Bernhard Riemann: Innovative Research in Topology

Riemann Surfaces and Research in Topology

Bernard Riemann surfaces, the Riemann function (for solving hyperbolic PDEs), and the Cauchy-Riemann equations in deriving an alternate physical plane where the independent variables are the stream function and a function analogous to the velocity potential function so that complex geometrical flow patterns are converted into ψ=constant and Φ=constant infinite rectangles so that I can set up my difference representations as well as integral formulae that can be used to solve any pseudo-Poisson equations. I also use Riemann’s advances in Analytic Continuation and much more.
 I often mention to my students that if one wants to become a Mathematical “superstar” overnight, then prove the Riemann Hypothesis: that the Riemann zeta function has its zeros that are only the negative even integers and complex numbers with real parts equal to 1/2. Bernhard proposed this in 1859 and is now the “holy grail” of Mathematical pursuits, which will give instant immortality to its conqueror.

Bernhard Riemann Contribution in Mathematics

I have been fortunate in my academic career to teach the Riemann integral (as well as the Lebesgue (1875-41) integral) and Riemann sums. I often marvel at the expressions of my students when they see that the integral of polynomials can easily be obtained using the well-known formulae for the sum of polynomials (which I also derive but not by induction) and Riemann sums.

Riemannian geometry

Riemann also founded Riemannian geometry in 1854 after Gauss (1777-55) asked him (as Riemann was his student), to prepare a Habilitationsschrift on the foundations of geometry, and this set the stage for Einstein’s (1879-55) general theory of relativity.
 
Riemannian geometry is a consistent non-Euclidean geometry (meaning that the fifth postulate of Euclid is relaxed); this step took centuries to occur due to the reverence held for Euclid’s (300 BC) work. Another such consistent geometry is that of Lobachevsky (1792-56).

Remembering Bernhard Riemann

I want to relay a little anecdote about Riemann and Gauss that I have been stating throughout my teaching career, spanning over 35+ years. When discussing Riemann, I always say to my students that it must have been very daunting for him to have Gauss as his Ph.D. advisor, only for one of my students at Oxford during a class reply, “It must have been scary for Gauss to have Riemann as his student.” That comment made me look at things from a completely different perspective (being a fan of Gauss, of course). As soon as I finished my Ph.D. and had more time on my hands, I purchased Gauss’ masterpiece Disquisitiones Arithmeticae (Latin “Arithmetical Investigations”), which deals with his passion for the primes and number theory, and this is a very interesting read.
 
I will end this discussion by stating how Stern (1807-94) described Riemann after he read Legendre’s (1752-33) masterpiece:
 
… “he already sang like a canary.”
 
Alas, I have run out of space again. Sorry, “old friend”.