# An Interview with Ann Varela- Great Mathematicians–Bernhard Riemann–Precursor to Einstein?

Michael F. Shaughnessy –

1) Ann, the name of Bernhard Riemann is somewhat linked to relativity. How close was he to being a precursor to Einstein?

Bernhard Riemann did extraordinary work on combining analysis with geometry. Riemann developed his theory of higher dimensions and delivered a lecture at Gӧttingen entitled *On the Hypotheses Which Underlie Geometry*. Although it was published two years after his death, it is now accepted as one of the most significant works in geometry.

Riemann’s work found the precise way to extend into *n* dimensions the differential geometry of surfaces. He was able to designate spatial relations as a series of number coordinates at every point in space. These coordinates described how much curvature was present.

2) Apparently, he also studied under Carl Friederich Gauss. Is this correct? (I often find it interesting to follow the lineage of individuals who have studied under other great minds.)

It is true! Riemann did study under the direction of Carl Friederich Gauss at Gӧttingen in 1846. Bernhard Riemann attended lectures given by Gauss for one year before he transferred to the University of Berlin. Apparently, Gauss only taught the lower level applied mathematics courses at Gӧttingen, so Riemann felt the need to transfer to another university. The University of Berlin afforded Riemann the opportunity to study under prodigious mathematicians such as Jacobi, Dirichlet, and Eisenstein.

In 1849, Riemann returned to Gӧttingen because Wilhelm Weber had returned as the chair of physics. This meant Riemann could work on his dissertation under the tutelage of Gauss. Riemann completed his dissertation in just over a year. The topic of Riemann’s dissertation was the foundations of complex variable analysis. Within the body of his dissertation, Riemann discussed the topics of conformal mapping and simple connectivity. The two main results of his dissertation were the Riemann mapping theorem and the introduction of Laurent series expansion for functions having poles and branch points.

3) He worked with hypergeometric differential equations. While I understand hyper to mean more or excessive- and I have some understanding of geometry- how does this fit into differential equations?

The Riemann differential equation is a simplification of the hypergeometric differential equation, permitting the regular singular points to occur anywhere on the Riemann sphere, not just at zero, one, and infinity. What is a regular singular point? *Singularity* is a point at which a given mathematical object is not defined. For example, when considering the real function * f(x) = 1/x*, we notice that the function has a singularity at * x = 0*. Therefore, we say the function is undefined at an x-value of zero. A *regular* singular point has solutions bounded by an algebraic function.

4) Riemann was involved in lecturing in 1854 and this apparently established the realm of Riemannian geometry- and many feel this was the foundation for Einstein’s theory of relativity- your thoughts?

Albert Einstein was working on his general theory of relativity in the early 1900’s. Supposedly, one imperfection of his theory was that gravitation, or acceleration, was omitted from his manuscript. Paul Ehrenfest discussed a curious observation with Einstein that might have assisted with including the overlooked terms. That observation pertained to a spinning disk, whose rim travels at a faster rate than its center, now known as the Ehrenfest paradox. See Figure 1. Einstein required more time to better understand gravity in terms of the curvature of space-time. He spent another ten years of research on his theory of relativity and collaboration with David Hilbert before it was ready to be published in 1915.

Figure 1. Ehrenfest Paradox.

So where does Riemann come into the picture? Well, Riemann had unique ideas on geometry, which included the case that the essential components for geometry are a space of points and a way of measuring distances along curves within the space. He proposed that the space did not need to be ordinary Euclidean space (where coordinates designate points) and that it could have any dimension. Riemann saw that geometry should not automatically deal with points, lines, or space in the usual sense.

Riemann’s work in geometry laid the mathematical foundation for Albert Einstein’s theory of relativity in that it explained the four-dimensional geometry of space-time. Riemann wished to support the notion of space being capable of transmitting forces such as electromagnetism and gravitation.

5) What is the Riemann mapping theorem?

The Riemann mapping theorem is used in complex analysis. The Riemann mapping theorem states that if* U* has elements in its set and is a *simply connected open subset* of the complex number plane *C *then there is a * biholomorphic* mapping from *U *onto the open unit disk. Now, let me try to break down some of this math jargon into more understandable terms. In order to be * simply connected*, an object must consist of one piece and cannot have any holes that pass all the way through it. Examples of *simply connected *objects include a disc and a ball. Objects like donuts and coffee carafes with handles are * non-simply connected *or *multiply connected*.

*Open subsets* can be described in fairly uncomplicated terms by using a shaded-in circle, as seen in Figure 2. The black outline of the circle represents the set of points (x, y) satisfying the equation of the circle x^{2 }+ y^{2 }= r^{2}. The grey-shaded disc represents the set of all points (x, y) satisfying x^{2 }+ y^{2 }< r^{2}. The grey set is an *open set*, the black set is its * boundary set*, and the union of the grey and black sets is a * closed set*.

Figure 2. Disc with Boundary.

©Ann F. Varela

*Biholomorphic* refers to a function that has a one-to-one correspondence between the elements of two sets. In other words, each element from one set pairs with only one element from another set. There are no unpaired elements. The function also consists of complex-values and at least one complex variable that has complex derivatives that consist of a collection of points within the set. Figure 3 shows how a set V in the plane is a neighborhood of a point P if a small disc around p is contained in V.

Figure 3. Neighborhood.

©Ann F. Varela

An illustration of the Riemann mapping theorem is shown in Figure 4.

Figure 4. Riemann Mapping Illustration.

https://upload.wikimedia.org/wikipedia/commons/e/e9/Illustration_of_Riemann_Mapping_Theorem.JPG

6) Set theory – and number theory–What did Riemann have to do with these?

In the subject of number theory, Riemann studied prime numbers, more specifically, the distribution of prime numbers. Riemann wanted to establish how many prime numbers are less than any given number *n*. His most renowned contributions to contemporary analytic number theory is the examination of the zeta function and how it relates to the distribution of prime numbers.

The work Riemann explored relating to set theory consisted of combining analysis with geometry. Riemann’s proposal was to introduce a collection of numbers at every point in space, which would describe how much it was twisted or warped. He found that a minimum of ten points were necessary to describe the properties of a manifold (a set of points, along with a set of neighborhoods for each point, satisfying a set of principles relating points and neighborhoods).

7) What are these Riemann surfaces- and why are they important?

Riemann surfaces are surface-like structures that cover the complex plane with several layers, which can have very complicated arrangements and interconnections. In Figure 5, the two horizontal axes represent the real and imaginary parts of the function’s dependent variable, *f(x),* and the vertical axis represents the real part of the function’s independent variable, the square root of *x*.

Figure 5. Riemann Surface for Function Representing the Square Root of * x*.

The role of the Riemann surface is to study complex surfaces. Riemann showed how such surfaces could be classified by a number that is determined by the greatest number of closed curves that can be drawn on the surface without breaking it into separate pieces.

8) What would you say are his most important contributions?

Riemann contributed to many interesting fields. His theory of surfaces outlines the foundation of topology and remains a critical element to understanding mathematical physics. His contribution to spatial relations is literally out of this world, as he introduced a series of number coordinates at every point in space, which represented how much curvature was present. It is also fascinating how Riemann was able to alter Gauss’s differential geometry of surfaces to work in multiple dimensions.

9) What is he most known for today?

Riemann’s hypothesis and zeta function are quite interesting and show that Gauss’s preliminary approximations of the distribution of prime numbers were more precise than imagined. Riemann’s zeta function and the relationship of its zeroes to the prime numbers continues to be of interest to scholars. In fact, the Riemann Hypothesis has not actually been proven all these years later. There is a one million dollar prize offering for the final solution.