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Mark Gerasimov
Mark Gerasimov

Handout: beta.html, beta.pdf, beta.png.Sources: Pensieve / Program: 2012-06 / Regina_Talk.Talk 1, Ben-Gurion University Colloquium on January 1Meta-Groups, Meta-Bicrossed-Products, and the Alexander PolynomialThis will be a repeat of a talk I gave in Regina in June 2012.Abstract. The a priori expectation of first year elementary schoolstudents who were just introduced to the natural numbers, if they would beready to verbalize it, must be that soon, perhaps by second grade, they'dmaster the theory and know all there is to know about those numbers. Butthey would be wrong, for number theory remains a thriving subject,well-connected to practically anything there is out there in mathematics.I was a bit more sophisticated when I first heard of knot theory. Myfirst thought was that it was either trivial or intractable, and mostdefinitely, I wasn't going to learn it is interesting. But it is, andI was wrong, for the reader of knot theory is often lead to the mostinteresting and beautiful structures in topology, geometry, quantumfield theory, and algebra.Today I will talk about just one minor example, mostly having to dowith the link to algebra: A straightforward proposal for a group-theoreticinvariant of knots fails if one really means groups, but works oncegeneralized to meta-groups (to be defined). We will construct onecomplicated but elementary meta-group as a meta-bicrossed-product (to bedefined), and explain how the resulting invariant is a not-yet-understoodyet potentially significant generalization of the Alexander polynomial,while at the same time being a specialization of a somewhat-understood"universal finite type invariant of w-knots" and of an elusive "universalfinite type invariant of v-knots".Handout: bh.html, bh.pdf, bh.png.Sources: 2012-08.There's also a paper in progress.Talk 2, Hebrew University Seminar on January 2Balloons and Hoops and their Universal Finite Type Invariant, BFTheory, and an Ultimate Alexander InvariantTalk cancelled due to illness.



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