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Topic: Riemann-Roch theorem

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Riemann–Roch theorem - Wikipedia, the free encyclopedia The Riemann–Roch theorem for curves was proved for Riemann surfaces by Riemann and Roch in the 1850s and for algebraic curves by F. In mathematics, specifically in complex analysis and algebraic geometry, the Riemann–Roch theorem is an important tool in the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles. An n-dimensional generalisation, the Hirzebruch-Riemann–Roch theorem, was found and proved by Friedrich Hirzebruch, as an application of characteristic classes in algebraic topology; he was much influenced by the work of Kunihiko Kodaira. en.wikipedia.org /wiki/Riemann-Roch_theorem   (1044 words)

Atiyah–Singer index theorem - Wikipedia, the free encyclopedia The theorem came at the end of more than 100 years' development on the theory of elliptic operators (such as Laplacians), going back to the Riemann-Roch theorem. Atiyah-Singer comment that the initial proof was based on that of the Hirzebruch-Riemann-Roch theorem (1954), and involved cobordism theory. In papers written or published in the period around 1962-1965 the theorem was stated and proved by Michael Atiyah, Raoul Bott and Isadore Singer. en.wikipedia.org /wiki/Atiyah-Singer_index_theorem   (902 words)

Re: Riemann-Roch Theorem But then this is not all that suprising since I have seen a derivation of the classic Riemann-Roch theorem from the Atyhia-Singe Index Theorem which is a generalization of the Hirzebuch Index Theorem. In Reply to: Riemann-Roch Theorem posted by DickT on November 24, 19101 at 13:26:47: The theorem which you quote is closely related to the Hirzebuch Index theorem which relates the index of a partial differential elliptic operator to the topology of the manifold on which it is defined. superstringtheory.com /forum/topboard/messages/127.html   (241 words)

PlanetMath: proof of Riemann-Roch theorem This is version 2 of proof of Riemann-Roch theorem, born on 2003-08-15, modified 2004-09-26. "proof of Riemann-Roch theorem" is owned by bwebste. Now, we need only confirm that the theorem holds for a single line bundle. planetmath.org /encyclopedia/ProofOfRiemannRochTheorem.html   (113 words)

Riemann–Roch theorem - Wikipedia, the free encyclopedia The Riemann–Roch theorem for curves was proved for Riemann surfaces by Riemann and Roch in the 1850s and for algebraic curves by F. In mathematics, specifically in complex analysis and algebraic geometry, the Riemann–Roch theorem is an important tool in the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles. An n-dimensional generalisation, the Hirzebruch-Riemann–Roch theorem, was found and proved by Friedrich Hirzebruch, as an application of characteristic classes in algebraic topology; he was much influenced by the work of Kunihiko Kodaira. en.wikipedia.org /wiki/Riemann-Roch_theorem   (1044 words)

Riemann-Roch theorem for oriented cohomology, by Ivan Panin Here we present the part devoted to the proof of Riemann-Roch type theorems stated in that preprint. The present article in particular contains proofs of Riemann-Roch type theorems stated in preprint Push-forwards in oriented cohomology theories of algebraic varieties, by I.Panin and A.Smirnov. Smirnov doesn't acknowledge his authorship for preprint 0542, and it was a mistake for me to post it without his approval, for which I apologize to you and to him. www.math.uiuc.edu /K-theory/0552   (195 words)

the infamous riemann-roch machine in math there's a famous theorem the riemann-roch theorem (roch is pronounced rock, for the ignorant americans out there.) we like bikes, we like math, the rest is obvious. www.math.sunysb.edu /~jthind/rrm.html   (65 words)

Riemann-Roch theorem Riemann-Roch theorem for oriented cohomology, by Ivan Panin... Quasi-algebraic geometry of curves I. Riemann-Roch theorem and Jacobian... An Adams-Riemann-Roch theorem in Arakelov geometry, Damian Roessler... www.scienceoxygen.com /math/734.html   (139 words)

Jeremy Gray Aspects of the history of the Riemann-Roch Theorem, from its discovery by Riemann and Roch to its use by Castelnuovo and Enriques. Two related theorems in complex function theory, the Riemann mapping theorem and the uni-formisation theorem, originate in the work of Riemann, Poincaré and Klein, and had their first rigorous proofs in a slew of papers (Koebe et al) around 1910. Revised throughout with four new appendices (Riemann's lectures and the Riemann-Hilbert problem, The uniformisation theorem, Picard-Vessiot Theory, The hypergeometric equation in several variables - Appell and Picard). puremaths.open.ac.uk /pmd_department/pmd_gray   (3149 words)

PlanetMath: Riemann-Roch theorem for curves This is version 7 of Riemann-Roch theorem for curves, born on 2001-12-12, modified 2005-03-18. "Riemann-Roch theorem for curves" is owned by mathcam. ($K$-theory :: Topological $K$-theory :: Riemann-Roch theorems, Chern characters) planetmath.org /encyclopedia/RiemannRochTheorem.html   (70 words)

Cornell Math - MATH 767, Fall 2003 Riemann-Roch is a great theorem, it has many applications, and also allows one to obtain function theory results and algebraic results using geometry (and vice versa). We will get to the point as quickly as possible where we can state and prove the Riemann-Roch theorem, and then we will spend the rest of the semester applying the Riemann-Roch theorem to obtain deep results in the algebraic geometry and function theory of curves. The theory of complex algebraic curves and Riemann surfaces is a beautiful theory at the crossroads of several fields surrounding algebraic geometry: complex analysis, commutative algebra, geometry, and topology. www.math.cornell.edu /Courses/GradCourses/FA03/767.html   (304 words)

Riemann Surfaces The Riemann-Roch theorem is a prime example of an ``index formula'' equating something analytic to something topological. We will extend the usual concepts of complex function theory to Riemann surfaces, defining holomorphic and meromorphic functions, and proving, for example, the identity theorem, the maximum principle, and the residue theorem. A Riemann surface is a surface with a complex structure; thus sufficiently small neighbourhoods of its points may by treated as open subsets of the complex plane. www.imf.au.dk /da/uddannelse/studord/older/F2003/node15.html   (200 words)

Grothendieck–Hirzebruch-Riemann–Roch theorem - Wikipedia, the free encyclopedia The Grothendieck–Hirzebruch-Riemann–Roch theorem sets both theorems in a relative situation of a morphism between two manifolds (or more general schemes) and changes the theorem from a statement about a single bundle, to one applying to chain complexes of sheaves. The Hirzebruch-Riemann-Roch theorem is (essentially) the special case where Y is a point and the field is the field of complex numbers. In mathematics, specifically in algebraic geometry, the Grothendieck–Hirzebruch-Riemann–Roch theorem (or Grothendieck–Riemann–Roch theorem) is a far-reaching result on coherent cohomology. en.wikipedia.org /wiki/Grothendieck-Riemann-Roch_theorem   (474 words)

Alexander Grothendieck - Article from FactBug.org - the fast Wikipedia mirror site The Grothendieck-Riemann-Roch theorem was announced by Grothendieck at the initial Arbeitstagung in Bonn, in 1957. Perhaps Grothendieck's deepest single accomplishment is the invention of the étale and l-adic cohomology theories, which explain an observation of André Weil's, that there is a deep connection between the topological characteristics of a variety and its diophantine (number theoretic) properties. Alexander Grothendieck (born March 28, 1928, Berlin) was one of the most important mathematicians active in the 20th century. www.factbug.org /cgi-bin/a.cgi?a=2042   (1645 words)

CONK! Encyclopedia: Category:Theorems Note that a theorem is distinct from a theory. www.conk.com /search/encyclopedia.cgi?q=Category:Theorems   (99 words)

Pure Group Publications We establish formulas for the equivariant Euler characteristic of locally free G-modules on a projective G-scheme X: We prove an Adams- Riemann-Roch theorem and, under a certain continuity assumption for the push-forward map, a Grothendieck-Riemann- Roch theorem in (higher) equivariant algebraic K-theory. In case of a flag variety G/B, the above continuity assumption is verified, and the Grothendieck-Riemann-Roch theorem for this situation yields a new proof of the Weyl character formula. Furthermore, we present the following applications: The Adams-Riemann-Roch theorem specializes to an interchanging rule between Adams operations and induction for representations. www.maths.soton.ac.uk /pure/viewabstract.phtml?entry=442   (102 words)

Talks in Mathematical Physics In the talk, a theorem of Riemann-Roch type is introduced which applies to the natural Bloch line bundle over the spectral curve of the heat equation with a space- and time-dependent periodic potential. A Riemann Roch Theorem for Infinite Genus Riemann Surfaces with Applications to Inverse Spectral Theory Abstract: Riemann surfaces of infinite genus arise in a natural way as spectral varieties of several partial differential equations in mathematical physics. www.math.ethz.ch /~felder/talks/talksSS98.html   (494 words)

Powell's Books - Annals of Mathematics Studies #127: Lectures on the Arithmetic Riemann-Roch Theorem by Gerd Faltings Annals of Mathematics Studies #127: Lectures on the Arithmetic Riemann-Roch Theorem Discontinuous Groups and Riemann Surfaces: Proceedings of the 1973 Conference at the University of Maryland Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference www.powells.com /biblio?isbn=0691025444   (478 words)

MAT1191HF - Topics in Algebraic Geometry: Grothendieck groups, Chow motives. The definition of the Grothendieck groups of an algebraic scheme was motivated by the expectation to generalize the classical Riemann-Roch theorem to the "relative case". Then, I will explain and comment the statement of the Grothendieck-Riemann-Roch theorem for a smooth, projective morphism of non-singular quasi-projective varieties. The theory of motives was concieved by A. Grothendieck in the 60's with the purpose to study (i.e. www.math.toronto.edu /~kc/1191hf.html   (243 words)

carberry_emma You mentioned earlier that Abel's Theorem tells us that a Riemann Surface embeds into its Jacobi variety. (I said that it lead to the result that a torus was analytically equivalent to its Jacobi variety, but this wasn't what he wanted, so he made his question more explicit.) Explain the statement of the theorem more concretely for the case g=1. Fefferman left the room for a while during this part of the exam, and Conway was absorbed in some notes he'd brought with him. www.math.princeton.edu /graduate/generals/carberry_emma   (792 words)

Pure Group Publications In this direction, Bernhard Koeck has proved an equivariant Adams-Riemann-Roch theorem and an equivariant Grothendieck-Riemann-Roch theorem. A very important and early result in equivariant Riemann-Roch theory is the classical Chevalley-Weil formula which computes the Galois module structure of the space of holomorphic differentials on a Riemann surface. One of the most fundamental and fruitful subjects in Algebraic Geometry is Riemann-Roch theory whose aim, very roughly speaking, is to compute Euler characteristics. www.maths.soton.ac.uk /pure/researchabstract.phtml?keyword=Riemann-Roch   (286 words)

PlanetMath: proof of Riemann's removable singularity theorem owned by pbruin proof of Riemann mapping theorem owned by rspuzio Riemann hypothesis (in Riemann zeta function) owned by djao planetmath.org /encyclopedia/R   (1614 words)

Gerd Faltings Proves Mordell's Conjecture (1983) History Summary Throughout time, the Riemann-Roch theorem has been transformed into something that encompasses other arithmetic areas of study as a result of dedicated scholars who have contributed to the discovery of new elements in algebra and geometry, adding new dimensions to original theorems. Lectures on the Arithmetic Riemann-Roch Theorem (Anal of Mathematics Studies, 127). His method of altering a familiar geometric theorem into algebraic terms led him to solve the complex geometric theorem proposed by Louis Mordell in 1922. www.bookrags.com /history/sciencehistory/gerd-faltings-proves-mordells-conje-scit-07123   (1045 words)

Riemann-Roch Theorem Lots of other books were consulted, and one resource opined that there are 27 different things that somebody calls a Rienamm-Roch Theorem. Now we have a connection on the manifold (recall that it is a 2-D Riemann surface) and the connection - an ordinary Riemannian one under all the fancy dress - generates Covariant Derivatives. The version I am going to discuss concerns Riemann Surfaces, and I am going to work from a conformally flat metric. superstringtheory.com /forum/topboard/messages/121.html   (585 words)

1 Model case of curves The topic of algebraic cycles has its origin in the theory of divisors on an algebraic curve, or compact Riemann surface. The Abel-Jacobi theorem (which yields the above isomorphism between If X is a non-singular projective curve over an algebraically closed field k, a divisor on X is an element of the free abelian group on the points of X; we denote this free abelian group by www.imsc.ernet.in /~kapil/papers/harishconf/node2.html   (382 words)

Grothendieck He gave an algebraic proof of the Riemann-Roch theorem. Grothendieck was always strongly pacifist in his views and campaigned against the military built-up of the 1960s. It is no exaggeration to speak of Grothendieck's years 1959-70 at the IHES as a 'Golden Age', during which a whole new school of mathematics flourished under Grothendieck's charismatic leadership. www-gap.dcs.st-and.ac.uk /~history/Mathematicians/Grothendieck.html   (516 words)

Search Results for theorem* A proof of the Riemann-Roch theorem is given, and the theory of Riemann surfaces and their topology is studied. Theorem 2 of Euclid's Phaenomena consists of four propositions with proofs for only three of them while the missing one is replaced by the remark "that this is the case has been shown elsewhere"; indeed theorem and proof are found as Theorem 10 in Autolycus's 'Rotating Sphere'. The theorem relating convergence almost everywhere and uniform convergence by D F Egorov, one of Bugaev's pupils, in 1911 is seen as marking the beginning of the Moscow school of the theory of functions of a real variable. www-groups.dcs.st-and.ac.uk /~history/Search/historysearch.cgi?SUGGESTION=theorem*&CONTEXT=1   (15447 words)

MAT1191HF - Topics in Algebraic Geometry: Grothendieck groups, Chow motives. The definition of the Grothendieck groups of an algebraic scheme was motivated by the expectation to generalize the classical Riemann-Roch theorem to the "relative case". Then, I will explain and comment the statement of the Grothendieck-Riemann-Roch theorem for a smooth, projective morphism of non-singular quasi-projective varieties. The theory of motives was concieved by A. Grothendieck in the 60's with the purpose to study (i.e. www.math.toronto.edu /kc/1191hf.html   (243 words)

The Grothendieck-Riemann-Roch theorem for group scheme actions, by Bernhard Koeck We establish formulas for the equivariant Euler characteristic of locally free G-modules on a projective G-scheme X: We prove an Adams- Riemann-Roch theorem and, under a certain continuity assumption for the push-forward map, a Grothendieck-Riemann- Roch theorem in (higher) equivariant algebraic K-theory. In case of a flag variety G/B, the above continuity assumption is verified, and the Grothendieck-Riemann-Roch theorem for this situation yields a new proof of the Weyl character formula. Let G be a group or a group scheme. www.math.uiuc.edu /K-theory/0108   (121 words)

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