Conference Agenda

This minisymposium highlights the use of computation inside algebraic geometry. Computations enter algebraic geometry in several different ways including numerical strategies, symbolic calculations, experimentation, and simply as a fundamental conceptual tool. Our speakers will showcase many of these aspects together with some applications.

The computation of the semigroup (or even the cone) of all effective divisor classes on a Calabi–Yau hypersurface of a toric variety is an important open problem, whose solution would have a number of applications in theoretical physics. This is a difficult computation, with no known effective algorithms. We present computational tools and algorithms for investigating this and related problems, and describe some results in this direction.

The task of determining the order zeta function for certain number rings (which is just a sophisticated form of counting subrings) gives rise to a particular kind of p-adic integrals. The domain of integration of these stubbornly withstands standard techniques, including even an out-of-the-box Hironaka-style resolution of singularities. However, choosing centers of blow-ups using the structural properties of the problem, a simultaneous monomialization of the conditions can be achieved, making the problem again accessible to usual methods. This talk is based on joint work with Josh Maglione, Bernd Schober, and Christopher Voll.

In a joint work with Aldo Conca, we look at what should correspond to Macaulay’s Theorem and Fröberg’s Conjecture for the Hilbert function of subalgebras of standard graded polynomial rings. Upper bounds correspond to generic forms and lower bounds correspond to strongly stable monomial ideals.

In this talk, the concept of singular value decomposition of complexes will introduced and applied to the computation of syzygies.