Mathematics
My Publications:
Abstract: We prove a Galois-type correspondence between compositions of purely inseparable field extensions (including infinite ones) and subalgebras of differential operators. This correspondence can be utilized to establish a connection between separable field extensions and purely inseparable field extensions. Specifically, it serves as a progression towards gaining a deeper comprehension of a foliation theory on varieties in positive characteristic.
Comment: My PhD thesis will contain an extension of this theory to varieties. In particular, one will find there a formula for pulling back a canonical divisor along arbitrary purely inseparable morphism with applications to study of changes of Kodaira dimension, and a formal Frobenius theorem for those morphisms.
Abstract: We explicitly compute canonical liftings modulo p^2 in a sense of Achinger--Zdanowicz of Dwork hypersurfaces. The computation involves studying a compatibility between Hodge filtrations and a crystalline Frobenius. In particular, remarkably, we explicitly compute a partial data of the crystalline Frobenius modulo p^2.
Comment: This paper covers material from my Master thesis.
Abstract: For a prime number q not equal 2 and r > 0
we study, whether there exists an isometry of order q^r
acting on a free Z^p^k-module equipped with a scalar product. We investigate, whether there exists such an isometry with no non-zero fixed points. Both questions are completely answered in this paper if p is not 2 and not q. As an application we refine Naik's criterion for periodicity of links in S^3. The periodicity criterion we obtain is effectively computable and gives concrete restrictions for periodicity of low-crossing knots.