In mathematical complex analysis, Rad\xf3's theorem, proved by Tibor Rad\xf3 (1925), states that every connected Riemann surface is second-countable (has a countable base for its topology)..
What is Riemann surfaces in complex analysis?
In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. An open subset of the Riemann surface for the multivalued function f(z) = log(z)..
Why are Riemann surfaces important?
Algebraic Geometry One important motivation for studying Riemann surfaces comes from algebraic geometry, which includes the study curves and surfaces defined by systems of polynomial equations. It turns out that the complexified version of an algebraic curve is often a Riemann surface..
Prerequisite courses: Topology (MA 231), Complex Analysis (MA 224), Introduction to Algebraic Topology (MA 232) or equivalent courses. Riemann surfaces are one-dimensional complex manifolds, obtained by gluing together pieces of the complex plane by holomorphic maps.
Complex geometry: Sheaf theory; several complex variables; Hodge theory. 6. Algebraic geometry: compact Riemann surfaces are the same as alge- braic curves.
Differential geometry: every Riemann surface carries a conformal met- ric of constant curvature. Einstein metrics, uniformization in higher dimensions. String
Dynamics. Unimodal maps exceedingly rich, can be studied by com- plexification: Mandelbrot set, Feigenbaum constant, etc. Billiards can be studied via Riemann
Complex analysis on riemann surfaces
German mathematician (1826–1866)
Georg Friedrich Bernhard Riemann was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rigorous formulation of the integral, the Riemann integral, and his work on Fourier series. His contributions to complex analysis include most notably the introduction of Riemann surfaces, breaking new ground in a natural, geometric treatment of complex analysis. His 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, is regarded as a foundational paper of analytic number theory. Through his pioneering contributions to differential geometry, Riemann laid the foundations of the mathematics of general relativity. He is considered by many to be one of the greatest mathematicians of all time.
In mathematics
Model of the extended complex plane plus a point at infinity
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value mwe-math-element> for infinity. With the Riemann model, the point mwe-math-element> is near to very large numbers, just as the point mwe-math-element> is near to very small numbers.
In mathematics
One-dimensional complex manifold
In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold.
