The Oppenheim conjecture, proved by Margulis in 1986, states that for a non-degenerate indefinite irrational quadratic form Q in $n \geq 3$ variables, the image set $Q(Z^n)$  of integral vectors is a dense subset of the real line. Determining the distribution of values of an indefinite quadratic form at integral points asymptotically is referred to as quantitative Oppenheim conjecture. The quantitative Oppenheim conjecture was established by Eskin, Margulis, and Mozes for quadratic forms in $n \geq 4$ variables. In this talk, we discuss the quantitative Oppenheim conjecture for ternary quadratic forms (n=3). The main ingredient of the proof is a uniform boundedness result for the moments of Margulis functions over expanding translates of a unipotent orbit in the space of 3-dimensional lattices, under suitable Diophantine conditions of the initial unipotent orbit.

Although vertex algebras originated in 2D conformal field theory, recent developments show that they also arise in 3D and 4D quantum field theories. Such vertex algebras often appear as chiralizations of symplectic singularities.

To construct new examples of these chiralizations, we study chiral differential operators on the basic affine space G/U, where G is a simple, simply connected algebraic group of type ADE, and U is a maximal unipotent subgroup of G. We show that the associated variety of these vertex algebras is isomorphic to the affine closure of G/U, which is a symplectic singularity, as conjectured by Ginzburg and Kazhdan and later proved by Jia and Gannon.

This is a joint work in progress with Xuanzhong Dai and Bailin Song.