for earlier work, when he developed what became known as the Bott periodicity theorem in 1959, the importance of which some mathematicians have compared to the discovery of the periodic table of the elements.
	www.nytimes.com /2006/01/08/national/08bott.html?ex=1294376400&en=3d26f04234002c60&ei=5088&partner=rssnyt&emc=rss (675 words)

	Not Even Wrong » Blog Archive » Raoul Bott 1923-2005
		Bott was one of the greatest mathematicians of the twentieth century; for some details about his life see the commemorative web-site set up by the Harvard math department.
		Bott was intimately involved with Clifford algebras and spinors from early on, and his extremely important paper with Atiyah and Shapiro shows how crucial these are for understanding K-theory.
		Bott was one of the leading mathematicians of his time, but he was also an inspirational teacher and a warm human being.
	www.math.columbia.edu /~woit/wordpress/?p=314 (1678 words)

	Bott biography
		Bott didn't fancy taking three more years to qualify in mathematics before starting a Master's degree in mathematics which is what he would have had to have done had he followed his initial choice to stay at McGill.
		Bott's career was greatly influenced by Weyl who was impressed with the results that Bott was producing and invited him to the Institute for Advanced Study in Princeton in 1949.
		Included is the famous Bott periodicity theorem (1956) and the Morse-Bott functions, an important generalization of Morse functions which Bott introduced in the course of this work.
	www-groups.dcs.st-and.ac.uk /~history/Biographies/Bott.html (2126 words)

	The Harvard Crimson :: News :: Math Professor Bott Dies at Age 82
		Bott was widely known as an outstanding figure in geometry and topology, contributing to many theories that became fundamental concepts in the field.
		Among these theories was the Bott periodicity theorem of 1959, a discovery which has been compared by mathematicians to the sciences’ discovery of the periodic table of elements.
		Bott is survived by his wife of 58 years, Phyllis, a son, Anthony, three daughters, Candace Bott, Jocelyn Scott, and Renee Bott, and nine grandchildren.
	www.thecrimson.com /printerfriendly.aspx?ref=510851 (600 words)

	Bott Wins Israel's Wolf Foundation Prize in Mathematics
		Bott will be honored with a reception at the offices of the Consulate General of Israel to New England in Boston on March 14.
		His first major contribution was the application of Morse theory to the topology of Lie groups, which led to the famous "periodicity theorems." He was a major contributor to the development of K-theory and also worked on Yang-Mills theory, moduli spaces of vector bundles, and elliptic genera.
		Bott is a member of the National Academy of Sciences and the London Mathematical Society.
	www.news.harvard.edu /gazette/2000/03.09/bott.html (461 words)

	The Harvard Crimson :: News :: Mathematics Professor Recounts Wartime Coming-of-Age
		Bott's most famous contribution to mathematics is the Bott Periodicity Theorem, which he discovered in 1956.
		His parents divorced soon after he was born, but Bott nonetheless led a childhood of affluence, since his new stepfather was a high-ranking manager in a sugar factory.
		Bott received his first rigorous education when he arrived in Canada for a year of high school after skipping two years and being elevated to his senior year for answering a trick mathematics question correctly.
	www.thecrimson.com /printerfriendly.aspx?ref=108082 (894 words)

	[No title]
		Bott's original proof of these beautiful results is based on the use of Morse t* *heory.
		Before proving the periodicity theorem, Bott had clearly demonstrated the power* * of Morse theory by using it to prove that there is no torsion in the integral homo* *logy of MG for any simply connected compact Lie group G [Bott56].
		Bott announced the periodicity theorem in [Bott57], and he gave two somewhat different proofs, both based on Morse theory, in [Bott58, Bott59a].
	www.math.purdue.edu /research/atopology/May/history.txt (14491 words)

	Bertie Bott Beans (Site not responding. Last check: 2007-10-09)
		Raoul Bott 1: '''Raoul Bott ''' (born September 24 1923) is a mathem 5: orked on the theory of electrical circuit s (Bott -Duffin theorem from 1949), then switched to 7: g methods from Morse theory, leading to the Bott periodicity theorem (1956).
		Bott periodicity theorem 1: homotopy theory which was discovered by Raoul Bott during the latter part of the 1950s, and proved 3: was still hard to compute with, in practice.
		What Bott periodicity offered was an insight into some high 5: ian in infinite dimensions), one formulation of Bott periodicity describes 11: Bott 's original proof used Morse theory ; subsequen
	www.vermontreview.com /edge/23247-bertiebottbeans.html (733 words)

	Clay Mathematics Institute
		Bott Periodicity in Topological, Algebraic and Hermitian K-Theory (Part I) Max Karoubi (University of Paris)
		In this first lecture about Bott periodicity, we summarize the history of the proof which leads naturally to topological K-theory of Banach algebras.
		We state the Lichtenbaum-Quillen conjecture which is the true analog of Bott periodicity in this context (through a "Galois descent").
	www.claymath.org /programs/outreach/academy/colloquium2005.php (1046 words)

	ANU - Mathematical Sciences Institute (MSI) - Seminars
		Bott Periodicity is a central result of K-theory, both for topological spaces and C*-algebras.
		It was originally used by Bott in order to study the homotopy groups of certain classical Lie groups.
		Bott periodicity finds application in the Atiyah-Singer Index theorem, which is of great importance in both commutative and non-commutative geometry.
	wwwmaths.anu.edu.au /calendar/04.08.21.cal.html?p=1 (675 words)

	[No title]
		Of course the heterotic string is a schizoid hybrid where the left-moving wiggles are described by 10d superstring theory and the right-moving wiggles are described by 26d bosonic string theory, but the 10d part is still very much the same as all the rest.
		Well, the easy part of Bott periodicity, which every particle physicist should know, is how Clifford algebras and spinors work in various dimensions - and how the pattern repeats with period 8.
		As for the K-theoretic aspects of Bott periodicity, you shouldn't mess with those too much until you're more in tune with the Tao of Mathematics.
	www.math.niu.edu /~rusin/known-math/00_incoming/MF (2652 words)

	Re: god-given bundles, Bott periodicity
		Is there a natural "periodicity theorem" in category theory...?" John Baez replied: "...
		The closest thing that comes to mind is Bott periodicity, which...
		Since C is a topological category it naturally becomes an omega-category, which we may also call C. Given an omega-category with a specified object x we can form a new omega-category hom(x,x), which again has a specified object, namely 1_x.
	www.lns.cornell.edu /spr/2001-12/msg0037494.html (374 words)

	The Reference Frame: The father of Bott periodicity died
		Via David G. Raoul Bott - a Harvard mathematician who was fighting against cancer in San Diego and who discovered, among other things, the Bott periodicity theorem in the late 1950s - died the night of December 19-20, 2005.
		His nanny was English which helped young Bott to learn authentic English.
		To summarize this paragraph: one should not be surprised that Bott hated foreign languages.
	motls.blogspot.com /2005/12/father-of-bott-periodicity-died.html (358 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		The method of proof involves the study of an appropriate generalization of Morse-Kirwan theory to infinite dimensions, and the construction of equivariantly perfect Morse-Kirwan functions on Hamiltonian LG-spaces.
		In the case of the smallest coadjoint orbit of the loop group, our function coincides with the energy function whose perfection plays a key role in the classical proof of Bott periodicity.
		In the case of Hamiltonian LG-spaces arising from Yang-Mills theory, the function we study is closely related to the Yang-Mills functional whose Morse-theoretic properties were first studied by Atiyah and Bott.
	www.math.technion.ac.il /~techm/20020103160020020103wei (224 words)

	[No title]
		Finally we describe a necessary conditions for the holomorphic K - theory of * a smooth variety to be Bott* periodic (i.e Khol(X) ~=Khol(X)[1_b]) in terms of the Hodge * *filtration of its cohomology.
		Stability of rational maps and Bott periodic holomorphic K - theory In this section we study the space of rational maps in the morphism spaces us* *ed to define holomorphic K - theory.
		We end by using the above results to give a necessary co* ndition for the holomorphic K - theory to be Bott* periodic, and use it to describe examples* * where peri- odicity fails, and therefore provide examples that have distinct holomorphic an* *d topological K - theories.
	www.math.purdue.edu /research/atopology/CohenR-Lima-Filho/holo-k-th.txt (9457 words)

	Not Even Wrong » Blog Archive » New This Week’s Finds
		John briefly mentions a relation of all this to Bott periodicity in topology, using a very abstract homotopy construction involving spectra.
		For the relation of Clifford algebras and K-theory, the standard refererence is the 1964 paper “Clifford Modules” by Atiyah, Bott and Shapiro published in the journal “Topology”.
		The crucial fact they describe is how the Thom isomorphism in K-theory (which is essentially the same fact as Bott periodicity) is related to the structure of Clifford modules.
	www.math.columbia.edu /~woit/wordpress/?p=173 (525 words)

	Breaking News: Mathematic Innovator Raoul Bott Dies - The Post Chronicle
		CARLSBAD, Calif. - Jan. 9, 2006 (UPI) -- Raoul Bott, a mathematician who made innovative contributions to differential geometry and topology, has died at the age of 82.
		The collaboration yielded the Atiyah-Bott fixed-point theorem, which in part shows that a mathematical map has a fixed point and also provides a means to count the number of fixed points on a given map, the newspaper said.
		Bott was also widely known for the Bott periodicity theorem in 1959, the importance of which some mathematicians have compared to the discovery of the periodic table of the elements.
	www.postchronicle.com /news/breakingnews/printer_2123123.shtml (215 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		Title: The Bott periodicity spaces and their homology operations.
		Abstract: Bott discovered that the iterated loop spaces of the stable orthogonal groups obey an eightfold periodicity.
		These Bott periodicity spaces could then be used to define a multiplicative generalized cohomology theory whose grade zero part is KO(X), the ring of virtual real vector bundles on X.
	www.math.rochester.edu /research/topology/bisson.html (112 words)

	University at Albany Mathematics Information Service (Site not responding. Last check: 2007-10-09)
		But this is what Bott periodicity does, somehow: it wraps things around so the most complicated thing is also the least complicated.
		Bott periodicity for O(infinity) was first proved by Raoul Bott in 1959.
		Bott is a wonderful explainer of mathematics and one of the main driving forces behind applications of topology to physics, and a lot of his papers have now been collected in book form:
	math.albany.edu:8000 /math/ug/bott-per.html (3022 words)

	A topological proof of Bott periodicity theorem and a characterization of - Kono, Tokunaga (ResearchIndex) (Site not responding. Last check: 2007-10-09)
		A topological proof of Bott periodicity theorem and a characterization of - Kono, Tokunaga (ResearchIndex)
		A topological proof of Bott periodicity theorem and a characterization of (1994)
		A topological proof of Bott periodicity theorem and a characterization of BU.
	citeseer.ist.psu.edu /9691.html (410 words)

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