Fourier analysis has been an extraordinarily powerful mathematical tool since its development 200 years ago, and currently has a wide range of applications in diverse scientific fields including digital image processing, forensics, option pricing, cryptography, optics, oceanography, and protein structure analysis. Like a prism that decomposes a beam of light into a rainbow of colors, Fourier analysis transforms signals into a mathematical spectrum of basic wave components of different amplitudes and frequencies, from which many hidden properties in the data can be deciphered. At the abstract mathematical level signals are represented by functions and their filtering and other operations on them by operators. From a functional analytical point of view, these objects are studied by decomposing them into elementary building blocks, some of which have wavelike behavior too. Decomposition techniques such as atomic, molecular, wavelet and wave-packet expansions provide a multi-scale refinement of Fourier analysis and exploit a rather simple concept: "waves with very different frequencies are almost invisible to each other". Many of these useful techniques have been developed around the study of some particular operators called singular integral operators and,  recently, similar techniques have been pushed to the analysis of new multilinear operators that arise in the study of (para) product-like operations, null-forms, and other nonlinear functional expressions. In this talk we will present some of our contributions in the study of multilinear singular integrals and function spaces, and their applications to the development of the equivalent of the calculus Leibniz rule to the concept of fractional derivatives.
 

Abstract: The study of regularity of the Cauchy-Riemann (d-bar) equation in several complex variables has a long history. In this colloquium, we will present several recent results on the d-bar problem. A common scheme for these solutions is to reproduce differential forms via integral representations. We will explain how the Stein and Rychkov extension operators for function spaces can be used to construct integral formulas. As an application, we will show that the integral representations can be used to study the stability of small deformation of complex structures on domains in a complex manifold.

The ideas of quantum physics have had a huge impact on the development of mathematics, all its fields have been influenced. Many notions have emerged, such as quantum groups and algebras, quantum calculus, many special functions. Numbers, the most elementary and ancient concept at the heart of mathematics since the Babylonians, should also have their place in the quantum landscape. This talk is an elementary and accessible overview of the emerging theory of quantum numbers, including motivations, first results, and the connection to other parts of mathematics.

Recall that a square matrix P is called a projection matrix if P^2 = P.  It makes sense to talk about projection matrices with coefficients in any commutative ring; the image of a projection matrix is called a projective module.  This seemingly innocuous notion intercedes in geometric questions in the same spirit as the famous Hodge conjecture because of Serre's dictionary: projective modules are ``vector bundles''. If X is a smooth complex affine variety, we can consider the rings of algebraic or holomorphic functions on X.  Which of the holomorphic vector bundles on X admit an algebraic structure?  I will discuss recent progress on these questions, using motivic homotopy theory, and based on joint work with Tom Bachmann and Mike Hopkins.