Stan Wagon (Macalester College) A Paradox Due to the Absence of the Axiom of Choice; and a Proof of the Tetrahedral Loop Conjecture
Apr. 11, 2017 2pm (MATH 350)
Lie Theory
Monica Vazirani (UC Davis) Representations of the affine BMW algebra Sponsored by the Meyer Fund
Apr. 11, 2017 3pm (Math 220)
Functional Analysis
Marty Walter (CU Boulder) Abelian Groups Revisited
Apr. 11, 2017 3pm (MATH 350)
Topology
Paul Lessard (CU Boulder) Symmetric Spectra
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Ever since the Banach-Tarski Paradox was discovered, people have wondered about eliminating the Axiom of Choice in favor of a system in which the paradox evaporates. But, as discovered by Mycielski and Sierpinski, the most common way of doing this -- taking all sets to be Lebesgue measurable -- leads to a different sort of paradox that is just as disturbing. I will also present a construction that shows how congruent regular tetrahedra can be chained together face-to-face so as to make a loop that is closed to within any preassigned error. This settles a question first raised and studied 60 years ago by Steinhaus and Swierczkowski.
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The BMW algebra is a deformation of the Brauer algebra, and has the Hecke algebra of type A as a quotient. Its specializations play a role in types B, C, D akin to that of the symmetric group in Schur-Weyl duality.
One can enlarge these algebras by a commutative subalgebra $X$ to an affine, or annular, version. Unlike the affine Hecke algebra, the affine BMW algebra is not of finite rank as a right $X$-module, so induction functors are ill-behaved, and many of the classical Hecke-theoretic constructions of simple modules fail. However, the affine BMW algebra still has a nice class of $X$-semisimple, or calibrated, representations, that don't necessarily factor through the affine Hecke algebra.
I will discuss Walker's TQFT-motivated 1-handle construction of the $X$-semisimple, or calibrated, representations of the affine BMW algebra. While the construction is topological, the resulting representation has a straightforward combinatorial description in terms of Young diagrams. This is joint work with Kevin Walker.
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An interesting connection between functional analysis and number theory.