Seminars: D. Sullivan, Finite algebraic models of processes in space

SEMINARS


Meetings of the St. Petersburg Mathematical Society September 23, 2013 11:00, St. Petersburg

Joint meeting of St. Petersburg Mathematical Society and Cohomology in Mathematics and Physics program at Euler Institute
Finite algebraic models of processes in space D. Sullivan Stony Brook University
Abstract: One can divide three space into unit cubes and call this the first subdivision. Subdivide once more by cutting each cube into eight identical smaller cubes. Each cell, that is, each vertex, each edge, each face and each cube of this second subdivision is canonically labeled by a pair of cells $[a,b]$ of the first subdivision where $a$ is contained in $b$ [with $a=b$ being allowed]. One can define a partial semigroup structure on such pairs by $[a,b] [b',c] = [a,c]$ iff $b=b'$. In this way the linear combinations of cells in the second subdivision becomes an associative algebra, where we define the product of generators to be zero if $b$ is different from $b'$ and by the semigroup product otherwise. If $d$ denotes the coboundary operator of algebraic topology, then d is a derivation of this associative algebra. This algebra is quite noncommutative. Its associated graded Lie algebra is solvable with commutator sub Lie algebra a three step nilpotent Lie algebra. We can continue to form a third subdivision in the same way by cutting up the second subdivision. We can again define an algebra structure as above using the labeling of the cells in the third subdivision by related pairs of cells of the second subdivision.We continue in this way subdividing and defining new algebra structures at each level. One can make everything finite dimensional at each level of subdivision by imposing triply periodic boundary conditions of length one. One obtains from the subdivisions an inverse system of finite dimensional real linear spaces of cochains related by cochain mappings. A cochain at a finer level defines a cochain at a coarser level by finite integration. These cochain mappings respect d but only preserve the algebra structures in an approximate sense. For example, at the level of cohomology the algebra structure is preserved. We say a sequence of cochains at various levels tending to infinity converges iff for each finite level the tail of the sequence projected to that level converges in the finite dimensional real vector space of cochains there. Question one: Give conditions on two convergent sequences that the products of the sequence elements [projected if need be] at the various levels also converges. One can show that a continuity condition on values of cochains on parallel faces of cubes at each level implies products converge. Moreover, the limit is graded commutative and is consistent with the wedge product of differential forms. Question two: Define and study the Poincare-Hodge star operator One can amalgamate the cells of the second subdivision one way to get the first subdivision. Amalgamating a second way gives the unit cube decomposition which is Poincare dual to the first subdivision. Note that this means that cutting up the unit size dual decomposition also leads to the second subdivision. Similarly two amalgamations of the third subdivision give both the second subdivision and its Poincare dual decomposition. And so forth... One has at each level the Poincare combinatorial star operator providing bijections between one cells and two cells [and between zero cells and three cells] of these dual pairs of amalgamations of the next subdivision. One is interested in convergence properties of these combinatorial star operators. One can use these operations, the products, the star operators and the differentials to make finite dimensional algebraic models of Riemannian manifolds and finite dimensional ODES modeling the evolution equations of fluid motion.