# The Geometry / Topology Group

## What is topology and geometry?

Would you like to understand the shape and structure of

• the physical universe that we live in,
• evolutionary processes in biology,
• networks in computer and information science,
• big data sets?

The industry needs people who can compute homology effectively! Our group offers courses and projects where students can learn methods to solve such challenges.

Homology is involved when medical doctors do brain scans, and when mobile-phone companies search for holes in their network coverage. The phones themselves use geometry to analyse the photos that you take with them.

Topology and geometry are fundamental disciplines of mathematics that are both concerned with the study of shapes and the concept of space. They are indispensable tools for research on these questions, and have found a multitude of applications.

Topological structures such as one-sided Möbius strips, knots and braids help engineers to construct more efficient conveyer belts, biologists to understand genes and evolution, and computer scientists to build quantum computers and plan the motion of robots.

#### Tell me more about topology!

Geometry has always been tied closely to mathematical physics via the theory of differential equations. It uses curvature to distinguish straight lines from circles, and measures symmetries of spaces in terms of Lie groups, named after the famous Norwegian mathematician Sophus Lie.

Topology, in contrast, is the study of qualitative properties of spaces that are preserved under continuous deformations. The spaces in question can be tame like a smooth manifold, or wild and hard as rock.

Topological ideas arise in practical problems, and research in topology still finds new applications, in particular to mathematical problems that are not directly phrased in terms of numbers and functions.

Homotopy theory (a subdiscipline of topology) has many applications within mathematics itself, in particular to algebra and number theory.

## Geometry and Topology at IMF

Our research in geometry and topology spans problems ranging from fundamental curiosity-driven research on the structure of abstract spaces to computational methods for a broad range of practical issues such as the analysis of the shapes of big data sets.

The members of the group are all embedded into a network of international contacts and collaborations, aim to produce science and scientists of the highest international standards, and also contribute to the education of future teachers.

• Second order Brunnian ring
• Lots of dots: Homology counts the circles that you see.

• The Klein bottle is a one-sided surface.

• An open-closed cobordism.

• Homological stability for the symmetric groups in a spectral sequence.