Some Things We Almost Know
Computer algebra systems have enabled us to conjecture more wildly and effectively than ever before about the plausibility of many mathematical statements. However, these systems have not been accompanied by a parallel increase in our problem solving abilities. This talk concerns a number of such statements drawn from a variety of disciplines: connection coefficients in quantum chemistry, Hankel determinants that arise in combinatorics, adjacency matrices and orthogonal polynomials on graphs, resultants of polynomials. A few of these statements can even be proved.