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Riemann–Roch theorem - Wikipedia, the free encyclopedia 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. The Riemann–Roch theorem for curves was proved for Riemann surfaces by Riemann and Roch in the 1850s and for algebraic curves by F. 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)

Geometry.Net - Scientists Books: Faltings Gerd Roch was Riemann's student and interpreted the quantity D + 1 - p as the dimension of the space of holomorphic integrands. The work of Riemann and Roch is readily seen to be related to the genus of the surface, if viewed in the light of the polygon of 4p sides. The modern view of the Riemann-Roch theorem in fact is naturally viewed as a generalization of a formula for the Euler characteristic, the latter of which involves the genus of a Riemann surface. www.geometry.net /scientists_bk/faltings_gerd.html   (632 words)

Atiyah–Singer index theorem - Wikipedia, the free encyclopedia In the mathematics of manifolds and differential operators, the Atiyah–Singer index theorem is an important unifying result that connects topology and analysis. 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. 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)

Riemann-Roch theorem   (Site not responding. Last check: 2007-10-26) The Riemann-Roch theorem for curves was proved for Riemann surfaces by Riemann-Roch in the 1850s and for algebraic curves by F. Schmidt in 1929. In algebraic geometry of dimension two such a formula was found by the geometers of the Italian school; a Riemann-Roch theorem for algebraic surfaces was proved (there are several versions, with the first possibly being due to Max Noether). 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. www.sciencedaily.com /encyclopedia/riemann_roch_theorem   (1048 words)

PlanetMath: proof of Riemann-Roch theorem Now, we need only confirm that the theorem holds for a single line bundle. "proof of Riemann-Roch theorem" is owned by bwebste. This is version 3 of proof of Riemann-Roch theorem, born on 2003-08-15, modified 2007-04-08. www.planetmath.org /encyclopedia/ProofOfRiemannRochTheorem.html   (106 words)

PlanetMath: Riemann-Roch theorem for curves "Riemann-Roch theorem for curves" is owned by mathcam. This is version 8 of Riemann-Roch theorem for curves, born on 2001-12-12, modified 2006-02-21. ($K$-theory :: Topological $K$-theory :: Riemann-Roch theorems, Chern characters) www.planetmath.org /encyclopedia/RiemannRochTheorem.html   (71 words)

Atiyah-Singer index theorem   (Site not responding. Last check: 2007-10-26) In the mathematics of manifolds and differential operators, the Atiyah-Singer index theorem is a basic general result that came at the end of a long development on the theory of elliptic operators (such as Laplacians), going back to the Riemann-Roch theorem. The precise statement of the Index Theorem requires K-theory, as well as the background in functional analysis and pseudo-differential operators in the manifold setting (sometimes called global analysis). 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; it served as a notable unification. www.theezine.net /a/atiyah-singer-index-theorem.html   (444 words)

RIEMANN-ROCH THEOREM   (Site not responding. Last check: 2007-10-26) Initially proved as Riemann's inequality, the theorem reached its definitive form for Riemann surfaces after work of Riemann's student Roch in the 1850s. In the theory of elliptic functions it is shown that this sequence is 1, 1, 2, 3, 4, 5... In algebraic geometry of dimension two such a formula was found by the geometers of the Italian school. www.websters-online-dictionary.org /definition/RIEMANN-ROCH+THEOREM   (829 words)

Re: Riemann-Roch Theorem 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. 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. I am only calling it the "Hirzebuch Index Theorem" because the only reference I know for it is in the Hirezbuch's book on PDE. I don't know who it is really due to. superstringtheory.com /forum/topboard/messages/127.html   (241 words)

Riemann-Roch theorem   (Site not responding. Last check: 2007-10-26) 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)

Cornell Math - MATH 767, Fall 2003 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. 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). Abel's theorem and the Jacobian of a curve www.math.cornell.edu /Courses/GradCourses/FA03/767.html   (304 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. The theorem offers one of the most instructive examples in the history of mathematics of how a result stays alive by admitting many interpretations: in complex function theory, and in both the algebraic and geometric styles of algebraic geometry. 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. puremaths.open.ac.uk /pmd_department/pmd_gray   (3149 words)

Journal of the American Mathematical Society One of the assumptions of the theorem is that the reduction is regular, so that the reduced space is a smooth symplectic manifold. C. Farsi: K-theoretical Index theorems for orbifolds, Quart. I. Satake: The Gauss-Bonnet theorem for V-manifolds, J. Math. www.ams.org /jams/1996-9-02/S0894-0347-96-00197-X/home.html   (464 words)

Riemann-Roch theorem for oriented cohomology, by Ivan Panin   (Site not responding. Last check: 2007-10-26) 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. 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. Here we present the part devoted to the proof of Riemann-Roch type theorems stated in that preprint. www.math.uiuc.edu /K-theory/0552   (195 words)

Riemann Surfaces   (Site not responding. Last check: 2007-10-26) 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. 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. Students who do not intend to take a degree in Mathematics or Statistics from the University of Aarhus, but wish to earn credits for a 2.dels course from the Department of Mathematics, should indicate at the beginning of the course that they wish to be examined. www.imf.au.dk /da/uddannelse/studord/older/F2003/node15.html   (200 words)

DOCUMENTA MATHEMATICA, Extra Vol. ICM III (1998), 811-822 The history of the Riemann-Roch Theorem, from its discovery by Riemann and Roch in the 1850's to its use by Castelnuovo and Enriques in from 1890 to 1914, offers one of the most instructive examples in the history of mathematics of how a result stays alive in mathematics by admitting many interpretations. Various mathematicians over the years took the theorem to be central to their researches in complex function theory, and in the study of algebraic curves and surfaces in a variety of algebraic and geometric styles. In surveying their interpretations and extensions of the theorem, the historian traces the creation of a general theory of complex algebraic curves and surfaces in the period, and uncovers lively agreements and disagreements. www.emis.de /journals/DMJDMV/xvol-icm/19/Gray.MAN.html   (181 words)

Gerd Faltings Proves Mordell's Conjecture (1983) History Summary   (Site not responding. Last check: 2007-10-26) 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. 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. Proving Mordell's conjecture led Faltings to further the study of Fermat's last theorem, long considered the greatest unsolved problem in mathematics until it was finally proven by Andrew Wiles (1953-) in 1994. www.bookrags.com /history/sciencehistory/gerd-faltings-proves-mordells-conje-scit-07123   (1045 words)

Pure Group Publications 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. While he has applied the first theorem to generalize a deep theorem of Burns and Chinburg concerning Adams operations on locally free classgroups in Galois module theory, he has applied the second theorem to give a new proof of the famous Weyl character formula. 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. www.maths.soton.ac.uk /pure/researchabstract.phtml?keyword=Riemann-Roch   (286 words)

Mikhail Kapranov at MSRI - A Riemann-Roch theorem for (higher) determinantal gerbes and differentiable manifolds   (Site not responding. Last check: 2007-10-26) Mikhail Kapranov at MSRI - A Riemann-Roch theorem for (higher) determinantal gerbes and differentiable manifolds Mikhail Kapranov - A Riemann-Roch theorem for (higher) determinantal gerbes and differentiable manifolds A PDF version of the lecture notes is available here. www.msri.org /publications/ln/msri/2002/hodgetheory/kapranov/1/index.html   (38 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. Furthermore, we present the following applications: The Adams-Riemann-Roch theorem specializes to an interchanging rule between Adams operations and induction for representations. 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. www.maths.soton.ac.uk /pure/viewabstract.phtml?entry=442   (102 words)

MAT1191HF - Topics in Algebraic Geometry: Grothendieck groups, Chow motives.   (Site not responding. Last check: 2007-10-26) 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)

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. The version I am going to discuss concerns Riemann Surfaces, and I am going to work from a conformally flat metric. 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. superstringtheory.com /forum/topboard/messages/121.html   (585 words)

1 Model case of curves   (Site not responding. Last check: 2007-10-26) The topic of algebraic cycles has its origin in the theory of divisors on an algebraic curve, or compact Riemann surface. 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 By abuse of notation we will denote the divisor class of a line bundle L by L also. www.imsc.ernet.in /~kapil/papers/harishconf/node2.html   (382 words)

[No title] Talk about the case when the divisor is of the form r.p (ie a single point). You mentioned earlier that Abel's Theorem tells us that a Riemann Surface embeds into its Jacobi variety. 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)

UC Berkeley Mathematics   (Site not responding. Last check: 2007-10-26) The so-called "Quantum Riemann-Roch formula" of Coates-Givental is a powerful identity in Gromov-Witten theory based on application of Grothendieck-Riemann-Roch theorem to universal families of holomorphic curves in Kähler target spaces. Respectively, a local version of Kawasaki-Riemann-Roch theorem should be developed instead of Grothedieck-Riemann-Roch. In the talk, we will explain the formulation of the Quantum-Riemann-Roch and its orbifold generalization, introduce the loop-space/loop-group formalism needed in these formulations, recall the Kawasaki-Riemann-Roch theorem participating in the proof and describe some applications (such as "Quantum Lefschetz formula" in genus-0 mirror theory). math.berkeley.edu /calendar-event266.html   (180 words)

LIPS   (Site not responding. Last check: 2007-10-26) Then using a theorem of Fitting and a recipe of Donkin we can calculate the number of irreducible modules for the Partition Algebras, and calculate their dimensions. The aim of this talk is first to define a partial space of the ordinary L-space in which it dimension does not depend on the value of the genus. This theorem which has simple combinatorial proof plays the role of the much deeper Riemann -Roch theorem in our partial theory. www.maths.qmw.ac.uk /~arhin/lips/lip.HTM   (1038 words)

Talks in Mathematical Physics Abstract: Riemann surfaces of infinite genus arise in a natural way as spectral varieties of several partial differential equations 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 conjecture for the trace of such endomorphisms is presented; it generalizes the Verlinde formula and is compatible with factorization. www.math.ethz.ch /~felder/talks/talksSS98.html   (494 words)

Köck: The Grothendieck-Riemann-Roch theorem for group scheme actions TAYLOR, Riemann-Roch type theorems for arithmetic schemes with a finite group action (J. Reine Angew. KÖCK, The Lefschetz theorem in higher equivariant K-theory (Comm. TAMME, The theorem of Riemann-Roch, in M. Rapoport, N. Schappacher and P. Schneider (eds.), Beilinson's conjectures on special values of L-functions (Perspect. www.numdam.org /numdam-bin/fitem?id=ASENS_1998_4_31_3_415_0   (568 words)

An Arithmetic Riemann-Roch Theorem for Singular Arithmetic Surfaces (Memoirs of the American Mathematical Society) An Arithmetic Riemann-Roch Theorem for Singular Arithmetic Surfaces (Memoirs of the American Mathematical Society) Finally, the author defines an intersection theory for arithmetic surfaces that includes a large class of singular arithmetic surfaces. This culminates in a proof of the arithmetic Riemann-Roch theorem. www.markcarey.com /tools/store/p/0821804073   (308 words)

The Riemann-Roch theorem for elliptic operators (ResearchIndex)   (Site not responding. Last check: 2007-10-26) 5.7%: The Riemann-Roch Theorem For Elliptic Operators And.. 1 Shubin: The Riemann-Roch theorem for general elliptic operat.. L² Riemann-Roch theorem for elliptic operators - Shubin citeseer.ist.psu.edu /432419.html   (162 words)

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