Describes numerous properties and applications of Bernoulli numbers and polynomials. These include summing powers of integers, approximate computation of integrals, values of the Riemann zeta function, and Stirling formula. 

A writeup, aimed at an undergraduate audience, of a result about generating functions for orthogonal polynomials. Shows that the only orthogonal polynomials with generating functions of a special form are the ultraspherical, Hermite, and Chebyshev polynomials of the first kind. Defines all the words above. 

Answers the question "Can we integrate x^{2} e^{ x2/2}?", and related questions from calculus. The method uses Hermite polynomials. These polynomials have numerous other interesting and important properties, and appear in many other parts of mathematics, so a survey of these properties is included. 
 
While rules for computing first derivatives of products, compositions, and inverses of functions are wellknown, the rules for higher derivatives are more complicated, and not easy to write down. In this talk, I explain how to use combinatorics to find formulas for the higherorder product, chain, and inverse function rules in calculus. Many pictures of combinatorial structures, such as partitions and trees. 


A talk on how to count moments using combinatorics. A number of examples of orthogonal polynomials are given. For all these examples, the corresponding moments are positive integers, and so count some combinatorial structures: pairings, partitions, and permutations. More complicated moments involve combinatorial structures with crossings. The last part of the talk defines linearization coefficients, which, miraculously, count exactly the same combinatorial structures when one only inhomogeneous objects are included. 

A more updodate version of the talk Orthogonal polynomials and counting permutations above, on the relation between orthogonal polynomials and permutations. 

Continued fractions related to orthogonal polynomials, especially the finite and periodic cases. In particular, explains Gaussian quadrature from numerical analysis. 

Moments, orthogonal polynomials, partitions, cumulants, Motzkin paths, etc., etc. Similar to the last undergraduate talk above, but somewhat more complicated. Ends with a discussion of the moment calculations for noncommutative polynomials. 


Different ways of looking at the free Brownian motion. In Combinatorics, it is related to counting Dyck paths by Catalan numbers. In Operator Theory, it is defined using free creation and annihilation operators. In Random Matrix theory, it appears as the limit of matrixvalued Brownian motions. It also has a realization as a limit in the group algebra of the Permutation Group. 