Donaldsons riemann surfaces is a lucid and modern account on riemann surfaces. More contemporary books are donaldsons 3 and schlags 5. Dealing with the main results on riemann surfaces from a variety of points of view. C2 z w2 is identified with the complex w plane by projection. Algebraic curves and riemann surfaces by miranda geoemtry of algebraic curves by arbarello, cornalba, gri ths, and harris curves and their jacobians by mumford. These surfaces were first studied by and are named after bernhard riemann.
Room 674, huxley building, department of mathematics, imperial college, queens gate, london sw7 2az telephone. Simon donaldson, riemann surfaces, oxford graduate texts in mathematics, 2011. Jun 23, 2019 ben mckay 14k 2 27 i do recommend the recent published book by donaldson on this subject. Riemann surfaces jwr wednesday december 12, 2001, 8. We denote by g kv the grassmannian of kdimensional subspaces of v and by pv g 1v the projective space of v. Students with more experience those who have studied complex manifolds in general, say will. The lectures will take place on wednesdays from 14. Riemann surfaces and algebraic curves jwr tuesday december 11, 2001, 9. There are much lecture notes published on authors on their pages. A riemann surface is a smooth complex manifold xwithout boundary of complex dimension one. The minicourse is dedicated to master students having. D donaldson, riemann surfaces, oxford graduate texts in mathematics.
It also deals quite a bit with noncompact riemann surfaces, but does include standard material on abels theorem, the abeljacobi map, etc. They involve maps from the complex plane to an algebraic torus, with prescribed discontinuities along a collection of rays, and are closely related to the riemannhilbert problems considered by gaiotto et al. By focusing on low dimensions, donaldson bridges the gap between elementary complex variables and the deeper topics of complex manifolds, cohomology, algebraic topology, vector bundles, sheaves and so forth. In higher dimensions, an almost complex structure is a pre requisite to the existence of a complex structure. We wont be so lucky in general, in the sense that riemann surfaces will not be identi able with their w or zprojections. Mathematics 9302 riemann surfaces uwo, math department. Riemann surfaces the department of mathematics western. In mathematics, particularly in complex analysis, a riemann surface is a onedimensional complex manifold. Ben mckay 14k 2 27 i do recommend the recent published book by donaldson on this subject. Another possibility is to study riemann surfaces as twodimensional real manifolds, as gauss 1822 had taken on the problem of taking a. It provides a model for a large number of more recent developments in areas including.
Riemannhilbert problems from donaldsonthomas theory. The books we will be using as reference for this course are farkas and kra riemann surfaces and donaldsons riemann surfaces. Integrable systems and riemann surfaces lecture notes preliminary version boris dubrovin april 15, 2009 contents. Among all modern expositions of the theory, this one is particularly close to riemanns. The topis is wide and well described in literature, so our aim was to bring some avour. For a complete list of my publications see mathscinet. Riemann surfaces, by simon donaldson, oxford university press. Donaldson, 9780198526391, available at book depository with free delivery worldwide. A view of 21st century mathematics mit, room 34101, august 28 september 2, 20 abstract. To master a number of topics related to riemann surfaces such as introductory di erential geometry, di erential topology, complex structures on. New version will be available soon notes lie groups and geometry.
Riemann surfaces university of california, berkeley. This is an outstanding introduction to the modern study of riemann surfaces. It is nice to see the chapter on elliptic functions and integrals. In this paper we study a class of riemannhilbert problems arising naturally in donaldsonthomas theory. Riemann surfaces can be thought of as deformed versions of the complex plane. I shall try to closely follow the newly published riemann surfaces, by simon donaldson, oxford university press, 2011. Donaldsons text, which emphasizes the connections of riemann surfaces with other mathematical. It is a culmination of much of traditional calculus, making. Later on we consider basically compact riemann surfaces and call c shortly the riemann surface of the curve c. A riemann surface sin c2 is nonsingular if each point z 0. It is an extremely useful part of mathematics, knowledge of which is needed by specialists in many other fields. The textbook for the course is donaldson riemann surfaces oxford graduate texts in mathematics. As a consequence we get the classi cation of riemann surfaces. Simon donaldson the theory of riemann surfaces occupies a very special place in mathematics.
It is particularly pleasing that the subject of riemann surfaces has attracted the attention of a new generation of mathematicians from newly adjacent fields for example, those interested in hyperbolic manifolds and iterations of rational maps and young. Cmcavalieri and miles, riemann surfaces and algebraic curves, london mathematical society student texts. It is then clear what a holomorphic function on s should be. Notes for course given 2007 and 2011 notes skeleton notes for an analysis course, c. It is a culmination of much of traditional calculus, making surprising conne. I recommend lectures on riemann surfaces by forster. Two vector bundles over the grassmannian g kv are the. Algebraic functions 11 let pz,w be a polynomial in two complex variables. A higgs bundle is a system composed of a connection aon a vector bundle eover a riemann surface. It is extremely wellwritten, but definitely more analytic in flavor. The topis is wide and well described in literature, so our aim was to bring some avour of it. Jan 01, 2011 the theory of riemann surfaces occupies a very special place in mathematics.
The point of the introduction of riemann surfaces made by riemann, klein and weyl 185119, was that riemann surfaces can be considered as both a onedimensional complex manifold and an algebraic curve. This is an introduction to the geometry of compact riemann surfaces, largely following the books farkaskra, fay, mumford tata lectures. The theory of riemann surfaces occupies a very special place in mathematics. Integrable systems and riemann surfaces lecture notes preliminary version boris dubrovin april 15, 2009 contents 1 kdv equation and schr odinger operator 2. Integrable systems and riemann surfaces lecture notes.
Simon donaldson royal society research professor address. The open set u is called the domain of the chart c. Among all modern expositions of the theory, this one is particularly close to riemanns original analytic approach based on potential the. Evidently, this is a generalization of the riemann mapping theorem. Geometry of higgs and toda fields on riemann surfaces. This book deserves very serious consideration as a text for anyone contemplating giving a course on riemann surfaces. It turnes out that all compact riemann surfaces can be described as compacti cations of algebraic curves. It deals almost only with compact riemann surfaces via an analytic approach, but also gives a discussion of resolution of singularities for complex.
Riemann surfaces uwmadison department of mathematics. Among all mod ern expositions of the theory, this one is particularly close to riemanns original analytic approach based on potential theory, pdes and ideas of mathematical physics. A sequel to lectures on riemann surfaces mathematical notes, 1966, this volume continues the discussion of the dimensions of spaces of holomorphic crosssections of complex line bundles over compact riemann surfaces. Thursday, october 14, 2010 week 41 this week we started our study of calculus on riemann surfaces.
It is an extremely useful part of mathematics, knowledge of which is. For monodromy and maps of riemann surfaces, see cavalierimiles cm or donaldson d or miranda. Analysis on riemann surfaces, meromorphic functions, uniformization theorem. Riemann surfaces there are some surfaces which we are interested in that were studied by riemann, which surprisingly have been named riemann surfaces. Whereas the earlier treatment was limited to results obtainable chiefly by onedimensional methods, the more detailed analysis presented here requires the use of various. In the case of riemann surfaces a direct proof for arbitrary rank which avoids introduction of the donaldson functional can be modeled on donaldsons proof of the narasimhanseshadri theorem in 18 such a proof was suggested in 57. It is a culmination of much of traditional calculus, making surprising connections with geometry and arithmetic. An authoritative but accessible text on one dimensional complex manifolds or riemann surfaces. In particular, it includes pretty much all the analysis to prove finitedimensionality of sheaf cohomology on a compact riemann surface.
Of course riemanns thesis and followup paper on theory of abelian functions is rather incredible as well. For riemann surfaces, a choice of almost complex structure is equivalent to a choice of complex structure. To put things into perspective and to reveal my bias, i have learnt most of what i know on riemann surfaces from the bers 19571958 lecture. Riemann surfaces, by simon donaldson, oxford graduate texts in mathematics. Prove the existence of the universal covering of a connected manifold x, and remark that in case xis a riemann surface, its universal covering is a riemann surface as well for91, i. Riemann surfaces oxford graduate texts in mathematics 9780199606740. Coverings of riemann surfaces and the universal covering study proper nonconstant holomorphic maps between riemann surfaces for91, 4. It is particularly pleasing that the subject of riemann surfaces has attracted the attention of a new generation of mathematicians from newly adjacent fields for example, those interested in hyperbolic manifolds and iterations of rational maps and young physicists who have been convinced certainly not by mathematicians that compact. Riemann surfaces arundebray may6,2016 thesenotesweretakeninutaustinsm392c riemannsurfaces classinspring2016,taughtbytimperutz. Forster gives a sheaf theoretic approach, and miranda gives an algebraic geometry approach. Lectures on riemann surfaces a very attractive addition to the list in the form of a wellconceived and handsomely produced textbook based on several years lecturing experience.