Mathematical precision through intuition

Intuition is a huge asset for people wanting to navigate their way around the world of mathematics. That, at any rate, is the opinion of Georgios Dimitroglou Rizell, researcher in mathematics at Uppsala University. As a Wallenberg Academy Fellow, he is tackling further development of symplectic geometry, a hot topic nowadays, with close links to physics and string theory.

Georgios Dimitroglou Rizell

PhD Mathematics

Wallenberg Academy Fellow 2016

Institution:
Uppsala University

Research field:
Contact and symplectic topology/geometry, in particular the study and classification of Lagrangian subspaces.

Rizell is ensconced in his office at the Ångström Laboratory. The view from his window stretches out over the plains extending towards the farmland close to a parish church. But his work as a researcher in mathematics has little to do with geographical location. The important events take place in his mind.

“One fascinating thing about mathematics is that it’s always with you. You only have to use your head to visit the world of mathematics,” Rizell says.

This fact has not prevented the young mathematician from sojourns in several international research environments after receiving his PhD from Uppsala University in 2012. Among other things, he has worked as a postdoc in Brussels and Paris, after which he spent a further two years in that role at Cambridge in England. The position was funded by the Knut and Alice Wallenberg Foundation under the auspices of the Wallenberg Academy Program in Mathematics.

“Cambridge is one of the most advanced environments in the world. A very high standard is maintained in all seminars, and you know that the ideas being discussed are always very close to the cutting edge of research.”

Now he has returned to his alma mater, Uppsala University. He stresses that the standard of research has also come on in Sweden, one reason being strategic initiatives to boost the quality of Swedish mathematics.

Concepts rooted in the 19th century

Rizell specializes in symplectic geometry, a term with roots in the 1800s, when the Irish mathematician Hamilton reformulated the equations for Newton’s classical mechanics, for example those governing planetary motion. The formulas were given a geometric language that made them easier to solve, and the characteristics of the solutions were made easier to describe.

When classical mechanics later evolved into quantum mechanics, researchers once against adopted the Hamiltonian approach to simplify the equations.

But it was not until the 1980s that people began to realize the huge potential in researching symplectic geometry for its own sake. A breakthrough was made when the Russian-French mathematician Gromov published his findings.

“Since then, most of us working in this field have used his method to try to understand as much as possible of the symplectic structure.”

Border zone between mathematics and physics

Rizell is researching on the borders of theoretical physics, in an area that is currently one of the most central and flourishing mathematical fields.

“Symplectic structures offer riches in several respects. They are deep, difficult to get a handle on, and have been found to decode important information about various mathematical phenomena. Modern symplectic geometry also has applications for understanding Hamiltonian dynamical systems, which are concerned with the original application of the theory.”

Among other things, study of symplectic spaces can aid understanding of how bodies move and the form that trajectories take. Rizell is also continuing to build on the theory by studying and classifying something known as Lagrangian subspaces. These are subspaces whose properties are central to the understanding of the surrounding symplectic space.

But there are also links to the string theory in physics, a model designed to explain what the world is made of. Here, there is a link to a mathematical discipline called topology, and the study of knots in particular.

Mathematical knots may be described as resembling tangled shoelaces or rope glued together at both ends. There is an infinite variety of knots, and one of the greatest challenges is to find new calculable methods to distinguish between them. There are numerous applications, including string theory and quantum theory in physics, as well as study of protein and DNA, for example.

“Physicists study the same spaces as we do, and there are certain physical principles that inform their mathematical reasoning. This can sometimes open mathematical doors. I see the close relationship to physics as a source of inspiration, and am constantly trying to learn more.”

“I’m delighted to have been admitted as a Fellow – and very grateful. It’s really gratifying personally to have been acknowledged, but most of all, it means a lot to receive funding to develop research at Uppsala. Among other things, we’ll be able to recruit more PhD students to the team, adding further strength to the research environment.”

Intuition a great asset

Mathematicians meet at seminars and conferences throughout the world, and new research articles are often authored by two or more colleagues working together.

“Once the problems to be solved have been identified, it’s a good idea to have access to many minds. This allows a collective approach, where the main goal is to achieve a common understanding of an entire mathematical field.”

But solitary rumination remains an essential part of research.

“Sooner or later you have to sit down with a pen and paper and give yourself the time just to think and outline ideas. Intuition is key in this process. Years of experience of the subject give you intuition. You get to know the world of mathematics, how things fit together, and what phenomena are controlled by other phenomena. If you follow the right path, you will ultimately be able to write down precisely formulated mathematical proof, where the argument can be followed line by line, just like in a computer program.”

Text Nils Johan Tjärnlund
Translation Maxwell Arding
Photo Magnus Bergström