In the first part we discuss Bjarni Jónsson's arguesian law for lattices, and its implications for lattice theory and for universal algebra. The geometric diagram in ALV2, discussed in December's talk, is explained (as was promised). We continue with some higher order arguesian identities first shown to us by Bill Lampe. The second part presents some new and old results about congruence varieties and congruence prevarieties. This part includes a discussion of commutator properties and Mal'tsev conditions. In the third part we discuss the importance of some unpublished notes of Alan Day on higher order Polin varieties and their applications to solving some problems on the collection of congruence varieties.
This talk will be devoted to certain generating functions associated with spaces of matrices. First, I will discuss how imposing linear relations subject to mild conditions ("playing board games") on modules of matrices does not affect associated generating functions. Secondly, we will relate Hadamard products of such rational generating functions to shuffle products of permutations. This is based on joint work with T. Rossmann, and work in progress with T. Rossmann and V. Moustakas.
Board and card games... and matrices Sponsored by the Meyer Fund
Apr. 25, 2023 3:30pm (MATH 3…
Topology
Rachel Chaiser (CU Boulder)
X
Matui's HK-conjecture proposes an in-principle computation of the K-theory of the reduced C*-algebra of a (nice enough) groupoid in terms of the homology of the groupoid. This can be viewed as a C*-algebraic analogue of the following phenomenon in algebraic topology. For a CW-complex of dimensiom at most 3, the Atiyah--Hirzebruch spectral sequence computes the K-theory explicitly as a direct sum of cohomology groups. However, this explicit computation fails to hold in general for spaces of dimension 4 and greater.
Given a flat manifold and an expanding self-cover, one can construct a groupoid where the invariants in the HK-conjecture are computed from the corresponding invariants of the manifold. Using this construction, a flat manifold for which the Atiyah--Hirzebruch spectral sequence fails to compute K-theory from cohomology ought to give a counterexample to the HK-conjecture. This method was introduced by Deeley to construct a counterexample given a flat manifold of dimension at least 9. This naturally leads to the question: is there a flat manifold of dimension 4 giving a counterexample?
