No abstract

Under iterated barycentric subdivision of a triangle, most triangles become flat in the sense that the largest angle tends to π. By analyzing a random walk on S L 2 ( ) we give asymptotics with explicit constants for the number of flat triangles and the degree of flatness at a given stage of subdivision. In particular, we prove analytical bounds for the upper Lyapunov constant of the walk.

We prove that every distinct covering system has a modulus divisible by either 2 or 3.

We use the method of Shintani, as developed by Taniguchi and Thorne, to prove the quantitative equidistribution of the shape of cubic fields when the fields are ordered by discriminant.

Enumerating integral orbits in prehomogeneous vector spaces plays an important role in arithmetic statistics. We describe a method of proving subconvexity of the zeta function enumerating the integral orbits, illustrated by proving a subconvex estimate for the Shintani ζ \zeta function enumerating class numbers of binary cubic forms.