Talk

Seminar webpage:  https://guests.mpim-bonn.mpg.de/bianchi/wiqi.html

The Milnor torsion is an invariant of unitary flat vector bundles on closed odd-dimensional manifolds. Its analytic counterpart was introduced by Ray and Singer and the equality of these torsions is the celebrated Cheeger-Müller theorem. In this talk, I will discuss how to extend the Milnor torsion to certain equivariant flat superconnections (or representations up to homotopy) for finite group actions and its relation to analytic torsion of the twisted de Rham complex.

The proof of Fermat’s Last Theorem pioneered a new approach to resolving families of ternary Diophantine equations using modularity of residual Galois representations attached to Frey curves. In the case of differing exponents, Darmon gave a framework for resolving generalized Fermat equations in one varying exponent using Frey varieties. In this talk, I will survey the methods and techniques of recent progress on Darmon’s program as well as the challenges that remain in the study of generalized Fermat equations.

We study generalised polynomials, that is, functions that can be expressed using standard algebraic operations and the floor function. Generalised polynomials have long been studied (both explicitly and implicitly) in number theory and dynamics, often using dynamical and ergodic-theoretic methods (especially dynamics on nilmanifolds). These methods enable one to deduce precise information about the average behaviour of generalised polynomials, but allow for complicated behaviour on special sets of density zero.

Let C be a complex projective curve and let L and M be two line bundles on C.  One can associate with L and M  some natural maps, which are called Gaussian maps, between spaces of global sections of certain sheaves on C. These maps have been classically studied in connection with extendability questions of curves on surfaces. In this talk I will focus on the case of Enriques surfaces, presenting some natural questions that arise in this situation.

The study of fully faithful functors, including equivalences, between derived categories of smooth projective varieties (or, more generally, smooth proper triangulated categories) is, in many ways, analogous to the study of rational contractions in the minimal model program. For a Fano manifold, homological mirror symmetry predicts that its derived category admits canonical semi-orthogonal decompositions (related by the braid group action) with remarkable properties, such as compatibility with rational contractions.