by John H. Tibbetts
Josh Pollitz, assistant professor in the Department of Mathematics, has been awarded a National Science Foundation grant for his project, “Homotopical methods and cohomological support in local algebra.”
Algebraic geometry is a central branch of modern mathematics, focusing on the study of systems of polynomial equations, which are objects that are fundamental throughout mathematics. But when polynomial equations reach higher dimensions with increased numbers of variables, the shapes of algebraic geometry become harder to visualize. Commutative algebra provides a framework or language for seeing and understanding the properties of shapes in higher dimensional algebraic geometry.
Pollitz will investigate singularities in commutative algebra through the lens of various homological constructions. Homological algebra is the study of certain structures in math and how they correspond.
A highly complex equation is easier to visualize when it is smooth. But a singularity, a sharp point or crossing in an otherwise smooth surface of a higher-dimensional mathematical object, makes visualizing the equation’s shape more difficult. Homological tools can help investigators understand how to measure the extremity of a singularity, extracting information and presenting it in the form of a visible mathematical object. This research could have potential applications in a variety of mathematical areas, as well as real-world applications through fields such as cryptography.
He plans to use the NSF grant to travel and speak about his research at universities and increase the visibility and strength of Syracuse’s Algebra Research Group, which meets weekly and brings in outside speakers with new ideas and encourages graduate students to travel and speak at conferences. This grant, which can be difficult for early-career investigators to secure, will support postdocs, graduate and undergraduate student research. A long-term goal of the grant is establishing Syracuse University as a hub for commutative algebra and algebraic geometry.