Michiel Hazewinkel is a Dutch mathematician, Emeritus Professor of Mathematics at the Centre for Mathematics and Computer and the University of Amsterdam known for his 1978 book Formal groups and applications and as editor of the Encyclopedia of Mathematics. Born in Amsterdam to Jan Hazewinkel and Geertrude Hendrika Werner, Hazewinkel studied at the University of Amsterdam, he received his BA in Mathematics and Physics in 1963, his MA in Mathematics with a minor in Philosophy in 1965 and his PhD in 1969 under supervision of Frans Oort and Albert Menalda for the thesis "Maximal Abelian Extensions of Local Fields". After graduation Hazewinkel started his academic career as Assistant Professor at the University of Amsterdam in 1969. In 1970 he became Associate Professor at the Erasmus University Rotterdam, where in 1972 he was appointed Professor of Mathematics at the Econometric Institute. Here he was thesis advisor of Roelof Stroeker, M. van de Vel, Jo Ritzen, Gerard van der Hoek. From 1973 to 1975 he was Professor at the Universitaire Instelling Antwerpen, where Marcel van de Vel was his PhD student.
From 1982 to 1985 he was appointed part-time Professor Extraordinarius in Mathematics at the Erasmus Universiteit Rotterdam, part-time Head of the Department of Pure Mathematics at the Centre for Mathematics and Computer in Amsterdam. In 1985 he was appointed Professor Extraordinarius in Mathematics at the University of Utrecht, where he supervised the promotion of Frank Kouwenhoven, Huib-Jan Imbens, J. Scholma and F. Wainschtein. At the Centre for Mathematics and Computer CWI in Amsterdam in 1988 he became Professor of Mathematics and head of the Department of Algebra and Geometry until his retirement in 2008. Hazewinkel has been managing editor for journals as Nieuw Archief voor Wiskunde since 1977, he was managing editor for the book series Mathematics and Its Applications for Kluwer Academic Publishers in 1977. Hazewinkel was member of 15 professional societies in the field of Mathematics, participated in numerous administrative tasks in institutes, Program Committee, Steering Committee, Consortiums and Boards.
In 1994 Hazewinkel was elected member of the International Academy of Computer Sciences and Systems. Hazewinkel has authored and edited several books, numerous articles. Books, selection: 1970. Géométrie algébrique-généralités-groupes commutatifs. With Michel Demazure and Pierre Gabriel. Masson & Cie. 1976. On invariants, canonical forms and moduli for linear, finite dimensional, dynamical systems. With Rudolf E. Kalman. Springer Berlin Heidelberg. 1978. Formal groups and applications. Vol. 78. Elsevier. 1993. Encyclopaedia of Mathematics. Ed. Vol. 9. Springer. Articles, a selection: Hazewinkel, Michiel. "Moduli and canonical forms for linear dynamical systems II: The topological case". Mathematical Systems Theory. 10: 363–385. Doi:10.1007/BF01683285. Archived from the original on 12 December 2013. Hazewinkel, Michiel. "On Lie algebras and finite dimensional filtering". Stochastics. 7: 29–62. Doi:10.1080/17442508208833212. Archived from the original on 12 December 2013. Hazewinkel, M.. J.. "Nonexistence of finite-dimensional filters for conditional statistics of the cubic sensor problem".
Systems & Control Letters. 3: 331–340. Doi:10.1016/0167-691190074-9. Hazewinkel, Michiel. "The algebra of quasi-symmetric functions is free over the integers". Advances in Mathematics. 164: 283–300. Doi:10.1006/aima.2001.2017. Homepage
David Eisenbud is an American mathematician. He is a professor of mathematics at the University of California and was Director of the Mathematical Sciences Research Institute from 1997 to 2007, he was reappointed to this office in 2013, his term has been extended until July 31, 2022. Eisenbud is the son of mathematical physicist Leonard Eisenbud and collaborator of renowned physicist Eugene Wigner. Eisenbud received his Ph. D. in 1970 from the University of Chicago, where he was a student of Saunders Mac Lane and, James Christopher Robson. He taught at Brandeis University from 1970 to 1997, during which time he had visiting positions at Harvard University, Institut des Hautes Études Scientifiques, University of Bonn, Centre national de la recherche scientifique, he joined the staff at MSRI in 1997, took a position at Berkeley at the same time. From 2003 to 2005 Eisenbud was President of the American Mathematical Society. Eisenbud's mathematical interests include commutative and non-commutative algebra, algebraic geometry and computational methods in these fields.
He has written over 150 books with over 60 co-authors. Notable contributions include the theory of matrix factorizations for maximal Cohen–Macaulay modules over hypersurface rings, the Eisenbud–Goto conjecture on degrees of generators of syzygy modules, the Buchsbaum–Eisenbud criterion for exactness of a complex, he proposed the Eisenbud–Evans conjecture, settled by the Indian mathematician Neithalath Mohan Kumar. He has had 31 doctoral students, including Craig Huneke, Mircea Mustaţă, Irena Peeva, Gregory G. Smith. Eisenbud's hobbies are music, he has appeared in Brady Haran's Numberphile web series. Eisenbud was elected Fellow of the American Academy of Arts and Sciences in 2006, he was awarded the Leroy P. Steele Prize in 2010. In 2012 he became a fellow of the American Mathematical Society. Eisenbud, David. Three-dimensional link theory and invariants of plane curve singularities. Annals of Mathematical Studies. 110. Princeton, N. J.: Princeton U. Press. Vii+171. ISBN 978-0-691-08381-0. Eisenbud, David.
Commutative algebra with a view toward algebraic geometry. Graduate Texts in Mathematics. 150. New York: Springer-Verlag. Xvi+785. ISBN 0-387-94268-8. MR 1322960. Eisenbud, David; the geometry of schemes. Graduate Texts in Mathematics. 197. Berlin. X+294. ISBN 978-0-387-98638-8. MR 1730819. Eisenbud, David; the geometry of syzygies. A second course in commutative algebra and algebraic geometry. Graduate Texts in Mathematics. 229. New York: Springer-Verlag. Xvi+243. ISBN 0-387-22215-4. David Eisenbud, Joseph Harris. 3264 and All That: A Second Course in Algebraic Geometry. Cambridge University Press. ISBN 978-1107602724. Eisenbud, David. "Progress in the theory of complex algebraic curves". Bull. Amer. Math. Soc.. 21: 205–232. Doi:10.1090/s0273-0979-1989-15807-2. MR 1011763. Eisenbud, David. "Cayley-Bacharach theorems and conjectures". Bull. Amer. Math. Soc.. 33: 295–324. Doi:10.1090/s0273-0979-96-00666-0. MR 1376653. Eisenbud–Levine–Khimshiashvili signature formula O'Connor, John J.. David Eisenbud at the Mathematics Genealogy Project Eisenbud's biographical page at MSRI
Joe Harris (mathematician)
Joseph Daniel Harris, known nearly universally as Joe Harris, is a mathematician at Harvard University working in the field of algebraic geometry. He attended college at and received his Ph. D. from Harvard in 1978 under Phillip Griffiths. During the 1980s he was on the faculty of Brown University, moving to Harvard around 1988, he served as chair of the department at Harvard from 2002 to 2005. His work is characterized by its classical geometric flavor: he has claimed that nothing he thinks about could not have been imagined by the Italian geometers of the late 19th and early 20th centuries, that if he has had greater success than them, it is because he has access to better tools. Harris is well known for several of his books on algebraic geometry, notable for their informal presentations: Principles of Algebraic Geometry ISBN 978-0-471-05059-9, with Phillip Griffiths Geometry of Algebraic Curves, Vol. 1 ISBN 978-0-387-90997-4, with Enrico Arbarello, Maurizio Cornalba, Phillip Griffiths William Fulton, Joe Harris.
Representation Theory, A First Course, Graduate Texts in Mathematics, 129, New York: Springer-Verlag, doi:10.1007/978-1-4612-0979-9, ISBN 978-0-387-97495-8, MR 1153249, with William Fulton Joe Harris. Algebraic Geometry: A First Course, New York: Springer-Verlag, ISBN 978-0-387-97716-4 David Eisenbud, Joe Harris; the Geometry of Schemes, Graduate Texts in Mathematics, 197, New York: Springer-Verlag, ISBN 978-0-387-98638-8, MR 1730819, with David Eisenbud David Eisenbud, Joseph Harris. 3264 and All That: A Second Course in Algebraic Geometry. Cambridge University Press. ISBN 978-1107602724. Moduli of Curves ISBN 978-0-387-98438-4, with Ian Morrison. Harris has supervised 50 Ph. D. students, including Brendan Hassett, James McKernan, Rahul Pandharipande, Zvezdelina Stankova and Ravi Vakil
Hermann Cäsar Hannibal Schubert was a German mathematician. Schubert was one of the leading developers of enumerative geometry, which considers those parts of algebraic geometry that involve a finite number of solutions. In 1874, Schubert won a prize for solving a question posed by Zeuthen. Schubert calculus was named after him. Schubert tutored Adolf Hurwitz at the Realgymnasium Andreanum in Hildesheim and arranged for Hurwitz to study under Felix Klein at University. Schubert cycle or Schubert variety Schubert polynomial Schubert, Kleiman, Steven L. ed. Kalkül der abzählenden Geometrie, Reprint of the 1879 original, Berlin-New York: Springer-Verlag, ISBN 3-540-09233-1, MR 0555576 Werner Burau and Bodo Renschuch, "Ergänzungen zur Biographie von Hermann Schubert," Mitt. Math. Ges. Hamb. 13, pp. 63–65, ISSN 0340-4358. O'Connor, John J.. Works by Hermann Schubert at Project Gutenberg Works by or about Hermann Schubert at Internet Archive Hermann Schubert at the Mathematics Genealogy Project
Encyclopedia of Mathematics
The Encyclopedia of Mathematics is a large reference work in mathematics. It is available in book form and on CD-ROM; the 2002 version contains more than 8,000 entries covering most areas of mathematics at a graduate level, the presentation is technical in nature. The encyclopedia is edited by Michiel Hazewinkel and was published by Kluwer Academic Publishers until 2003, when Kluwer became part of Springer; the CD-ROM contains three-dimensional objects. The encyclopedia has been translated from the Soviet Matematicheskaya entsiklopediya edited by Ivan Matveevich Vinogradov and extended with comments and three supplements adding several thousand articles; until November 29, 2011, a static version of the encyclopedia could be browsed online free of charge online. This URL now redirects to the new wiki incarnation of the EOM. A new dynamic version of the encyclopedia is now available as a public wiki online; this new wiki is a collaboration between the European Mathematical Society. This new version of the encyclopedia includes the entire contents of the previous online version, but all entries can now be publicly updated to include the newest advancements in mathematics.
All entries will be monitored for content accuracy by members of an editorial board selected by the European Mathematical Society. Vinogradov, I. M. Matematicheskaya entsiklopediya, Sov. Entsiklopediya, 1977. Hazewinkel, M. Encyclopaedia of Mathematics, Kluwer, 1994. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 1, Kluwer, 1987. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 2, Kluwer, 1988. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 3, Kluwer, 1989. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 4, Kluwer, 1989. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 5, Kluwer, 1990. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 6, Kluwer, 1990. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 7, Kluwer, 1991. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 8, Kluwer, 1992. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 9, Kluwer, 1993. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 10, Kluwer, 1994. Hazewinkel, M. Encyclopaedia of Mathematics, Supplement I, Kluwer, 1997. Hazewinkel, M. Encyclopaedia of Mathematics, Supplement II, Kluwer, 2000.
Hazewinkel, M. Encyclopaedia of Mathematics, Supplement III, Kluwer, 2002. Hazewinkel, M. Encyclopaedia of Mathematics on CD-ROM, Kluwer, 1998. Encyclopedia of Mathematics, public wiki monitored by an editorial board under the management of the European Mathematical Society. List of online encyclopedias Official website Publications by M. Hazewinkel, at ResearchGate
In mathematics in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts transitively. The elements of G are called the symmetries of X. A special case of this is when the group G in question is the automorphism group of the space X – here "automorphism group" can mean isometry group, diffeomorphism group, or homeomorphism group. In this case, X is homogeneous if intuitively X looks locally the same at each point, either in the sense of isometry, diffeomorphism, or homeomorphism; some authors insist. Thus there is a group action of G on X which can be thought of as preserving some "geometric structure" on X, making X into a single G-orbit. Let X be a non-empty set and G a group. X is called a G-space if it is equipped with an action of G on X. Note that automatically G acts by automorphisms on the set. If X in addition belongs to some category the elements of G are assumed to act as automorphisms in the same category.
Thus, the maps on X effected by G are structure preserving. A homogeneous space is a G-space. Succinctly, if X is an object of the category C the structure of a G-space is a homomorphism: ρ: G → A u t C into the group of automorphisms of the object X in the category C; the pair defines a homogeneous space provided ρ is a transitive group of symmetries of the underlying set of X. For example, if X is a topological space group elements are assumed to act as homeomorphisms on X; the structure of a G-space is a group homomorphism ρ: G → Homeo into the homeomorphism group of X. Similarly, if X is a differentiable manifold the group elements are diffeomorphisms; the structure of a G-space is a group homomorphism ρ: G → Diffeo into the diffeomorphism group of X. Riemannian symmetric spaces are an important class of homogeneous spaces, include many of the examples listed below. Concrete examples include: Isometry groupsPositive curvature:Sphere: S n − 1 ≅ O / O; this is true because of the following observations: First, S n − 1 is the set of vectors in R n with norm 1.
If we consider one of these vectors as a base vector any other vector can be constructed using an orthogonal transformation. If we consider the span of this vector as a one dimensional subspace of R n the complement is an -dimensional vector space, invariant under an orthogonal transformation from O; this shows us. Oriented sphere: S n − 1 ≅ S O / S O Projective space: P n − 1 ≅ P O / P O Flat:Euclidean space: An ≅ E/ONegative curvature:Hyperbolic space: Hn ≅ O+/O Oriented hyperbolic space: SO+/SO Anti-de Sitter space: AdSn+1 = O/OOthersAffine space: An = Aff/GL. Grassmannian: G r = O / Topological vector spaces From the point of view of the Erlangen program, one may understand that "all points are the same", in the geometry of X; this was true of all geometries proposed before Riemannian geometry, in the middle of the nineteenth century. Thus, for example, Euclidean space, affine space and projective space are all in natural ways homogeneous spaces for their respective symmetry groups; the same is true of the models found of non-Euclidean geometry of constant curvature, such as hyperbolic space.
A further classical example is the space of lines in projective space of three dimensions. It is simple linear algebra to show that GL
Phillip Augustus Griffiths IV is an American mathematician, known for his work in the field of geometry, in particular for the complex manifold approach to algebraic geometry. He was a major developer in particular of the theory of variation of Hodge structure in Hodge theory and moduli theory, he received his B. S. from Wake Forest College in 1959 and his Ph. D. from Princeton University in 1962 working under Donald Spencer. Since he has held positions at Berkeley, Harvard University, Duke University. From 1991 to 2003 he was the Director of the Institute for Advanced Study at New Jersey, he has published on algebraic geometry, differential geometry, geometric function theory, the geometry of partial differential equations. Griffiths serves as the Chair of the Science Initiative Group, he is co-author, with Joe Harris, of Principles of Algebraic Geometry, a well-regarded textbook on complex algebraic geometry. In 2008 he was awarded the Brouwer Medal. In 2012 he became a fellow of the American Mathematical Society.
Moreover, in 2014 Griffiths was awarded the Leroy P. Steele Prize for Lifetime Achievement by the American Mathematical Society. In 2014, Griffiths was awarded the Chern Medal for lifetime devotion to mathematics and outstanding achievements. Griffiths, P. A.. "On certain homogeneous complex manifolds". Proc Natl Acad Sci U S A. 48: 780–783. Doi:10.1073/pnas.48.5.780. PMC 220851. PMID 16590943. Griffiths, P. A.. "Some remarks on automorphisms, analytic bundles, embeddings of complex algebraic varieties". Proc Natl Acad Sci U S A. 49: 817–820. Doi:10.1073/pnas.49.6.817. PMC 300013. PMID 16591103. Griffiths, Phillip A.. "On the differential geometry of homogeneous vector bundles". Trans. Amer. Math. Soc. 109: 1–34. Doi:10.1090/s0002-9947-1963-0162248-5. MR 0162248. Griffiths, P. A.. "The residue calculus and some transcendental results in algebraic geometry, I". Proc Natl Acad Sci U S A. 55: 1303–1309. Doi:10.1073/pnas.55.5.1303. PMC 224316. PMID 16591357. Griffiths, P. A.. "The residue calculus and some transcendental results in algebraic geometry, II".
Proc Natl Acad Sci U S A. 55: 1392–1395. Doi:10.1073/pnas.55.6.1392. PMC 224330. PMID 16578635. Griffiths, P. A.. "Some results on locally homogeneous complex manifolds". Proc Natl Acad Sci U S A. 56: 413–416. Doi:10.1073/pnas.56.2.413. PMC 224387. PMID 16591369. "A transcendental method in algebraic geometry". Actes, Congrès intern. Math. 1970. Tome 1. Pp. 113–119. Griffiths, Phillip A.. "Periods of integrals on algebraic manifolds". Bull. Amer. Math. Soc. 76: 228–296. Doi:10.1090/s0002-9904-1970-12444-2. MR 0258824. Deligne, Pierre. "Real homotopy theory of Kähler manifolds". Inventiones Mathematicae. 29: 245–274. Doi:10.1007/BF01389853. MR 0382702. With Joe Harris: Griffiths, Phillip. "A Poncelet theorem in space". Comment. Math. Helvetici. 52: 145–160. Doi:10.1007/bf02567361. With S. S. Chern: "Abel's Theorem and Webs". Jber. D. Dt. Math.-Verein. 80: 13–110. 1978. Griffiths, Phillip A.. "Complex analysis and algebraic geometry". Bull. Amer. Math. Soc. 1: 595–626. Doi:10.1090/s0273-0979-1979-14640-8. MR 0532551. Griffiths, Phillip A..
"Poincaré and algebraic geometry". Bull. Amer. Math. Soc.. 6: 147–159. Doi:10.1090/s0273-0979-1982-14967-9. MR 0640942. Mumford–Tate groups and domains: their geometry and arithmetic, with Mark Green and Matt Kerr, Princeton University Press, 2012, ISBN 978-0-691-154251 Exterior differential systems and Euler-Lagrange partial differential equations, with Robert Bryant and Daniel Grossman, University of Chicago Press, 2003, ISBN 0-226-07793-4 cloth.