### Mathematical Research at FSU

#### Algebraic Geometry and Number Theory

Many fields of mathematics and physics, ranging from number theory to geometry to string theory, share common underlying algebraic structures. Historically, algebraic geometry began as the study of varieties, the zero sets of polynomials. The study of these geometric objects is inherently tied with the algebraic study of the polynomials which define them. Many aspects of modern algebraic geometry are concerned with more abstract properties whose roots are connected to varieties. These include commutative algebra, intersection theory, and arithmetic geometry.

An algebraic link associated to a plane curve singularity.

On the most basic level, number theory is the study of the integers, focusing on prime numbers. There are many connections between number theory and other fields, notably algebra, analysis and algebraic geometry. Some fundamental questions in number theory are centered around the distribution of prime numbers, the structure of number fields and their rings of integers, and the structure of integral and rational points on varieties. Influential results and conjectures in modern mathematics, such as Falting's theorem (the Mordell conjecture), Siegel's theorem, and the Birch and Swinnerton-Dyer conjecture, bring together number theory and algebraic geometry.

Several department members engage in research related to the study of algebraic structures and of their various manifestations. Amod Agashe works in arithmetic geometry (a subfield of number theory), and studies abelian varieties (which include elliptic curves), especially using the theory of modular forms. Ettore Aldrovandi works in homological algebra and category theory, and in geometric constructions of algebraic objects such as elements of nonabelian cohomology. Paolo Aluffi studies algebraic geometry, with emphasis on enumerative geometry and the theory of singularities, and applications to questions in combinatorics and string theory. Mark van Hoeij concentrates on computational aspects, for example in the solution of difference equations, in factorization algorithms, and in number theory. Matilde Marcolli's interests range very broadly on the theory of motives in algebraic geometry, number theory, and physics, and in the field of noncommutative geometry. Kathleen Petersen studies number theory and applies tools from algebraic geometry and number theory to the study of manifolds. Sam Ballas studies algebraic varieties that arise as representations of groups arising in low dimensional topology and how they relate to deformations of geometric structures on manifolds.