I will illustrate a comparison between two diffeomorphic moduli spaces, the Hitchin moduli space of holomorphic Higgs bundles and the de Rham moduli spaces of irreducible connections over a smooth projective curve, in terms of Lagrangians filling up the entire space.
I will survey some results on cycles using Griffiths’ infinitesimal invariants of normal functions.
It is a very natural question to understand the difference between Kähler varieties and projective varieties. On the other hand there has been a lot of recent progress on the bimeromorphic geometry of Kähler varieties. In particular the minimal model program exists for some threefolds and fourfolds and we have abundance for threefolds. We review this progress and its relation to the difference between Kähler and projective varieties.
A semiglobal field is the function field of a curve over a complete discrete valued field. We discuss local-global principles for homogeneous spaces under connected linear algebraic groups with respect to its divisorial discrete valuations. There is a reduction to the patching setting developed by Harbater, Hartmann and Krashen, for projective homogeneous spaces and principal homogeneous spaces leading to a Hasse principle in generality for projective homogeneous spaces. In this talk, we shall discuss the progress in this direction. (Joint with Philippe Gille.)
Thanks to advances in logarithmic Gromov-Witten theory, we can now construct mirror partners canonically in the generality that birational geometry suggests. In the talk I will try to quickly introduce this intrinsic mirror construction and then comment on recent developments in proving both the original enumerative predictions and homological mirror symmetry in this setup.
I will report on joint work with Hannah Larson, and joint work in progress with Jim Bryan, in which we try to make sense of Bott periodicity from a naively algebro-geometric point of view.
In this talk I’ll discuss joint work with Sarah Frei, Lena Ji, Soumya Sankar and Bianca Viray on the problem of determining when a geometrically rational variety is birational to projective space over its field of definition. Our main interest is the rationality problem for conic bundles over P2 with quartic discriminant curve, where there exist both rational and irrational examples over nonclosed fields. I will discuss our perspective on the failure of the intermediate Jacobian torsor obstruction to characterize rationality, which leads us to study conic bundles that differ by a constant Brauer class.