# Has the 140-Year-Old Schwarz-Christoffel Math Problem Finally Been Solved?

• Diffy
In summary, a researcher at Imperial College London has solved the Schwarz-Christoffel formula, which has been a challenge for mathematicians for 140 years. This formula maps a complex plane region to a polygon in a conformal manner, but was previously limited to simply connected polygons. The new formula allows for the mapping of multiply connected polygons, such as regions with holes. The paper detailing this breakthrough can be found at the provided links.
Diffy
I am not entirely sure about the details but thought I would share...

A problem, the Schwarz-Christoffel formula which has defeated mathematicians for almost 140 years has been solved by a researcher at Imperial College London.

More detail:
he Schwarz-Christoffel mapping maps a suitably "nice" (such as the interior of the unit circle, or the upper half plane) region of the complex plane (numbers of the form x+iy, mapped as (x,y) in cartesian coords, where i^2 = -1) to the interior of a general polygon (in the complex plane) in a conformal manner (angles are preserved). However, the polygon must be "simply connected" (any loop can be smoothly shrunk to a point, without getting "caught" around a hole); a few "multiply connected" polygons were possible, but not in the general case. Now it appears the formula can be extended to include polygons that include such a hole (such as perhaps the area outside a small square, but inside a larger one), to be mapped to some member well-described family of regions like the exterior of several unit disks
http://sinews.siam.org/old-issues/2008/januaryfebruary-2008/breakthrough-in-conformal-mapping

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Here is the paper: http://wwwf.imperial.ac.uk/~dgcrowdy/PubFiles/Paper-20.pdf

Schwarz–Christoffel mappings to unbounded multiply connected polygonal regions
Abstract said:
A formula for the generalized Schwarz–Christoffel conformal mapping from a bounded multiply connected circular domain to an unbounded multiply connected polygonal domain is derived. The formula for the derivative of the mapping function is shown to contain a product of powers of Schottky–Klein prime functions associated with the circular preimage domain. Two analytical checks of the new formula are given. First, it is compared with a known formula in the doubly connected case. Second, a new slit mapping formula from a circular domain to the triply connected region exterior to three slits on the real axis is derived using separate arguments. The derivative of this independently-derived slit mapping formula is shown to correspond to a degenerate case of the new Schwarz–Christoffel mapping. The example of the mapping to the triply connected region exterior to three rectangles centred on the real axis is considered in detail.

## 1. What is the 140 year old math problem?

The 140 year old math problem refers to the "Twin Prime Conjecture" which was proposed by French mathematician Alphonse de Polignac in 1849. It states that there are infinitely many pairs of prime numbers that differ by 2, such as 3 and 5, 17 and 19, and 41 and 43.

## 2. Why is this math problem important?

The Twin Prime Conjecture is important because it is a fundamental problem in number theory and has been unsolved for over 140 years. It also has applications in cryptography and computer science.

## 3. Who solved the 140 year old math problem?

In 2013, mathematicians Yitang Zhang and James Maynard independently proved that there are infinitely many pairs of primes that differ by at most 70,000,000. This was a significant step towards solving the Twin Prime Conjecture.

## 4. How was the 140 year old math problem solved?

Zhang and Maynard used a combination of techniques from analytic number theory and combinatorics to solve the problem. They developed new methods and refined existing ones to make progress towards the conjecture.

## 5. Is the 140 year old math problem completely solved?

No, the Twin Prime Conjecture is not completely solved. While Zhang and Maynard's work proved the existence of infinitely many pairs of primes that differ by at most 70,000,000, the conjecture still remains open for an exact bound. However, their breakthrough has opened up new avenues for research and brought us closer to solving this long-standing problem.