World leading mathematicians to solve mysteries at the interface of geometry and topology
An array of fundamental and unanswered questions in mathematics lie at the intersection of geometry and topology. With a 60 million kroner grant from the Danish National Research Foundation to establish a new research center, University of Copenhagen mathematicians hope to solve a few of these decades-old problems.
For a bird, there is only one shortest route between Copenhagen and New York, while there are infinitely many shortest routes between the North and the South Pole. This is because the earth is round, not flat. But what happens if one instead can only travel by boat or road? Or, what if "shortest" is to be interpreted as "cheapest" or "least CO2-emitting"? These are a few of the questions that a group of mathematicians from the University of Copenhagen in Denmark will strive to answer.
In abstract terms, it is about optimizing a geometric problem: "the shortest path". And the example of shortest paths from the North Pole to the South Pole shows that the answer is closely related to the basic "form" or topology of the object considered.
A team of world leading mathematicians from the University of Copenhagen’s Department of Mathematical Sciences, headed by Professor Nathalie Wahl, will, with a 60.2 million kroner grant from the Danish National Research Foundation, establish a new research center geared to address fundamental scientific problems such as these that lie at the interface of the fields of geometry and topology.
From forest fires to airbags
Despite being fundamental mathematical research, there are plenty of potential applications. The 43-year-old Oxford-educated mathematician Nathalie Wahl explains:
"While these mathematical questions are abstract in essence, they are also applicable in practice in numerous contexts. For example, if you seek to calculate how a forest fire will spread, how an airbag inflates, or how space expands – each of these things can be considered as geometric objects that change shape over time. They are all controlled by the same types of equations."
The combination of geometry and topology has seen groundbreaking results over the past decade – not least by Nathalie Wahl and her colleagues at the University of Copenhagen.
"The University of Copenhagen is already among the world’s best in the field of topology. This grant gives us the chance to attain a similar level in geometry. By bringing the disciplines nearer to one another, we expect to achieve significant progress in solving fundamental problems at the overlap of these two branches of mathematics," says Nathalie Wahl.
The research team will include the topologist Søren Galatius who was recently recruited from a full professorship at Stanford and the geometer Tobias Holck Colding, who is the Cecil and Ida Green Distinguished Professor at MIT.
"The grant will also allow us to attract the best PhD students and postdocs from around the world, to work in a fun and exciting research environment that can compete with the biggest players like Harvard, MIT, Stanford, and Oxford."
The Copenhagen Center for Geometry and Topology (GeoTop) will open in 2020. It will be located at the Department of Mathematical Sciences at the University of Copenhagen.
- The new center is a so-called Center of Excellence funded by the Danish National Research Foundation (DNRF). The foundation has granted 60.2 million kroner to support the center over a six-year period, with the possibility of an extension.
- The core research team consists of the topologists Nathalie Wahl, Søren Galatius and Oscar Randal-Williams, as well as the geometers Tobias Holck Colding and Niels Martin Møller.
- Geometry is concerned with the study of forms, taking into account characteristics such as distances, angles and curvature.
- Topology is concerned with the study of more basic properties of forms, such as holes, that are preserved when the form is stretched or bent.
- The researchers aspire to answer fundamental questions in geometry and topology. Their research objectives are structured around three mathematical concepts: moduli, geodesics, and singularities.