CAREER: Tropical and Diophantine Geometry

Ohio State University, The, Columbus OH

Investigators

Abstract

The PI will apply tropical geometry techniques to resolving long-standing problems in combinatorics and diophantine geometry. Tropical geometry is a method for transforming questions in algebraic geometry, which is the study of polynomial systems of equations, into questions in combinatorics, which is the mathematics of discrete structures. The combinatorial problems addressed involve understanding the counts of certain discrete geometric objects. Diophantine geometry is the study of rational solutions to polynomial systems. By degenerating the polynomial systems according to combinatorics, one can bound the number of solutions. The PI will engage in educational and outreach activities including the following: the broadening of his department's honors program; community outreach presentations; mentorship of students from underrepresented groups; graduate advising; and exposition. The research in this project focuses on open problems in combinatorics and diophantine geometry. With collaborators, the PI will incorporate algebraic geometric methods (particularly Hodge theory, positivity, and genericity) into geometric combinatorics with the goal of understanding the possible face numbers of certain polyhedral complexes. By refining established techniques, he will investigate sharper inequalities among face number which will elucidate combinatorial structures. He will also deepen the commutative algebraic underpinnings of his recent work on matroids. In Diophantine geometry, he will study bounds on the number of rational and torsion points on curves. Following the methods of Buium and Kim, with collaborators, he will examine functions on curves of bad reduction that vanish on points of arithmetic interest and bound their zeroes. This involves elaborating analogies between classical and p-adic analytic geometry. 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.