Complex dynamics and exotic geometry receive funding from mathematics program

Distinguishing simple from complex dynamics

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Eusebio Gardella, Senior Lecturer at the Division of Analysis and Probability Theory, receives funding for a project that deals with C*-algebras obtained from certain types of dynamical systems. “The workgroup in C*-algebras at our department is very active and has been growing significantly during the last three years. This grant from the KAW will allow me to hire an additional postdoc, which will make our workgroup one of the largest ones worldwide. This is an excellent opportunity for the mathematics department at GU/Chalmers to gain visibility and attract future talent. The proposed candidate, Julian Kranz, was a guest at the department for four months last year and would without a doubt make connections to the research of several members of the group.”

Many mathematical tools have been developed to describe change. These are often dynamical systems, which provide a mathematical description of how something changes or moves over time. The field is rooted in Newtonian mechanics, which can be used to describe the motion of the planets. For atoms and particles, the laws of physics are different from those of the Newtonian mechanics that governs celestial bodies – for example, it is not possible to simultaneously specify exact values for a particle’s position and its velocity. This and other laws that apply to the microworld are described by physics’ quantum mechanics, which has a mathematical formulation that differs from classical mechanics. Operators replace classical quantities such as position and speed.

One field of mathematics that provides quantum mechanics with a fitting theoretical framework is operator algebras, and particulary C*-algebras. Although they were originally concerned with atoms and particles, the theory of C*-algebras has evolved to include purely theoretical and highly abstract phenomena. One problem that is of vital importance for the theory of C*-algebras is to determine when a C*-algebra obtained from a dynamical system belongs to one of the types for which a great deal of progress has been made recently. A primary objective in the project is understanding the new classification of C*-algebras that arise from dynamical systems. What do we need to know about the dynamical system to be able to classify it?

Exotic geometry will shed light on a century-old conjecture

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Christian Johansson, Senior Lecturer at the Division of Algebra and Geometry, receives funding within the Langlands Program. “I am very much looking forward to starting work on this project, and it will be extra fun to do it in collaboration with a new postdoc. I think it will give the project a big boost and I really enjoy collaborating, it always brings new perspective.”

In January 1967, Robert Langlands, then in his early thirties, composed a letter to a fellow mathematician, André Weil, where he outlined an ambitious project in which he linked two completely different mathematical fields – number theory and geometry. He formulated a great many conjectures, which are hypotheses that have not yet been proven, although most experts believe they could be true.

Christian’s plan is to study L-functions, a vital component of the Langlands Program. L-functions provide information about Diophantine equations, for which only integer solutions are allowed for polynomials with integer coefficients. Despite being known since Antiquity, Diophantine equations are notoriously difficult, but not always impossible, to solve. Nor is it possible to determine in advance whether a given equation has a solution at all, which the Russian mathematician Yuri Matiyasevich proved in 1970. One way of using L-functions to approach Diophantine equations is to explore the 100-year-old but still unsolved Artin’s conjecture. This deals with basic properties of L-functions and was an important inspiration for Langlands’ ideas. The project aims to use exotic geometry to make progress in the exploration of Artin’s conjecture in particular, and the Langlands Program in general.

Grants to researchers at the department employed by Chalmers

Klas Modin, Assistant Professor at the Division of Applied Mathematics and Statistics, receives funding for a project which aims to gain a deeper understanding of the fundamental mechanisms underlying large-scale atmospheric and oceanic flow patterns that affect life on Earth.

Read more about Klas Modin’s research at the web of Chalmers University of Technology >>

Jiacheng Xia, who received his doctoral degree in mathematics from Chalmers University of Technology in 2021, will hold a postdoctoral position at the University of Wisconsin-Madison, USA. One of the project’s main aims is to study the geometric rigidity of certain objects with high symmetry, which is of interest in number theory.

Read more about Jiacheng Xia’s research at the web of Chalmers University of Technology >>