Georgia Institute of Technology College of Sciences

The main result to be discussed will be the boundedness from $L^\infty \times L^\infty$ into $BMO$ of bilinear pseudodifferential operators with symbols in a range of bilinear H\"ormander classes of critical order. Such boundedness property is achieved by means of new continuity results for bilinear operators with symbols in certain classes and a new pointwise inequality relating bilinear operators and maximal functions. The role played by these estimates within the general theory will be addressed.

Define the middle layer graph as the graph whose vertex set consists of all bitstrings of length 2n+1 that have exactly n or n+1 entries equal to 1, with an edge between any two vertices for which the corresponding bitstrings differ in exactly one bit. The middle levels conjecture asserts that this graph has a Hamilton cycle for every n>=1. This conjecture originated probably with Havel, Buck and Wiedemann, but has also been (mis)attributed to Dejter, Erdos, Trotter and various others, and despite considerable efforts it remained open during the last 30 years. In this talk I present a proof of the middle levels conjecture. In fact, I show that the middle layer graph has 2^{2^{\Omega(n)}} different Hamilton cycles, which is best possible. http://www.openproblemgarden.org/op/middle_levels_problem and http://www.math.uiuc.edu/~west/openp/revolving.html

A brief presentation, followed by an informal discussion.

The importance of the 2D Fourier transform in mathematical imaging and vision is difficult to overestimate. For instance, the impulse response of an optical system can be defined in terms of diffraction integrals, that are in turn Fourier transforms of a function on a disk. There are several popular competing approaches used to calculate diffraction integrals, such as the extended Nijboer-Zernike (ENZ) theory. In this talk, an alternative efficient method of computation of two dimensional Fourier-type integrals based on approximation of the integrand by Gaussian radial basis functions is discussed. Its outcome is a rapidly converging series expansion for the integrals, allowing for their accurate calculation. The proposed method yields a reliable and fast scheme for simultaneous evaluation of such kind of integrals for several values of the defocus parameter, as required in the characterization of the through-focus optics.