Commutative Algebra and Algebraic Geometry: The Fellowship of the Ring: Markov chains from algebraic complexes

Seminar | October 1 | 3:45-4:45 p.m. | 748 Evans Hall

 Perci Diaconis, Stanford University

 Department of Mathematics

There are many natural examples where a graded vector space can be associated to a combinatorial objet or variety. This comes with boundary maps and so homology and a Laplacian(down&up + up&down). Surprisingly often the Laplacian has all positive coefficients in a natural basis and a Markov chain can be found. Now, algebra and probability can be combined to get sharp analysis of the rate of convergence of the chain to stationarity. For example, the poset homology of the Boolean lattice yields ‘random adjacent transpositions’ in S(n).