Single Review on past events

At the International Congress of Mathematicians in 1900 in Paris, the famous mathematician David Hilbert presented a list of 23 unsolved mathematical problems. One of these problems – number 16 – is among those still unsolved to date. It asks: “What are the possible configurations of the connected components of an algebraic curve in the real projective plane”?

To approach this question, one has to deal with Algebraic Geometry, like Daniel Plaumann does. Algebraic Geometry studies algebraic equations in several variables. As the name suggests, it combines algebra and geometry. On the algebraic side, equations can be simplified through symbolic computation. But even when the equations are as simple as possible, it may happen that, on the geometric side, the solutions come in several groups called connected components. These connected components are a purely geometric concept and cannot be easily detected on the level of algebra by analyzing the equations, since they usually do not correspond to a factorization of the equations. For a smooth curve of even degree in the real plane, each component looks like a deformed oval. Hilbert’s 16th Problem asks how these ovals can be arranged with respect to each other.

According to Daniel Plaumann, a major difficulty lies in the fact that connected components are not well represented on the algebraic side. “One approach to Hilbert’s 16th problem is to come up with constructive ways of producing a curve that realizes some prescribed configuration of ovals. Mathematician Axel Harnack already determined the maximal number of ovals in 1876 and gave a method of constructing curves with that maximal number (but without giving any information about the configuration). The most general constructive approach to date is an ingenious method called patchworking, developed by Oleg Viro since the late 1970s.

At the end of his talk Daniel Plaumann discussed a related question from his own research: curves with maximally nested ovals. This particular case can be fully understood on the algebraic side. “The shape of the real curve is reflected in a particular shape of its equation. This was conjectured by Peter Lax (1958) and proved by Helton and Vinnikov (2007). It has applications to a number of other pure and applied problems, for example in control theory. In joint work with Cynthia Vinzant, we gave a new constructive proof of this result”.