Lecture - 14:15
Most of mathematics is concerned at some level with setting up and solving equations. Algebraic geometry is the mathematical discipline handling solution sets of polynomial equations. By relating these sets to ideals in polynomial rings, problems concerning their geometry can be translated into algebra. As a consequence, algebraic geometers have developed a multitude of often highly abstract techniques for the qualitative and quantitative study of the solution sets, without, in the first instance, considering the equations.
Modern computer algebra algorithms, on the other hand, allow us to manipulate the equations and, thus, to study explicit examples. In this way, algebraic geometry becomes accessible to experiments. The experimental method, which has proven to be highly successful in number theory, is now also added to the toolbox of the algebraic geometer.
In my talk, starting from intuitive examples, I will explain some concepts of the geometry-algebra dictionary, with particular emphasis on computatioonal aspects. In addition to basic algorithms, I will also touch more advanced applications such as parametrizing rational curves or computing resolutions of singularities. If time permits, I will show some examples of computing cohomology.
Colloquium - 16:00
Sturmfels showed that to any matroid one can associate its Bergman fan. This is a rational polyhedral fan in Euclidean space which can be obtained from the matroid via its lattice of flats and a choice of integer basis (Ardila and Klivans). These fans are the simplest examples of tropical linear spaces. So although many matroids do not arise from linear spaces over fields, any matroid is realizable by a linear space over the tropical semi-field.
Considering the Bergman fan of a matroid as a tropical variety we show how to define an intersection product on tropical cycles contained in the matroidal fan. One can show that this has all of the properties of an intersection theory, with some benefits. For example, its definition is on the cycle level; there is no need to pass to any kind of equivalence classes of cycles. We show two interesting cases of this intersection product. First, when it is restricted to cycles coming from matroidal quotients. Secondly when it is restricted to so called tropical Chern cycles, which are supported on the skeleta of the fan.
Matroidal fans exhibit properties analogous to those of smooth spaces in classical geometry, and thus are considered as the local building blocks of tropical manifolds. These are abstract polyhedral complexes glued from matroidal fans with an additional integral affine structure. The intersection product defined above can be extended easily to tropical manifolds. Borrowing from classical geometry, we derive some conjectured formulae for tropical manifolds relating their Euler characteristics and products of Chern cycles. These formula come from specific cases of the Hirzebruch-Riemann-Roch formula. In dimension one this formula is simple, and in dimension 2 it has been proved in certain cases.
02.02.2015 | 14:15
@TU MA 041