Collaborative Research: Linkage, syzygies, and multiplicities theory

About This Grant

The award supports research in pure mathematics, more precisely in Commutative Algebra. The Principal Investigators will address issues related to the structure of geometric objects that arise as solution sets of systems of polynomial equations in several variables. These objects, known as varieties, are essential in both mathematics and applications to science and engineering. A classical problem dating back to the nineteenth century is the classification of varieties, and an important tool for this classification is linkage theory. One of the goals of the project is to identify new necessary and sufficient conditions for two varieties to belong to the same linkage class. Minimal free resolutions provide a method for describing complex algebraic objects, such as modules, through a sequence of matrices. While these resolutions generally require an infinite sequence of matrices, the researchers aim to identify finite patterns in such resolutions. The implicitization problem is a classical problem in pure mathematics that is of significant interest in geometric modeling and computer-aided design. Given any variety, the goal is to find the system of polynomial equations that has the geometric object as a solution set; knowing these “implicit” equations greatly enhances the understanding of the variety. The Principal Investigators actively engage with the mathematical community by serving on editorial and scientific boards, delivering lectures, and preparing expository notes for both national and international events. They will promote scientific exchange by organizing international conferences and national online seminars. Additionally, they will provide research training opportunities for students, postdoctoral scholars, and early-career researchers. The Principal Investigators aim to identify new properties preserved by linkage, focusing on licci ideals, the ideals linked to a complete intersection in a finite number of steps. The PIs also propose sufficient conditions for an ideal to be licci, based on the interplay between shifts in free resolutions and graded Betti numbers. Additionally, they intend to prove that perfect ideals of codimension three are licci if they are strongly nonobstructed, a property that arises in deformation theory. Advances in linkage theory have applications to Hilbert schemes. The PIs seek structural results about syzygies and finite patterns in infinite resolutions. In this vein, they propose a conjecture about the direct sum decomposition of the syzygy modules of the residue field of a Golod ring. The Behrend function has been used effectively to address enumerative problems in algebraic geometry, but is challenging to compute. The investigators plan to apply their expertise on blowup algebras to compute this function for subschemes of affine space that are either zero-dimensional, or defined by monomial ideals. Additionally, they aim to tackle the classical problem of determining the implicit equations that define the graph of a rational map between projective spaces, by leveraging their recent work on Jouanolou duality and relating the Rees algebra of an ideal to its Fitting ideals. In equisingularity theory, the Principal Investigators intend to establish fiber-wise, multiplicity-based, criteria for families of analytic spaces to be Whitney equisingular by characterizing integral dependence of modules through the constancy of multiplicities. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.