Computational Algebra Seminar
Maximilian Kreuzer died on 26th November after many months fighting against a severe illness. I want to take this opportunity to talk about one of his most important contributions to toric geometry: the classification of the four dimensional reflexive polytopes that he produced with Skarke in 2000.
This classification of 473,800,776 polyhedra provides most famous source of Calabi-Yau threefolds (giving 30,108 distinct pairs of Hodge numbers), and is obviously of considerable importance in string theory. I'll attempt to sketch the methods used by Kreuzer and Skarke to obtain this result, and how the combinatorial data should be interpreted.
Some families of Diophantine equations, such as quadratic forms, have a very useful property: if an equation has solutions in the real numbers and in each p-adic field, then it has a rational solution. Such families of equations are said to satisfy the Hasse principle. In general the Hasse principle does not hold, but many violations are described by the so-called Brauer-Manin obstruction. This obstruction was first defined by Manin and is based on the Brauer group of the variety. I will talk about how to compute the Brauer-Manin obstruction for certain classes of varieties, using many ingredients from algebraic geometry and number theory.