On Curves and Planes

Tropical geometry is a fairly recent research field in mathematics named in honour of its initiator, the Brazilian mathematician and computer scientist Imre Simon. It is a piecewise linear version of algebraic geometry. The complicated shapes of algebraic geometry are
replaced by polyhedra, objects with flat sides. The theory often serves to translate difficult problems in algebraic geometry into much simpler problems in combinatorics.

The most important applications of tropical geometry in mathematics so far have been in the enumerative geometry of curves. This is a classical subject of algebraic geometry, studied at least since the 19th century. In recent years it was revolutionized by the input coming from theoretical physics, in particular from string theory.

One would like to know, for example, how many algebraic curves satisfy certain suitable conditions; say, how many curves of degree d in the plane have a given number of singularities (self intersections) passing through a certain number of general points. For instance, what is the number of cubics (curves of degree 3) in the complex plane with one singularity passing through 8 general points? It is known that this number equals 12. It was proved by Mikhalkin that the number of such algebraic curves is equal to the corresponding number of tropical curves, which are just piecewise linearly embedded graphs in the plane and hence much easier to understand.

ICTP mathematician Lothar Göttsche has introduced refined enumerative invariants of curves giving finer invariants than just the numbers of curves, first in algebraic geometry and then in tropical geometry. He was asked to give one of this semester's Bernoulli lectures at the Ecole Polytechnique Federale de Lausanne's (EPFL) Bernoulli Center in Lausanne, Switzerland, on May 15, on this topic.

The Bernoulli Center is one of the primary research meeting places in mathematics in Europe. Their thematic programs consist of a six month period of concentrated activity in a specific area of current research interest in the mathematical sciences. The activities during each of these thematic semesters include three special Bernoulli lectures given by experts in the field. The current semester's thematic program is on "Tropical geometry in its complex and symplectic aspects", organized by Ilia Itenberg and Grigory Mikhalkin.

Gottsche has previously been invited to give two-week lecture courses on this material at IMPA, the top mathematics research institute in Brazil, from 14 to 24 2013, and at the graduate school on new aspects on singularity theory at ICMAT, Madrid, Spain, from 16 to 27 September 2013.