The Singularity Theory and Algebraic Geometry group in the Basque Center for Applied Mathematics is looking for a PhD candidate to work with Ilya Smirnov in the general area of local algebra. The research area of the Fellowship will be local algebra, a branch of commutative algebra that originates from the study of singularities of the system of polynomial equations. More generally, it studies local rings from the viewpoint shaped by singularity theory and algebraic geometry as the results can be applied to local rings formed by the germs of functions at a point. The specific project is to be determined in discussions with the Fellow, the potential areas are multiplicity theory and positive characteristic methods in commutative algebra. Multiplicity theory is one of the oldest parts of local algebra, but it recently had an influx of new ideas largely focused on Lech’s inequality. The inequality can be improved in various ways and this work is yet to be fully completed. “Positive characteristic methods” refer to the algebraic study of singularities over a field of positive characteristic based on the use of the Frobenius endomorphism: there are various classes of so-called F-singularities and an even greater number of numerical invariants. This field is closely connected to algebraic geometry, but it has also purely algebraic open problems.
Last updated: 13 August 2022
Apply by 29 September 2022