Multiplicity refers to the number of times a factor appears in a polynomial equation: A very simple example would be that x2 = 0 has a higher multiplicity than x = 0, though both equations are represented by a vertical line on the y axis. Grifo focuses on a variation of this problem: Finding equations with high multiplicity that pass through those points. Interpolation, a classical problem in geometry, involves determining the equations of a curve that pass through a given set of points. Grifo also studies higher order interpolation through the lens of symbolic powers, which is an algebraic approach. P-derivations, which stem from the idea of derivatives in calculus, may be a mechanism for solving longstanding questions in commutative algebra. She, along with collaborators Jack Jeffries of Nebraska and Alessandro De Stefani of the University of Genova, were the first group to apply a strategy from arithmetic geometry, called p-derivations, to commutative algebra. The polynomial equation on the right has a singularity at the highlighted corner that crosses the (0,0) point.Ī hallmark of Grifo’s research is importing tools from other fields of mathematics to commutative algebra. With no visual points of “bad behavior,” this solution set has no singularities. The polynomial equation on the left has a solution set that cuts out a circle across the x- and y-axis. We want to be able to tell you about the singularities of a 200 or 300D figure.” “We want to understand this bad behavior and be able to classify it and distinguish between different types of bad behavior. “To us, it’s these points of ‘bad behavior’ that are interesting,” Grifo said. The solution is to convert the problem from one of geometry to one of algebra: To learn about a shape’s singularities, Grifo probes the algebraic properties of the corresponding equations. “For a 200-dimensional object, I can’t draw it, and you can’t see it, so whatever interesting information I want to extract from it is not something that I can visualize geometrically,” she said. Grifo is interested in singularities of shapes that are more complex. By contrast, the curvy, V-like shape that results from the equation x2 – 圓 = 0 has a singularity, located where the two sides of the V meet. A circle, represented by the polynomial equation x2 + y2 = 1, has no such points. Visually, singularities look like a sharp point, a corner or a crinkle. Her work focuses on points of irregularity, called singularities, in the geometric shapes described by a system of polynomial equations. With the five-year, $425,000 CAREER grant, Grifo will expand on her work in commutative algebra, particularly as it relates to applications of p-derivations, symbolic powers and cohomological support varieties. A University of Nebraska–Lincoln mathematician has received a grant from the National Science Foundation to advance her work in commutative algebra, an area of abstract algebra that provides a framework for visualizing and understanding the properties of shapes in higher dimensions that is key to solving real-world problems in robotics, statistics, physics and beyond.Įloísa Grifo, assistant professor of mathematics, is the first woman from the Nebraska U mathematics department - and the second Husker mathematician overall - to receive funding from NSF’s Faculty Early Career Development Program.
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