Why are soap bubbles spherical? For centuries mathematicians have wondered about such questions. In the last few decades, the field of geometric analysis, in which properties like volume and surface are investigated, has thrived. Franz Schuster from the Institute of Discrete Mathematics and Geometry at the Vienna University of Technology has now received an “ERC Starting Grant” from the European Research Council ERC, to the value of approximately one million Euros. With this grant, Schuster can expand his research team and further investigate the connections between the theory of geometric inequalities and other important areas of mathematics.
A Round Garden does not Need a Long Fence
Legend has it that the Phoenician Queen Dido was granted as much land as she could enclose with a leather band. Which shape was she supposed to choose in order to obtain as large an area as possible? In Mathematics, the solution to this problem is expressed by the “isoperimetric inequality”: the area of a two dimensional shape is always smaller or equal to the square of the circumference divided by four pi. Equality holds precisely for the circle – it is the shape with maximum area for a given circumference. In higher dimensions, spheres are the optimal solution. That is why soap bubbles are spherical: they have to enclose a given volume of air with a surface of minimal area.
The Beauty of Mathematics
“In Mathematics, the most beautiful results are often also the most important ones”, says Franz Schuster. “And the isoperimetric inequality is definitely one of the most beautiful and most important results in geometry.” In his research project, Schuster is working on far-reaching generalizations and strengthened versions of the isoperimetric inequality. To this end he uses methods from integral geometry, a branch of mathematics whose strong connection to isoperimetric problems has only recently been discovered – in large part due to important contributions from the research group for Convex and Discrete Mathematics at the Vienna University of Technology.
Useful for Science
The research, however, is not only concerned with beauty: the results, obtained by Franz Schuster, have important implications for several different scientific areas. “In science, most processes, such as fluid dynamics or quantum dynamics, are described by partial differential equations”, says Schuster. “Unfortunately, the solutions of these equations cannot usually be expressed in a simple formula, they can only be approximated.”
Geometric estimates are needed in order to find out whether such solutions even exist, whether they are uniquely determined and how well certain approximations work. This geometric core is often closely related to isoperimetric inequalities.
“With Great Power Comes Great Responsibility”
The ERC Starting Grant is not the first substantial grant Franz Schuster has received. Only recently, in June 2012, he was awarded the START-prize from the Austrian Science Fund FWF. “This success was made possible only because of the excellent scientific environment in the research group for Convex and Discrete Geometry”, Schuster points out. “Since the seventies, the research group for Convex Geometric Analysis - founded by Professor Peter Gruber – is has been among the most important centers worldwide in this area. Only two years ago, Professor Monika Ludwig, one of the world’s leading scientists in that field, was able to return to Austria after three years as Full Professor in New York.”
Franz Schuster considers himself and his team as a part of an important tradition of Convex Geometry at the Vienna University of Technology. “This means that these grants come with a great responsibility to continue this tradition and to set a course for the future”, says Schuster.
Dr. Franz Schuster
Convex and Discrete Mathematics
Wiedner Hauptstraße 8
Phone: +43 1 58801 10465