Calendar

Abstract:  The uniqueness of infinity plays a crucial role in understanding solutions to geometric PDEs and the geometric and topological properties of manifolds with Ricci curvature bounds. A major progress is made by Colding—Minicozzi who studied one type of infinity, called the asymptotic cones, of a Ricci flat manifold with Euclidean volume growth and proved the uniqueness of asymptotic cones if one cross section is smooth. Their result generalizes an earlier uniqueness result by Cheeger—Tian which requires integrability of the cross section among other things. 

In this talk I will talk about some recent results on the study of the similar uniqueness problem of another type of infinity called the asymptotic limit spaces for Ricci flat manifold with linear volume growth, following the path of Cheeger—Tian and Colding—Mincozzi for cones. Here one considers translation limits instead of rescaling limits and the limits are no longer cones but cylinders. I will highlight the similarities and differences between the two settings. This is joint work with Zetian Yan.

Abstract: I will discuss two results on totally geodesic submanifolds.  In joint work with Minju Lee, we show that in a geometrically finite rank one manifold of infinite volume, every maximal totally geodesic submanifold of dimension at least two in the convex core is properly immersed, has finite volume, and there are only finitely many of them.  In joint work with Subhadip Dey, we construct the first higher rank examples where rigidity fails: we exhibit a Zariski dense surface subgroup $\Gamma<SL(3,\mathbb Z)$ such that the locally symmetric space $\Gamma\backslash SL(3,\mathbb R)/SO(3)$ contains a sequence of geodesic planes whose closures are fractal, with Hausdorff dimensions accumulating to 2.

Complex abelian varieties are compact complex tori that can be embedded as complex subvarieties of the complex projective space. They are among the most fundamental algebraic varieties and their study goes back to Riemann. In the 50’s Hodge introduced a necessary condition for a cohomology class of a smooth algebraic variety to be represented by a linear combination of classes of algebraic subvarieties. He conjectured that the condition is sufficient, i.e., that Hodge classes are algebraic. The Hodge conjecture is known in dimension at most three, but until recently it was not known even for 4-dimensional abelian varieties. Furthermore, Weil identified Hodge classes on 4-dimensional abelien varieties with complex multiplication and proposed them in the late 70’s as test cases for the Hodge conjecture. In the late 90’s it was realized that the algebraicity of the Weil classes would imply the Hodge conjecture for abelian varieties of dimension 4 and 5. We will explain the terms  above and the further developments that were needed for the proof earlier this year that the Weil classes on abelian 4-folds are algebraic.