Grants for string theory, geometric objects and the Langlands program
The Mathematics Program of Knut and Alice Wallenberg Foundation this year grants SEK 35 million to 16 researchers, three of them doctoral students at the Department of Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg. Rolf Andreasson, Douglas Molin and Ludvig Svensson all receive international postdoctoral positions and funding for two years after they return to Sweden.
Finding the correct measure for abstract geometric objects
Rolf Andreasson will hold a postdoctoral position with Professor Cristiano Spotti at Aarhus University, Denmark. In his project geometric equivalents of the solutions to algebraic equations will be investigated through one of their metrics, the Weil–Petersson metric.
– I got in touch with the group in Aarhus and Cristiano Spotti through the seminar series we have had between the universities in recent years. The expertise there both overlaps and complements my research interests in a good way. It means incredibly much to have the opportunity to work there, and I am really looking forward to it.
Algebraic geometry is the study of varieties, which are the geometric equivalents of the solutions to algebraic equations. A simple variety is a circle with radius r, given by the solutions to the equation x²+y²=r². However, varieties can be considerably more abstract geometric objects. One way of organising them is to assign to each object a point in a moduli space, that parametrizes all objects of the same type and where similar objects are close to each other.
One tool for investigating varieties is to study their metrics – ways of measuring distance in space. One well-known metric is the Kähler–Einstein metric, which can also be used to define a natural metric on the moduli space: the Weil–Petersson metric. However, in most cases neither the Kähler–Einstein nor the Weil–Petersson metric can be described explicitly.
The project will develop a method for constructing an explicit approximation of the Weil–Petersson metric, inspired by a previous approximation of the Kähler–Einstein metric. The method originates in statistical mechanics, the study of large systems of interacting particles. As the number of particles increases, the Kähler–Einstein metric emerges from the system’s collective behaviour.
The aim is to show how the Weil–Petersson metric can also arise from the same particle system. The partition function, a fundamental concept in statistical mechanics, plays a central role. This function gives rise to a new and more explicit metric on the moduli space.When the number of particles increases, there is reason to believe that this metric approximates the Weil–Petersson metric, thus providing a new way of understanding it.
New contributions to the Langlands program
Douglas Molin will hold a postdoctoral position with Professor James Newton at Oxford University, United Kingdom. The planned research is part of one of the most comprehensive and influential projects in mathematics, the Langlands program.
– It will of course be a lot of fun to spend two years in Oxford. It is also great that several of us from Gothenburg have received scholarships.
The Canadian mathematician Robert Langlands formulated in a letter from 1967 a web of profound and far-reaching conjectures, which linked apparently disparate areas of mathematics, such as algebra, geometry, harmonic analysis and number theory. Ever since, it has been known as the Langlands program.
One decisive breakthrough in the Langlands program arrived with the proof of a famous conjecture from the 17th century, which was formulated by the French mathematician Pierre de Fermat. He claimed that the equation xn+yn=zn,has no positive integer solutions for n greater than 2, but did not provide any proof. Over 350 years passed until Andrew Wiles, with vital help from his former doctoral student Richard Taylor, succeeded in proving Fermat’s theorem. This built upon a deep link between elliptic curves in algebraic geometry and modular forms in complex analysis – two objects whose symmetries play an important role in the proof.
One natural continuation is to attempt to unite these two types of symmetries for more general objects in geometry and analysis. On one hand, arithmetic symmetries are studied through Galois representations, algebraic structures that are named after the 19th-century French mathematician Évariste Galois; on the other, modular forms are generalised to automorphic forms – analytical functions with intricate, almost kaleidoscopic symmetry properties.
Many mathematicians have worked on these generalisations of Wiles’ results. The research in the planned project deals with more intricate generalisations, where methods from homotopy theory have proved to be useful. In the spirit of the Langlands program, the aim is to establish new connections in situations that are far beyond the classical link between elliptic curves and modular forms.
Models for the hidden dimensions of string theory
Ludvig Svensson will hold a postdoctoral position with Professor José Ignacio Burgos Gil at the Institute of Mathematical Sciences, Madrid, Spain. The project’s research area is complex analysis and its intersection with mathematical physics. Special geometric objects, Calabi–Yau manifolds, are of particular interest, and have been shown to provide solutions to the field equations in Einstein's general theory of relativity.
– I am delighted and grateful, and I am really looking forward to becoming part of the research group at ICMAT. The opportunity to work in an inspiring environment with strong expertise in mathematical physics and arithmetic geometry creates excellent conditions for synergies with my own research.
Calabi–Yau manifolds gained widespread attention through their role in modern physics’ string theory, which aims to unite quantum physics with general relativity. However, string theory predicts a ten-dimensional spacetime, while our world is essentially four-dimensional: three space dimensions and one time dimension. The remaining six dimensions of string theory are therefore thought to be so tiny that they are hidden to us. Calabi–Yau manifolds describe the geometry of these hidden extra dimensions. The manifolds also display mirror symmetry: each manifold has a mirror partner, and even if the two manifolds in the pair are geometrically different, they give rise to the same physics.
Of particular interest in mirror symmetry are situations in which the geometry of Calabi–Yau manifolds degenerates. This can be investigated using mathematical objects called period integrals. The period integrals associated with a Calabi–Yau manifold contain rich information about its geometry and arithmetic.
When the Calabi–Yau geometry degenerates, the corresponding period integrals often become divergent, meaning they are infinite. Despite this, it is possible to extract a finite part from them. The project aims to investigate these finite parts of divergent Calabi–Yau period integrals and determine the extent to which they still have interesting arithmetic and/or geometric information, comparable to that of their convergent counterparts.
The mathematics program is a long-term investment funded by Knut and Alice Wallenberg Foundation where the Royal Swedish Academy of Sciences assesses the candidates. SEK 650 million will be granted to Swedish mathematics research between 2014 and 2030. Every year, a number of mathematicians are given the opportunity to conduct research at prestigious institutions outside Sweden, whilst excellent mathematicians from abroad come to Sweden.