A Riemannian manifold ${\displaystyle M,g)}$ is said to be {\it curvature-homogeneous} if it is homogeneous to second order, i.e., if, for any two points in ${\displaystyle M}$ the Riemannian curvature tensors are equivalent under some isometry of the two tangent spaces.  Of course, a locally homogeneous metric is curvature-homogeneous, but the converse is not true in dimensions greater than ${\displaystyle 2}$. (For a surface (i.e., in dimension ${\displaystyle 2}$) curvature-homogeneity is equivalent to having constant Gaussian curvature, and such metrics are, of course, all classified locally and they are locally homogeneous.)  Already in dimension ${\displaystyle 3}$, there are many questions about the existence and generality of curvature homogeneous metrics, even locally.  In this case, curvature-homogeneity is equivalent to having the eigenvalues of the Ricci curvature be constant, which is a system of partial differential equations on the metric.

In this talk, I will review what is known about such metrics in dimension ${\displaystyle 3}$, particularly the work of O. Kowalski and his collaborators during the 1990s. I will show that, for certain values of the eigenvalues of the Ricci tensor, these partial differential equations are integrable by the Darboux’ method, which yields some surprising relations with classical subjects, such as the theory of holomorphic curves in the complex projective plane and the geometry of contact curves in the ${\displaystyle 3}$-sphere.

Let F be a binary form with integer coefficients, non-zero discriminant and degree d with d at least 3.. We shall discuss the problem of estimating the number of square free and more generally k-free integers of absolute value at most Z represented by F. This is joint work with Stanley Xiao.

If ${\displaystyle f}$ is a complex-valued continuous function on the unit circle ${\displaystyle 𝕋}$ that is even (i.e., ${\displaystyle f(e^{i\theta })=f(e^{-i\theta })}$ for all ${\displaystyle \theta \in ℝ}$), and all we know about ${\displaystyle f}$ is its average value on the ${\displaystyle n}$-th roots of unity for each ${\displaystyle n\in \mathbb{N}}$, then is it possible to recover ${\displaystyle f}$ itself from this information? By applying basic Fourier analysis and exploiting properties of the Möbius function and the divisor function, we show that the question has an affirmative answer if ${\displaystyle f}$ is assumed to be ${\displaystyle C^2}$ on ${\displaystyle 𝕋}$ and to have mean ${\displaystyle 0}$ (i.e., ${\displaystyle {\int }_0^{2\pi }f(e^{i\theta })d\theta =0}$). When ${\displaystyle f}$ is not even, we describe how ${\displaystyle f}$ may be recovered from the average values of appropriate phase shifts of ${\displaystyle f}$ on roots of unity.

Philip Engel: The Hurwitz theory of elliptic orbifolds


Consider a tiling of a compact, oriented surface by triangles, squares, or hexagons. Enumeration of such tilings can be phrased in the language of Hurwitz theory of an elliptic orbifold. There is a combinatorial curvature associated to each vertex. We will outline a proof that the generating series for the number of tilings with a given number of tiles and specified nonzero curvatures lies in a ring of quasi-modular forms. The asymptotics as the number of tiles goes to infinity allows one to derive formulas for the volumes of strata of cubic, quartic, and sextic differentials---in particular, these volumes are a power of pi times a rational number.  This generalizes work of Eskin and Okounkov in the case of abelian and quadratic differentials