Even prize money of one million dollars and 17 years of time have only been sufficient to solve one of the seven “millennium problems” in mathematics, which says a great deal about the difficulty of the prize-winning problems posedby the renowned Clay Mathematics Institute in Oxford, United Kingdom. Mathematicians want recipes for transforming complex problems into simpler ones. The new LOEWE research unit “Uniformized Structures in Arithmetic and Geometry”, in which teams of mathematicians at the Technische Universität Darmstadt and the Goethe University Frankfurt are combining their expertise, is dedicated to reaching this goal. “We want to achieve a critical mass together in order to produce fantastic results in less time”, says Professor Dr. Jan Hendrik Bruinier from the Department of Mathematics at TU Darmstadt. “The additional manpower will also help us to gain more visibility globally”, adds Professor Dr. Martin Möller from the Institute of Mathematics at the Goethe University.
Doughnuts and other objects
Mathematicians want to replace tricky geometric objects with significantly simpler ones – replacing a doughnut or a little more complicated shapes like a pretzel with a plane in each case. These examples can still be imaged quite vividly. Yet a geometric object can have many more dimensions for a mathematician than the three spatial dimensions that are conceivable to humans. Simplifying these shapes is an extremely complicated undertaking that will keep the renowned mathematicians in the LOEWE research unit busy for quite some time.
Yet why make all this effort? “Complex geometric objects represent solutions to difficult problems”, explains Bruinier. Roughly speaking, you could say that simplifying the geometry also makes the solutions that they represent easier to access. One example are so-called elliptic curves. They are graphical representations of difficult to solve equations that are given by so-called polynomials. If one considers instead of the polynomial equation y=x2 of the parabola the equation y2=x3+1 for example, one obtains an elliptic curve.
Elliptic curves are used millions of time a day for encryptions on the Internet, which are hard to crack due to their complexity. The first millennium problem the “Birch and Swinnerton-Dyer conjecture” also deals with elliptic curves. The geometric representation of such curves are circles that rotate on a doughnut through the hole in the middle. The researchers hope to use uniformization to find out more about “solution sets for polynomial equations” according to Bruinier.
Complexity of a billiard table
The opposite process of transforming supposedly simple planes into complex geometries can also prove fruitful. This is the case if the apparently complex geometry is well studied and mathematicians have already developed tools for dealing with it. The geometry of a billiard table is relatively straightforward: a flat plane. A ball runs across it on physically calculable paths. Nevertheless, it can quickly become complicated. For example, if one asks whether the path taken by a ball that has been hit once and continues rolling endlessly will evenly cover the table or not. The question becomes even more complicated if the surface of the table takes on another form such as an “L-shape” instead of its usual rectangular form.
This problem retains its symmetries if you turn the billiard table into a kind of pretzel: a tube with three handles. “We know of many processes in these types of “pretzel spaces”, says Möller about one advantage of this increased geometric complexity. The “pretzels” are also more flexible, according to Möller. They can be subjected, for example, to a shearing transformation without changing the processes significantly, , while such a transformation would change drastically the trajectories taken by a billiard ball.
The LOEWE research unit will also contribute to understanding these complicated spaces even better says Möller. The “pretzels” can take on many different forms and yet remain pretzels as long as they only have three holes. “But we still know very little about what other characterises the pretzels share with one another when viewed as a whole”, explains Möller.
The mathematicians working together with Bruinier and Möller now want to conquer this unexplored territory. Both are fascinated by numbers, arithmetic and how this subdiscipline of mathematics flows into other areas within their field of research. They even dare to hope that they can come closer to finding a solution to one of the millennium problems posed by the Clay Mathematics Institute.
What is uniformization?
A doughnut is a complex shape for a mathematician, if only because it has a hole in it. You can travel around the ring in two ways: around the edge of the hole and through the hole. In both cases you arrive back at the starting point. A flat plane would be much easier: no curvature, no holes and unlimited freedom of movement in all directions. Such a flat plane can now be made out of the doughnut – the “torus” in specialist jargon – by making two cuts: a first cut lengthwise around the hole and a second cut, transverse to the first, along the loop going through the hole. If the torus is now folded apart, it produces a parallelogram. read on…
If one imagines a small creature sitting on the torus, its direct environment will not be changed significantly by this unfolding process: The creature can still move in a plane in two directions as before. Yet if it moves further away, it will now reach one end instead of arriving again at its starting point as before. However, the circling process on the initial torus can be simulated on the plane by duplicating the parallelogram and then placing the copies next to each other like a tiled floor so that an endless surface is created. If the creature now crosses the boundary between two such tiles, it will reach a point on the new tile at some stage that corresponds to its starting point on the first tile.
Due to the mirror image nature of the tiles, the plane takes into account the complexity of the torus. The end result is a simpler surface , due to its uncurved nature, whose symmetry describes the paths the small creatures can take to arrive at its starting point again. For mathematicians, the plane and its symmetries are an adequate replacement for the torus – and represent its uniformization.
Even more complex geometrical objects that are studied by mathematicians in order to solve extremely difficult equations can also be simplified through uniformization. Examples are a double torus with two holes or a type of abstract pretzel, a curved surface with three holes. To a small creature, a double torus with two holes would look like an L-shaped billiard table, on which a calculation for how a billiard ball behaves can be an extremely complex task. chme
“Our research could help to improve IT security”
Interview with Professor Jan Hendrik Bruinier
The mathematical functions that you want to investigate using uniformization already have applications. What are they?
The secure encryption of data is based on the complexity of these functions and the error correction processes, as utilised when playing scratched DVDs or transferring data via satellite. Mathematics is the common language used by many sciences and thus some of the methods that we are researching also play a role in biology, such as in genealogy research, or in optimisation models in economics. read on…
How could your future results flow into these applications?
We conduct basic research and are still one step before the application. Yet our research could also be utilised for practical purposes. For example, the aim is to further improve encryption processes to increase the security of small mobile devices, bank cards or digital passports. New error correcting coding processes are constantly being developed as mentioned previously. Our methods could be used to investigate the efficiency of these processes. The sciences could also use our results as new mathematical tools.
How will potential users be made aware of your results?
We have good contacts here in Darmstadt with colleagues from the fields of computer sciences and the CYSEC profile area here at TU Darmstadt, such as with the cryptographers working with Johannes Buchmann. Some of our specialist colleagues with whom we publish our research also work for potential users such as Microsoft. The close links with physicists in Heidelberg, who also use the mathematical functions we investigate, could also simplify the transfer of knowledge for practical applications. Interview: Christian Meier
Jan Hendrik Bruinier, born in 1971, received his doctorate and qualified as a lecturer in Heidelberg. He was a professor in Cologne from 2003. Jan Bruinier has been a professor at TU Darmstadt since 2007. His research focuses on number theory, modular forms and algebraic geometry.
Physics can benefit from uniformization
Interview with Professor Nils Scheithauer
Professor Scheithauer, as a former theoretical physicist you are working on applications of uniformization in physics. Where do you see possible applications?
Mainly in string theory, in which elementary particles are replaced by 1-dimensional strings.
And in which physicists put great hope to unify the four fundamental forces?
Yes, that’s correct. A challenge string theorists face is to find a precise formulation for string theory, as we have for example for electromagnetism. We construct and classify the mathematical objects with which string theorists work, so-called vertex algebras, and restrict the playing field in this way. read on…
How is your work linked to uniformization?
Uniformization is a fundamental tool in the construction and classification of vertex algebras. Therefore, we benefit enormously from the exchange of ideas and knowledge with our colleagues both here in Darmstadt and in Frankfurt. Our integration into the LOEWE research unit will strengthen this exchange of ideas.
How will your results be disseminated to the string theory community?
We already have contact with several string theorists, for example in Hamburg and Zurich. In addition, we also interchange ideas at joint conferences. I will organize conferences as part of the LOEWE research unit to intensify the contacts. Interview: Christian Meier
Nils Scheithauer, born in 1969, obtained his Diploma in Mathematics and Physics and his Ph.D. in Theoretical Physics in Hamburg. After habilitating at the University of Heidelberg he became a Professor of Algebra at the TU Darmstadt in 2008. His main fields of research are automorphic forms, Lie algebras and vertex algebras.