Algebra and Cryptography Seminar, Spring 2019

Organizers: Delaram Kahrobaei, Vladimir Shpilrain, Robert Gilman, and Alexei Myasnikov


2:30-3:30 pm
Room 3307, CUNY Graduate Center
365 Fifth Avenue at 34th Street

March 15 : Robert Laugwitz (Rutgers University), Hopfological algebra and categorifications of cyclotomic rings
Abstract: Hopfological algebra is an approach to categorification of small quantum groups suggested by Khovanov in 2005. Here, stable categories of modules over finite-dimensional Hopf algebras are used and replace the usual homological algebra of differentially graded algebras. Based on this approach, sl2-quantum groups were categorified for prime roots of unity by Khovanov-Qi and Elias-Qi through constructing p-differentials on nil-Hecke algebras.
Most of this talk will be an introduction to the algebraic structures involved in Hopfological algebra. Later parts will report on joint work with You Qi (Caltech) on the categorification of general rings of cyclotomic integers.

May 10: Nora Szakacs (University of York, U.K.), Inverse monoids with tree-like Schutzenberger graphs
Abstract: Inverse monoids are monoids equipped with an involutive inverse operation that satisfies the identities mm^{-1}m=m and mm^{-1}nn^{-1}=nn^{-1}mm^{-1}. They form a variety (in the sense of universal algebra), therefore free inverse monoids, and inverse monoid presentations exist. The Cayley graph of an inverse monoid need not be strongly connected. Its strongly connected components are called Schutzenberger graphs, and in these, edges do occur in inverse pairs, hence one can consider them as metric spaces in the usual way. There is a corresponding notion of Schutzenberger automata, which are just Schutzenberger graphs with a chosen initial and terminal vertex. By results of Stephen, solving the word problem is equivalent to deciding the language of all Schutzenberger automata. We show that if all Schutzenberger graphs of an inverse monoid are quasi-isometric to trees, then it satisfies some nice algorithmic properites:
- in any Schutzenberger graph, the language of geodesics starting at a given point is rational,
- the language of the Schutzenberger automata are context-free,
- the word problem for such inverse monoids is uniformly decidable.
This is joint work with Pedro Silva.

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