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We present a list of problems in arithmetic topology posed at the June 2019 PIMS/NSF workshop on Arithmetic Topology. Three problem sessions were hosted during the workshop in which participants proposed open questions to the audience and engaged in shared discussions from their own perspectives as working mathematicians across various fields of study. Participants were explicitly asked to provide problems of various levels of difficulty, with the goal of capturing a cross-section of exciting challenges in the field that could help guide future activity. The problems, together with references and brief discussions when appropriate, are collected below into three categories: 1) topological analogues of arithmetic phenomena, 2) point counts, stability phenomena and the Grothendieck ring, and 3) tools, methods and examples.
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We construct a functor from the smooth 4-dimensional manifolds to the hyper-algebraic number fields, i.e. fields with non-commutative multiplication. It is proved that that the simply connected 4-manifolds correspond to the abelian extensions. We recover the Rokhlin and Donaldsons Theorems from the Galois theory of the non-commutative fields.
Recent important and powerful frameworks for the study of differential forms by Huber-Joerder and Huber-Kebekus-Kelly based on Voevodskys h-topology have greatly simplified and unified many approaches. This article builds towards the goal of putting Illusies de Rham-Witt complex in the same framework by exploring the h-sheafification of the rational de Rham-Witt differentials. Assuming resolution of singularities in positive characteristic one recovers a complete cohomological h-descent for all terms of the complex. We also provide unconditional h-descent for the global sections and draw the expected conclusions. The approach is to realize that a certain right Kan extension introduced by Huber-Kebekus-Kelly takes the sheaf of rational de Rham-Witt forms to a qfh-sheaf. As such, we state and prove many results about qfh-sheaves which are of independent interest.
Arithmetic dynamics is the study of number theoretic properties of dynamical systems. A relatively new field, it draws inspiration partly from dynamical analogues of theorems and conjectures in classical arithmetic geometry, and partly from $p$-adic analogues of theorems and conjectures in classical complex dynamics. In this article we survey some of the motivating problems and some of the recent progress in the field of arithmetic dynamics.
We study the relationship between the local and global Galois theory of function fields over a complete discretely valued field. We give necessary and sufficient conditions for local separable extensions to descend to global extensions, and for the local absolute Galois group to inject into the global absolute Galois group. As an application we obtain a local-global principle for the index of a variety over such a function field. In this context we also study algebra
This is an English translation of Nikolai Chebotaryovs paper Die Probleme der modernen Galoisschen Theorie from 1932. An excerpt from this paper was given as a lecture at the International Congress of Mathematicians in Zurich in 1932. With the lecture being given to commememorate the centennial of Evariste Galois death, the paper is a broad survey of various contemporary problems in Galois Theory the author found represented the culminations of work done by Galois and his successors.
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