# Tobias Ekholm tells about his research in symplectic geometry

Tobias Ekholm, born 1970, was awarded his PhD in 1998 for the thesis Immersions and their self intersections which deals with how to extract deformation properties of geometric objects by studying how they cross themselves.

“As a kid, I was interested in maths and physics. At the same time, it has never been a matter of course for me to be in mathematics, but rather it feels like a series of lucky circumstances have led me to the point where I am today.”

When Tobias began his university studies, he wanted to study physics or become a civil engineer.

“I started with mathematics and was taken by the subject. You could say I was caught by the way of thinking and reasoning, it was like coming home.”

Tobias devoured everything he came across in mathematics. In 1994, he began his doctoral studies. His supervisor was the Russian mathematician Oleg Viro. Viro had worked in the US and brought the subject area of geometric topology, “rubber space geometry”, to Uppsala University.

“It is geometry without lengths and angles, which may sound a bit strange, but it is about very basic geometric structures.”

After his dissertation, Tobias received a postdoctoral position at Stanford University, where Yakov Eliashberg was his supervisor.

“In connection with that, I started working on symplectic geometry, which changed the direction of my research. The area has close links with physics, such as string theory and quantum field theory.”

The next expansion of Tobias’ research came in 2012, when two physicists, Cumrun Vafa and Mina Aganagic, found a polynomial in knot theory that contains surprisingly much information about a knot’s deformation class. Tobias and the American mathematician Lenny Ng applied symplectic geometry in knot theory and found the same polynomial. Together, the four researchers succeeded in explaining the phenomenon both mathematically and physically.

“That collaboration brought me further toward the boundary between mathematics and physics. For me, they are not different areas, but different aspects of the same thing. Many of these things are interconnected. Areas in mathematics are very well connected, and in my work, I use large parts of mathematics.”

In 2014, Tobias started working with Maxim Zabzine at the Department of Astronomy and Physics with the Wallenberg project Geometry and Physics, which involves a close collaboration between mathematics and physics. The project is ongoing until 2020.

“I am still working on aspects of what I did in my dissertation. Nothing disappears, rather it is constantly being built on and expanded. I am grateful that Uppsala University has given me the opportunity to build a world-class environment in symplectic geometry.”

## Symplectic geometry

Symplectic geometry has its roots in Hamilton’s geometric description of classical mechanics. In the mid-1980s, the area was revolutionized by Gromov and Floer using holomorphic curve techniques, where geometric information is obtained through qualitative properties of solution spaces for generalized Cauchy-Riemann equations, which are certain systems of first order partial differential equations. Holomorphic curves are also encountered in topological string theory, and the boundary between geometry and physics’ string and gauge theories is an area of rapid development. Looking ahead, the development of topological M-theory and topological aspects of AdS/CFT are central areas with crucial issues remaining to be resolved.