Asymptotic growth of symbolic powers, mixed multiplicities, and convex bodies

Asymptotic growth of symbolic powers, mixed multiplicities, and convex bodies Asymptotic growth of symbolic powers, mixed multiplicities, and convex bodies This project is in commutative algebra which is the area of mathematics that studies properties of polynomial equations in several variables. Various phenomena across science and engineering can be modeled by systems of polynomial equations, which leads to natural interactions with other fields such as applied mathematics, biology, computer science, and mathematical physics. For systems of polynomial equations, one studies its sets of solutions, called varieties, and the corresponding shapes they describe in high-dimensional spaces. As the number of equations and variables grow larger, one useful approach to understand these shapes is to study the algebraic functions defined on them. Using this approach, many properties such as smoothness and degrees of varieties can be translated into algebraic terms. One such strategy is to study properties of varieties by analyzing the asymptotic behavior of sequences of algebraic objects called ideals. This project contributes to this line of research with problems originating from two directions that involve the study of these properties of ideals. The PI will also be involved in mentoring students and in organizing academic events, focusing on benefiting graduate students and underrepresented groups in mathematics. The symbolic powers of an ideal encode important algebraic and geometric information of the ideal and the variety it defines. The PI will investigate the asymptotic behavior of homological invariants of symbolic powers such as, the number of generators, the Castelnuovo-Mumford regularity, and the projective dimension. For the sequence of numbers of generators, the PI intends to investigate if it always has polynomial complexity, building of previous results by the PI and his collaborators. A general result in this direction would have important consequences on the arithmetic rank, Frobenius complexity, and Kodaira dimension of divisors. For the other two sequences, regularities and projective dimensions, the PI will focus on regular rings of positive characteristic. In such rings, a new class of ideals is defined for which it has been previously shown the limit of these sequences exist; this class includes several types of determinantal ideals, as well as the square-free monomial ideals. The PI will investigate the existence of these limits for more general classes of ideals. The relation between multiplicities and convex bodies is an important research topic lying in the interaction of Commutative Algebra, Algebraic Geometry, and Combinatorics. In recent years, this line of research has gained much activity due to the introduction of Newton-Okounkov bodies and mixed multiplicities of filtrations of ideals. The PI and his collaborators intend to find conditions for the non-vanishing of mixed multiplicities of multigraded algebras. The PI will also investigate a notion of mixed multiplicities for filtrations of not necessarily zero dimensional ideals and the relation of these new multiplicities with the mixed volumes of certain Newton-Okounkov bodies.