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During the solar eclipse of May 29, 1919, Arthur Eddington and his collaborators were able to observe a change in position of the stars as they passed by the sun matching the then recent theory of Albert Einstein of general relativity. The light from the stars curved around the sun. However, this description is not completely correct. The actual statement of general relativity is that space and even time will be curved around heavy objects such as stars. The light is actually following a straight line in a curved space.

Thinking about how our own space that we live in curves, is something that is very hard to image. Einstein was able to picture it and describe it ... but only because earlier work by mathematicians Bernhard Riemann who had already in 1854 started developing ways of understanding curved spaces in dimension three or higher. And again, his understanding was based on the work of his teacher Carl Friedrich Gauss.

The idea of Gauss can roughly be described as follows. If you where living in a lower dimensional space, would you be able to detect how your space is shaped? It turns out that if you are one-dimensional, you have no way of knowing the shape of the space you live in. However, if you are living in a two-dimensional space, you can actually determine your shape from things you can see from inside your space. For example, if you draw a triangle of what to you looks like straight lines, then in a flat space all the angles sum up to 180 degrees. However, if your space is shaped more like a sphere the sum will be larger while it is less for something shaped more like a saddle. This idea was later used in 2000 in the BOOMERanG experiment to determine how our universe curves by looking at angles in a triangle made by the cosmic background radiation.

Although the influence of Gauss and Riemann on Einstein's theory is clear, it is only one application of their work. In many complex problems that appear in physics, robotics or even in finance, we can view them as a simpler problem just happening in a more complex curved space. With this shift, we can understand solutions from studying shapes.

The objective of the project is to study such application of geometry to equations. There will be a particular focus towards application to equations where there exists some noise or random input, meaning that we have to combine geometry with probability theory. Our research is focused on spaces where there is some restriction to how our objects or noise can move. In this setting, many of the classical ideas of Riemann will not be available, and we need to develop a brand new toolbox to solve the problems we are considering.

Finally, the project will also focus om making computers more comfortable with working on shapes. Computers really like to break everything down to many, many small computations using adding and multiplication. But how can we then teach them to do computations on shapes, when these things are not really available? For example, there is no real way to add two points on a sphere and obtain a new point on the sphere as their sum. We therefore need new algorithms and ideas that help our digital friends work on shapes without destroying them.