PRE2023- Symmetries and Invariants in Geometry and Arithmetic (PID2022-142024NB-I00)

There are many possible PhD projects in Arithmetic Geometry, supervised by D. Macías (sites.google.com/site/danielmaciascastillo), or in Algebraic Topology, supervised by F. Cantero (matematicas.uam.es/~federico.cantero). Arithmetic Geometry studies both number fields and function fields of curves over finite fields, and rational points on varieties defined over such fields, or even on Galois representations. This study relates them to special L-values. An important example is the Birch and Swinnerton-Dyer Conjecture for elliptic curves or abelian varieties. A possible PhD problem is to investigate this conjecture through the ideas of Iwasawa theory. Recently there has also been much interest in Drinfeld modules over functions fields, a source of Galois representations and L-values in positive characteristic. There are also many possible PhD problems in this new and exciting direction. Algebraic Topology is an area of mathematics that investigates spaces using algebra. Classifying the many ways a string can sit in the Euclidean space is the subject of knot theory, a very active area of research. One of the most promising tools to study them is Khovanov homology, which was used recently by Lisa Piccirilo to prove that the Conway knot is not slice. This invariant was connected to the field of stable homotopy theory ten years ago by Lipshitz and Sarkar. The research of the student may also lie in between these two areas.