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I am in master program of mathematics, specialized in PDE and numerical analysis. Now I am trying to decide which classes to take for next semester. Of course I want to become an expert in my field, but I am also interested in Geometry. I learned some differential geometry in my bachelor class, and I want to take the next tier, which is Riemann surfaces in my university program. But I am not very sure this will be the right choice, since if I don't take this, I will have some time to take more numerical analysis courses which directly connect to my area. But in same time I feel like I can self-study some numerical methods in the future. So now, the question is:

  1. Can taking a Riemann surface course be more beneficial (which is a very vague concept) over taking more numerical analysis course?
  2. If I want to do research in my future career (i.e. proceed to PhD) will there be any topics in analysis related to Riemann surfaces?
Jingeon An-Lacroix
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    Analysis is a very broad topic... If you want to have anything to do with functions of complex variables, then knowing the bases of Riemann surfaces is a must. – abx May 05 '20 at 06:48
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    I recall several speakers at the last ICM in Brazil saying that their research in applied mathematics drew ideas from the study of the dynamics of geodesic flow on Riemann surfaces, i.e. constant curvature surfaces. Perhaps not the usual material in a course on Riemann surfaces, but related by the close connection between dynamics and the Laplacian operator. – Ben McKay May 05 '20 at 08:09
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    @BenMcKay Would you remember any names of those people ? Just curious to see applications of the topic – Piyush Grover May 05 '20 at 15:59
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    The question as titled is trivial. (If you're ever wondering whether it will be beneficial to learn any new piece of math, then the answer is 'yes'.) The more focussed questions in the body are much better. Maybe re-phrase the title as "How can learning Riemann surfaces …"? – LSpice May 05 '20 at 16:08
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    @LSpice I agree, I tried to rephrase. – Jingeon An-Lacroix May 05 '20 at 16:19
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    The topic of Riemann surfaces is beautiful, and connected to so many things. They can be looked at from so many different perspectives. Do not get fooled by how easy they may seem. Some of the deepest questions in Mathematics have to do with them (the Riemann hypothesis for one). I know this does not answer your question, but I wanted to write it as a comment. – Malkoun May 05 '20 at 17:15
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    @Malkoun Thanks, maybe I just needed some motivations. – Jingeon An-Lacroix May 05 '20 at 17:17
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    I should add that they are also the easiest examples of moduli spaces. Many times, you are looking at the set of all solutions, whether algebraic or analytic, of some equations (algebraic or differential). Often the general solution depends on some parameters. In case, there is only one complex parameter, and the general solution depends holomorphically on it, then the natural habitat for such a parameter is a Riemann surface. – Malkoun May 05 '20 at 17:21
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    @Malkoun Thank you so much for the detailed comment. Can you recommend any introductory book with regarding the topic you stated? – Jingeon An-Lacroix May 05 '20 at 17:43
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    It is a tough question to answer in general. It depends on one's taste and background. I would for instance recommend Otto Forster's book "Lectures on Riemann surfaces". It is a good one. – Malkoun May 05 '20 at 17:55
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    Riemann surfaces connect to many areas of geometry, dynamics, analysis and physics. – Hollis Williams May 05 '20 at 18:56
  • My wife used to study "Minimal Surfaces". Is there a connection between "Minimal Surfaces" and "Riemann Surfaces" ? – JosephDoggie May 05 '20 at 20:05
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    Not really, although one can have Riemann's minimal surface and Riemannian surfaces :). You can think of a minimal surface as something which always wants to minimize its surface area at every point (like a soap bubble). A Riemann surface is a one-dimensional complex manifold, but complex manifolds are closer to objects called algebraic varieties than they are to the objects which you think of when you say 'surface', hence many of the applications in algebraic geometry. – Hollis Williams May 06 '20 at 19:31
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    As another application, in string theory (if you are interested in learning a bit more about that) the quantum field theory which one studies lives on a Riemann surface. – Hollis Williams May 06 '20 at 19:36
  • You might enjoy the edited volume Computational Approach to Riemann Surfaces, Eds Bobenko & Klein, Springer 2011. I've left this as a comment instead of an answer, as it may not address the question in the title. That said, given your interests in PDEs and numerical analysis, it could be worth taking a look at. If you'd like me to convert this comment into an answer, let me know. – J W Jan 07 '22 at 11:50

2 Answers2

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Let me answer the question in the title. The answer is definitely yes.

Just to mention an important topic, the proof of Uniformization Theorem for Riemann surfaces requires to construct at least one holomorphic or meromorphic form with prescribed singularies. All known proofs use some Analysis, and none of them is simple.

In fact, you will be led to study deep properties of elliptic operators on the surface (aka "Hodge Theory"), and this will surely boost your analysis skills.

Francesco Polizzi
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Painleve - It came to appear that, between two truths of the real domain, the easiest path quite often passes through the complex domain.

Hadamard - It has been written that the shortest and best way between two truths of the real domain often passes through the imaginary one."

Read first "The Magic Wand Theorem of A. Eskin and M. Mirzakhani" by Anton Zorich for motivation on Riemann sufaces.

And, "A Singular Mathematical Promenade" by Etienne Ghys (particularly p. 87-93 and glance at the Wikipedia article on monodromy).

More prosaically:

To understand integral transform solutions (Mellin, Laplace, Fourier) to pdes, you need to understand poles and branch cuts of complex functions.

Solutions to Laplace's equation in two dimensions are called harmonic functions that are the real and complex components of a complex function and give mutually orthogonal contour lines on the complex plane and Riemann sphere.

From "THE THEOREM OF RIEMANN-ROCH AND ABEL’s THEOREM" by Siu:

The theorem of Riemann-Roch and Abel’s theorem could be interpreted as answering the question: for which configuration of charges, dipoles, or multipoles on a compact Riemann surface of genus ≥ 1 would the flux functions (whose level curves are the flux lines and which are the harmonic conjugates of the electrostatic potential functions) in the case of the theorem of Riemann-Roch, or their exponentiation after multiplication by 2πi in the case of Abel’s theorem, be single-valued on the Riemann surface so that the flux lines are closed curves?

At a more basic level for numerical analysis, in understanding convergence of real power series and, therefore, series solns. to pdes, you need to understand singularities (poles and branch cuts in the complex domain) and these involve Riemann surfaces.

Same for Newton (finite difference) and sinc function (Nyquist-Shannon) interpolations of sequences of real/complex numbers and their numerical analytic continuations and for asymptotic series a la Poincare. (Norlund, Poincare, and Berry wrote well on these topics.)

Helps in understanding convolutions, Dirac Delta functions and their derivatives, and, therefore, fractional calculus and operational calculus.

Necessary in understanding basic string theories.

The list is endless. Without such knowledge, you live (perhaps blissfully) in Abbott's Flatland.

The examples really suggest that you may be imposing a gratuitous, restrictive dichotomy--there is plenty of synergy between the study of numerical analysis and Riemann surfaces and both provide paths to other intriguing areas of the grand, evolving tapestry of mathematics, engineering, and science. (Of course, if you are looking where the money is in America, well I suggest a medical degree or starting a munitions factory.)

Tom Copeland
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    Electromagnetics involves solving Maxwell's equations (pdes), of course. – Tom Copeland May 05 '20 at 15:14
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    Haha, I liked you mentioned the Flatland. Of course I understand the importance of the Riemann surfaces in PDEs, but what hesitates me is 'I don't know taking a course is beneficial compare to take other analysis course'. I guess you are saying knowing the Riemann surfaces is (almost) mendatory, but do you think I have to take a course? No way of self-study when I needed? – Jingeon An-Lacroix May 05 '20 at 15:44
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    @JingeonAn Most math professors, engineers, and scientists are specialists and go through the motions when teaching. Take courses under the rare motivated, conscientious generalist and/or study good textbooks and articles on your own and use Q&A sites and MathCad to check your analysis symbolically or numerically. – Tom Copeland May 05 '20 at 16:07
  • Thank you for the great answer! – Jingeon An-Lacroix May 05 '20 at 16:17
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    Examples: The two papers by McMullen "Advanced Complex Analysis" and "Riemann surfaces, dynamics, and geometry" and the books "Visual complex analysis" by Needham and "On Riemann's Theory of Algebraic Functions and Their Integrals" by Klein. – Tom Copeland May 05 '20 at 16:43
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    For basic numerical analysis, I'd guess understanding the role of Riemann surfaces in analytic continuation of finite difference and sinc (cardinal series) interpolation and summability of series and in understanding asymptotic series is a good start at least. Norlund, Poincare, and Berry wrote well on these topics. – Tom Copeland May 05 '20 at 17:20
  • Thank you so much. Do you have any book recommendation for self-study on the subject? – Jingeon An-Lacroix May 05 '20 at 17:44
  • I no longer have Klein's book, but because I have a background in mathematical physics and he was one of the last great mathemages (all keenly interested in physics), it's worth looking at. Klein very much promoted visual insight--see comments in my response to https://mathoverflow.net/questions/32479/what-are-some-mathematical-sculptures/252321#252321 – Tom Copeland May 06 '20 at 19:21
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    Check out the refs I've already given. Also https://math.stackexchange.com/questions/262677/what-was-klein-working-on-when-he-replaces-his-riemann-surface-by-a-metallic-su and https://mathoverflow.net/questions/313254/references-for-riemann-surfaces – Tom Copeland May 07 '20 at 18:56
  • @Copeland Thank you so much! – Jingeon An-Lacroix May 07 '20 at 20:42
  • Speak of the devil. Klein and Riemann surfaces are disucussed in the recent AMS article "Higgs Bundles—Recent Applications" by Laura Schaposnik (https://arxiv.org/abs/1909.10543). – Tom Copeland May 09 '20 at 17:59
  • and https://www.ams.org/journals/notices/202005/rnoti-p625.pdf – Tom Copeland May 09 '20 at 18:06
  • See also "Divergent series: taming the tails" by Berry and Howls for a glimpse at the interplay of numerical analysis, pdes, and Riemann sheets. – Tom Copeland May 11 '20 at 03:05
  • In support of my claim of interconnectedness, see my answer and the Papadopoulos and Elizalde refs in the comments in https://mathoverflow.net/questions/127601/does-the-derivative-of-log-have-a-dirac-delta-term/364456#364456. (Riemann was a master at numerical computations as well as an amazing theoretician.) – Tom Copeland Jul 02 '20 at 04:56
  • For superb visualizations and conceptual analysis starting from square one, see "Exploring Visualization Methods for Complex Variables" by Andrew J. Hanson and Ji-Ping Sha. – Tom Copeland Jul 31 '20 at 15:09
  • Nice motivating forward and mix of math and history in "Uniformization of Riemann Surfaces: Revisiting a hundred-year-old theorem" by Henri Paul de Saint-Gervais, et al. – Tom Copeland Apr 20 '21 at 01:44