Monday, February 6, 2023
02/06/2023 - 4:00pm
A well-known conjecture of Margulis predicts the existence of a uniform lower bound on the systole of any irreducible arithmetic locally symmetric space. In joint work with F. Thilmany, we proved that this conjecture is equivalent to a weak version of the Lehmer conjecture, a well-known problem from Diophantine geometry.
02/06/2023 - 4:30pm
This is a report on work of my graduate student Cristian Rodriguez. A Q-Fano 3-fold is a complex projective variety with mild singularities such that its 1st Chern class is positive. Q-Fano 3-folds with b_2=1 arise as end products of Mori's minimal model program. Thousands of families are expected, whereas there are only 17 in the smooth case. We will describe mirror symmetry for Q-Fano 3-folds in terms of the Strominger-Yau-Zaslow conjecture and Kontsevich's homological mirror symmetry conjecture, building on work of Auroux. The mirror of a Q-Fano 3-fold is a K3 fibration over the affine line such that the total space is log Calabi--Yau and some power of the monodromy at infinity is maximally unipotent. In 95 cases the Q-Fano is realized as a hypersurface in weighted projective space and we describe the mirror K3 fibration explicitly.