Conference Agenda

This mini-symposium will focus on practical computational methods in intersection theory and their applications. At its most basic, intersection theory gives a means to study the geometric and enumerative properties of intersections of two varieties within another. These questions are fundamental to both algebraic geometry and its applications. Fulton-MacPherson intersection theory provides a powerful tool-set with which to study these intersections; however, many mathematical objects which are needed in this framework have long been computationally inaccessible. This barrier has limited the use of these ideas in computations and applications. In recent years several new and computable expressions for Segre classes, Polar classes, Euler characteristics, Euler obstructions, and other fundamental objects in intersection theory have been developed. This has led to a variety of computationally effective symbolic and numeric algorithms and opened the way for ideas from intersection theory to be applied to solve both mathematical and scientific problems. Some of this recent work will be highlighted in this mini-symposium. The first talk in the session will be an introductory talk, which will demonstrate the natural relations between intersection theory and numerical algebraic geometry and will highlight how intersection theory can be applied to solve classical problems such as testing ideal membership (without computing a Groebner basis). Subsequent talks will explore computational aspects of intersection theory in more detail and will highlight their practical applications.

In this talk we discuss new effective methods to test pairwise containment of arbitrary (possibly singular) subvarieties of any smooth projective toric variety and to determine algebraic multiplicity without working in local rings. These methods may be implemented without using Gröbner bases; in particular any algorithm to compute the number of solutions of a zero-dimensional polynomial system may be used. The methods arise from techniques developed to compute the Segre class s(X,Y) of X in Y for X and Y arbitrary subschemes of some smooth projective toric variety T. In particular, this work also gives an explicit method to compute these Segre classes and other associated objects such as the Fulton-MacPherson intersection product of projective varieties. These algorithms are implemented in Macaulay2 and have been found to be effective on a variety of examples. This talk is based on joint work with Corey Harris (University of Oslo).

The talk is within the area of Algebraic Geometry of Data. Bottlenecks are pairs of points on a variety joined by a line which is normal to the variety at both points. These points play a special role in determining the appropriate density of a point-sample of the variety. Under suitable genericity assumptions the number of bottlenecks of an affine variety is finite and we call it the bottleneck degree. We show that it is determined by invariants of the variety, such as polar classes and Chern classes. The talk is based on joint work with D. Eklund and M. Weinstein.

For a local ring (A, m) and an ideal I such that A/I has finite length, the Hilbert-Samuel polynomial P(n) of I is a polynomial such that P(n)=length(A/I^n) for large n. The leading coefficient and degree of this polynomial are important invariants of the ideal and encode information about the singularities of its coordinate ring. We present methods for computing this polynomial for arbitrary ideals and give various geometric examples involving hypersurfaces, determinantal ideals, etc. Moreover, we will share progress on local computation of other invariants such as the Bernstein-Sato polynomial and multiplier ideals, which measure singularities of varieties.

Inspired by the problem of Resolution of Singularities, we define a series of invariants for points of maximum multiplicity of algebraic varieties. We do this using arcs and the Nash multiplicity sequence. The invariants that we obtain capture the contact that arcs in a variety X can have with the subset of points of X where the multiplicity reaches its maximum value. In addition, they are strongly related to an important invariant for constructive resolution of singularities.