Geometry is the study of spaces and objects within them. The word symplectic is of Greek origin and means complex. However, in mathematics, the word complex is already used in many other contexts, e.g. in the term complex numbers. Symplectic spaces were devised to formulate classical physics.

Symplectic geometry can be used to understand the orbits of celestial bodies, whose movements follow the laws of physics. In this context, the coordinates of the symplectic space describe not only the position of the body but also its velocity. This space is always of an even dimension, since as many coordinates are needed to describe velocity as position.

“The fascinating thing is that symplectic geometry has been found to be relevant for understanding many other mathematical phenomena, especially within other types of geometries, not just for understanding physical systems,” says Dimitroglou Rizell, an associate professor of mathematics at Uppsala University.

His office is on the fourth floor of the original Ångström Laboratory building and offers a panoramic view over the Uppsala plain. On one of the walls hangs a modern digital blackboard that “remembers” what he and the researchers in his team have drawn on it. The board is used to look at drawn visualizations of symplectic spaces.

Dimitroglou Rizell enjoys thinking in images.

“It feels natural for me to tackle problems in a visual way.”

In classical geometry, shapes – circles, triangles, and rectangles – are rigid. They have fixed linear forms. But in topology, the niche in which Dimitroglou Rizell conducts his research, geometric structures can also be soft or elastic – as though they were made of rubber.

Solving problems

One example of objects that can be embedded in and studied through symplectic spaces is knots. Geometric knots resemble a tied and tangled string, whose ends have been joined to form a closed loop.

This knowledge can be used as a model to understand cells and proteins and their structures.

“A protein sometimes resembles a tangled knot. For instance, a biologist might need to understand how the properties of this knot affect the protein’s function and require mathematical tools to do so.”

Symplectic spaces appear not only naturally in molecular biology but also in other research fields.

For researchers, it is essential to understand the spaces and their various properties. But identifying all structures within these spaces is a major challenge, since they can be extremely complex.

Standard geometry, in which areas, curves, and angles are calculated, often isn’t enough.

“Geometric structures are often complicated and difficult to describe and understand. So, for example, if we want to study classical knots using symplectic spaces, the question remains how we are to understand the symplectic space,” he says.

Using algebra

One way to do so is to use algebra – a branch of mathematics that utilizes formulas, equations and algorithms. One of Dimitroglou Rizell’s goals is to describe the spaces in a computable way using the language and methods of algebra.

At present we don’t fully understand four-dimensional spaces, but we understand even less about six-dimensional symplectic spaces and above. I would like to find tools to understand more about these spaces in higher dimensions as well. But that’s a whole other mystery to solve.

“It’s not easy, but algebra is essentially more computable than a space.”

Specifically, he and his team are exploring an algebraic invariant – a kind of constant that measures properties of the space – called the Fukaya category.

This is a theoretical tool that researchers can use to perform calculations to distinguish different symplectic spaces.

“If two symplectic spaces are equivalent, their Fukaya categories are also equivalent. This constitutes an invariant that allows us to determine whether two symplectic spaces differ. In this way, we can study whether two spaces are equivalent or not as an algebraic problem.”

Dimitroglou Rizell’s project as a Wallenberg Scholar is to further develop the invariant to measure and distinguish more aspects and properties of these spaces.

Numerous applications

New tools to understand more symplectic spaces are needed, not least in modern mathematics and cell biology.

But Dimitroglou Rizell’s vision includes tackling the truly major research problems that remain unsolved in his field.

The spaces in symplectic geometry are abstract mathematical objects whose dimensions do not necessarily correspond to any tangible worldly phenomenon. Symplectic spaces have an even dimension, and in low dimensions – i.e., two and four – we understand much more than we do in higher-dimensional spaces when it comes to classification, for example.

“At present we don’t fully understand four-dimensional spaces, but we understand even less about six-dimensional symplectic spaces and above. I would like to find tools to understand more about these spaces in higher dimensions as well. But that’s a whole other mystery to solve.”

Text Monica Kleja
Translation Maxwell Arding
Photo Magnus Bergström