A study of hypersurface singularities

University Of Kansas Center For Research Inc, Lawrence KS

Investigators

Abstract

This project will study subtle geometric aspects of systems of polynomial equations, while at the same time training the next generation of scientists, both in academics and industry. A polynomial is any equation that can be formed using the fundamental operations of addition, subtraction, and multiplication, and solving a system of polynomial equations means solving multiple polynomial equations simultaneously. Polynomial systems, though simple to define, are rich enough to describe, or model, many interesting phenomena. They are ubiquitous in many different areas of science, including computer science, biology, chemistry, physics, and engineering, and play a critical role in the applications of these areas to industry. A polynomial system can involve many different unknowns, or variables, and in applications, the number of these variables is often quite large (e.g., having thousands of variables in a system is not rare). Unfortunately, the technique of graphing, which allows us to visualize and study polynomial equations, is no longer available to us once we are dealing with four or more variables. This project is focused on overcoming this obstacle by developing systematic ways to describe and precisely measure the complexity of polynomial systems, no matter the number of variables. Furthermore, this will done in an algebraic, discrete manner; that is, using methods that can be translated into the language of a computer. In this pursuit, the PI will train, mentor, and support students. He will train graduate students to conduct technical mathematical research, to implement their results using open-sourced computing languages, and to effectively communicate their results to a wide audience. This will prepare such students for careers in academic research and instruction, as well as in industry. At the undergraduate level, we will mentor students, especially those from under-represented groups, to prepare them for successful careers in industry, and also for graduate school in a STEM, or adjacent, field. More precisely, The PI will study the singularities of algebraic varieties, or systems of polynomial equations, through understanding invariants associated to them. Invariants are objects (such as a number, or an ideal in a ring, or a polyhedral shape) derived from a variety in a consistent manner, and that can meaningfully, precisely, quantify subtle properties of that variety. The invariants studied are both discrete (e.g., defined using the Frobenius morphism in prime characteristic), and continuous (e.g., defined in terms of integrability conditions, or resolution of singularities). The PI will employ many concrete techniques, including some based on convex geometry, polyhedral geometry and variants of linear optimization problems, and will produce effective algorithms to explicitly compute many interesting invariants of singularities. These algorithms will be implemented in open-source computer algebra systems such as Macaulay2. This project is jointly funded by the Algebra and Number Theory program and the Established Program to Stimulate Competitive Research (EPSCoR). 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.