# Geometric Analysis of Optimal Shapes

#### What

Optimal shapes show up everywhere in nature from in the architecture of colorful bird feathers to on the edge of black hole geometries, and in explaining why stars, planets and soap bubbles are round like spheres. However, it turns out that the mathematics of finding such optimal surfaces is not so easy. One very promising technique is to start with a given surface and then attempt to flow it to an optimal one. This does run into serious troubles, known as singularities. The project is about studying how to get past such singularities to construct optimal surfaces for use in modern theoretical physics. This involves both giving examples of things we can become singular and finding conditions under which no singularities can occur, all with rigorous mathematical proofs.

#### Why

Soap bubbles are useful even when they live only in our imagination, as theoretical tools to study all possible topological shapes of space - such as the very universe we inhabit. They also have deep applications in defining the center of mass of a physical universe. You might think that you could do this with just your school physics formulas. But in modern physics, the world is not so easy to cut into pieces and add back up. Ever since Einstein, we are dealing with what mathematically speaking is called a nonlinear problem with boundary conditions. One clever workaround is to replace the desired center with a sphere collection far away. In more general spaces, the spheres are then replaced by optimal sphere-like shapes, which is why this project includes constructing such surfaces.

#### How

We work theoretically, with chalk on the blackboard or pen and paper. This method is needed to discover and establish new truths in mathematics. One does not get far by just checking a few tens of thousands of examples on a computer, when one needs to prove solid theorems about geometry, which will remain valid for all possible surfaces including in all of the most unimaginable situations. Mathematically speaking, there are several quite different ways to study the geometry and topology of surfaces. In this project we approach such problems via partial differential equations. Many of these ideas, valuable for purely mathematical reasons, are inspired by and informed by physics intuition. One can really say that the interdisciplinarity flows both ways and ends up going in loops too.

#### SSR

Geometry is quite literally everything we see around us. But there's also lots of hidden geometry in the shape of spaces, which may not at first glance seems like spaces at all - such as sets of coordinates in a data set, carved out by constraint equations. The by far most successful models of anything in our observable physical world are also cooked up in terms of differential geometry and concepts of energy, which lead to (partial) differential equations. This is in a precise sense witnessed by physical laws ranging from Galilei and Newton over Maxwell's electromagnetic theory of light to Einstein's general relativity, quantum field theory and The Standard Model of particle physics. Even the famous Higgs boson was discovered theoretically first, via advanced tools in geometry and differential equations known as principal fiber bundles and gauge theory. This is ultimately why we are interested in developing such new geometric descriptions of spaces and optimal shapes to study their structures: For understanding our world much more in depth, which down the long road will allow us to interact in this world in brand new ways, for great, as of yet unknown, benefits to humankind and society.