Mathematics creates order in the universe (interview)

ATS, 17 May 2022 – Mathematician Franc Forstnerič, a professor at the Faculty of Mathematics and Physics in Ljubljana, is one of three Slovenians to have won the prestigious ERC Advanced Grant for Established Researchers this year. Mathematics is about creating order in the universe, says the winner of the first ERC project for mathematics in Slovenia.

Forstnerič received a five-year European Research Council (ERC) grant worth almost €1.5 million for his project titled Holomorphic Partial Differential Relations, which aims to deliver new methods and findings in the domain of Oka varieties and more generally of oriented holomorphic varieties. systems.

In 1985, Forstnerič completed his doctoral thesis on holomorphic maps in several complex variables at the University of Washington in Seattle, USA.

During his studies, he was introduced to the Oka-Grauert principle, which deals with the existence and properties of holomorphic maps of certain classes of complex manifolds. At the time, he lacked the “mathematical maturity” he now needs to build on the principle, he told the STA.

After obtaining his doctorate, Forstnerič first returned to Ljubljana and then undertook several extended research stays abroad. In 1997, he returned to Ljubljana, determined to work intensively on the Oka-Grauert principle.

“In 1989, an important article on the subject was published by the eminent Franco-Russian mathematician Mikhael Gromov, winner of the Abel Prize in 2009. Gromov put the theory on new bases, introduced new techniques and suggested other possible developments, but he did not present detailed evidence.

“Gromov is a brilliant mathematician who has contributed key new ideas in a number of mathematical fields, but he often leaves the detailed arguments and further development to others,” Forstnerič said.

He involved his doctoral student Jasna Prezelj in the research, and together, within a few years, they managed to make a crucial breakthrough in understanding Gromov’s ideas.

Forstnerič then worked on the problem of characterizing the class of complex varieties to which the results of the theory apply. In 2006 he characterized this class by a simple “convex approximation property” and by a number of other properties which were obviously not equivalent to each other.

“This manifold property means that any holomorphic mapping of a convex set in a complex Euclidean space can be approximated by holomorphic mappings of the entire Euclidean space in a given manifold,” he explains. This solved one of Gromov’s key problems, and within a few years a complete theory emerged.

Based on this, Forstnerič introduced a new class of complex varieties into the literature in 2009, which he named Oka varieties after the originator of the theory, Japanese mathematician Kiyoshi Oka (1901-1978).

Manifolds are geometric objects such as curves and surfaces. “The world we live in is a variety,” notes the Slovenian mathematician, adding: “We live on a sphere; the sphere, the galaxies, the universe, these are all varieties.

Complex manifolds always have an even number of dimensions. “There is an additional structure that defines a special class of mappings between these varieties – holomorphic mappings.”

One of the reasons holomorphic mappings are important is that they occur naturally in physical problems. “For example, if you want to design an airplane wing, you have to study laminar flow. The wing is located in an airflow, this air will bounce, the wing will change direction, and this will cause buoyancy.

“It’s what keeps the plane in the air. But when you want to model how that airflow is going to flow around the wing, you draw a shape and then you have to calculate what’s going to happen. is simpler to use conformal mapping to map this wing shape to a circle. After you map it to a circle, you have explicit laminar flow solutions that avoid the circle. Then you map those solutions using the conformal mapping. It is a simple application of these mappings” says Forstnerič.

His theory of Oka received significant recognition in 2020. “Every 10 years, the American Mathematical Society renews the classification of mathematical domains in cooperation with the German journal Zentralblatt fur Mathematik. There was no suitable domain for this theory, so we proposed it and it was accepted.

“They introduced a new field called Oka Theory and Oka Manifolds. This is my contribution to classification. As far as I know, this is the second such case in mathematics in Slovenia,” he noted. .

His work is also fascinating in that he helped bring the theory of this type of complex manifolds back to Japan after 80 years. “My main contribution has been to conceptualize the theory and thus make it more widely applicable.”

The ERC project awarded to Forstnerič will enable him to expand his research in this area and pave the way for the existence of solutions to a number of complex problems in analysis and geometry as well as other areas of mathematics and beyond.

This will also allow him to build an international team that will include three or four other researchers. The project will be carried out at the Faculty of Mathematics and Physics in Ljubljana.

Forstnerič’s work inspired the Japanese mathematician Yuta Kusakabe, who managed to make important breakthroughs in this field in his doctoral thesis in 2020.

“I invited him to Slovenia. He has a young family, so he can’t come at the moment, but as the project will last for five years, I hope that during this time he can take a sabbatical and come here. .,” said Forstnerič, adding that he was happy to be joined by another established researcher, Rafael Andrist.

In science, it is very important to introduce a new concept at the right time, he said. “Examples may have been discussed before, but once you introduce a relevant concept and show that it has many different characterizations that all lead to the same goal, then it can become the seed of a new theory. .”

This requires good knowledge of a specific scientific field, the ability to detect and abstract key features, and a bit of serendipity, he added.

“Mathematics somehow creates order in the universe. It’s not just calculations. You have to establish a concept and, based on that concept, develop a theory. Once you have the right concept , you can develop it further, but until you figure it out, everything is a bit hazy,” Forstnerič said.

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