Nagroda Instytutu Matematycznego PAN za wybitne osiągnięcia naukowe w zakresie matematyki
Uroczyste wręczenie Nagrody Naukowej IM PAN odbyło się dnia 21 czerwca 2018 r.
Laureatką nagrody w 2018 r. została prof. dr hab. Agata Smoktunowicz z School of Mathematics, University of Edinburgh, za wybitne osiągnięcia w algebrze.
Laureatka Nagrody wygłosiła referat:
Nil rings and braces
Abstract: In this talk we will consider applications of noncommutative ring theory in other research areas, namely geometry, braces, skew braces and set-theoretic solutions of the quantum Yang-Baxter equation. Circa 2007, Rump presented some surprising connections between nil and nilpotent rings (more generally, Jacobson radical rings) and solutions of the quantum Young-Baxter equation. Rump introduced a structure which he called ”a brace”, to describe all involutive, non-degenerate set-theoretic solutions of the quantum Yang-Baxter equation. In particular, he showed that Jacobson radicalrings are in one-to-one correspondence with two-sided braces, and that every Jacobson radical ring yields a solution to the Young-Baxter equation. Because of their similarity to rings, braces quickly captured the attention of algebraists.It turns out that braces can be investigated using ring theory methods. One might say that ring theory is enjoying a ’second youth’ because of braces! We will look at some new and older results on Jacobson radical rings and nil rings and the corresponding results in brace theory. Recall that a ring is nil if every element to some power is zero. We also mention applications of ring theory in geometry to investigate superpotential algebras, and consider some open questions related to braces.