4,064 research outputs found
Poincare-Birkhoff-Witt Theorems
We sample some Poincare-Birkhoff-Witt theorems appearing in mathematics.
Along the way, we compare modern techniques used to establish such results, for
example, the Composition-Diamond Lemma, Groebner basis theory, and the
homological approaches of Braverman and Gaitsgory and of Polishchuk and
Positselski. We discuss several contexts for PBW theorems and their
applications, such as Drinfeld-Jimbo quantum groups, graded Hecke algebras, and
symplectic reflection and related algebras.Comment: 30 pages; survey article to appear in Mathematical Sciences Research
Institute Proceeding
Global surfaces of section for Reeb flows in dimension three and beyond
We survey some recent developments in the quest for global surfaces of
section for Reeb flows in dimension three using methods from Symplectic
Topology. We focus on applications to geometry, including existence of closed
geodesics and sharp systolic inequalities. Applications to topology and
celestial mechanics are also presented.Comment: 33 pages, 3 figures. This is an extended version of a paper written
for Proceedings of the ICM, Rio 2018; in v3 we made minor additional
corrections, updated references, added a reference to work of Lu on the
Conley Conjectur
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