The Department of Mathematics at Northwestern University plans to hold an emphasis year in Algebraic and Smooth Microlocal Analysis in the 2011-12 academic year. This entails having a program of visitors in the field at all levels and for varying durations and holding four conferences in the field during the course of the year. Microlocal analysis is analysis in phase space. It is the mathematics underpinning the semi-classical limit in quantum mechanics, as well as having applications in other areas of physics and geometry. The proposed project would bring together researchers using microlocal methods in widely disparate contexts, ranging from quantum mechanics to representation theory, complex geometry, and mirror symmetry. The central feature of the emphasis year is to be a program of four workshops, as well as mini-courses from visiting mathematicians. The topics of the workshops will be: (i) Algebraic Microlocal Analysis (D-modules and deformation quantization); (ii) Analytic Microlocal Analysis (microlocal analysis in the complex domain); (iii) Spectral and Scattering theory in the C-infinity setting; (iv) Nonlinear Evolution Equations. These four workshops represent quite distinct areas of microlocal analysis, reflecting the major areas of concentration in the field at Northwestern. The workshop on Analytic Microlocal Analysis is intended to unify and bridge the different areas. Many of the technical triumphs in microlocal analysis lack approachable expositions, and the emphasis year should help make microlocal tools more accessible to researchers in many fields.
Microlocal Analysis is a part of analysis and geometry that studies functions, differential equations, and other objects on a space in terms of its phase space. A point of the phase space describes a position of a particle on the original space together with its momentum. In particular, its dimension is double that of the original space. The phase space plays a fundamental role in classical and quantum mechanics, as well as in optics (as the space of light rays) and other parts of physics. There are more general types of phase spaces, called symplectic manifolds. In particular, large classes of complex manifolds, or geometric spaces based not on real but on complex numbers, are in this class. There are other, less well-understood, connections between symplectic manifolds and complex manifolds; they are expressed by the mirror symmetry, an area of both string theory and geometry. All this suggests that microlocal methods play crucial role in many areas of mathematics and physics. In fact these areas spread all the way from quantum field theory to differential geometry to complex analysis to number theory. The goal of the Emphasis Year at Northwestern is to bring together leading researchers working in these different areas. More information can be found on the website: www.math.northwestern.edu/~dbaskin/spscconf/
Microlocal analysis is a discipline that studies differential equations and their solutions not only in terms of the space on which it is defined but in terms of another space, of dimension twice that of the original one, which is called the cotangent bundle. In physics, points of this new space are pairs (q,p) where q is the point of our physical space and p is a momentum of a particle. The cotangent bundle is also called the phase space. A physical particle may be viewed as moving in the phase space, since both its position and momentum are changing with time. Microlocal methods proved to be remarkably successful in differential equations and their numerous applications. Intellectual merit. Microlocal analysis and its methods and ideas are being developed in various contexts: real smooth (when our spaces are multi-dimensional spaces over real numbers); complex analytic (when we consider spaces over complex numbers); algebraic (when our spaces are algebraic, i.e. consist of solutions of algebraic equations). There are other versions of microlocal analysis. One is sheaf theoretical, where instead of solutions of differential equations one considers different but related geometric objects called sheaves. Another studies microlocal methods of analysing spaces of solutions of algebraic equations modulo a prime number p. All these different contexts, while related, are studied by different schools using quite different methods. Putting them in a unified context was the main aim of the project. The for meetings, four lecture courses, and a volume of proceedings united different methods and ideas and made them available to experts all over the field, as well as those who work in applications. Those applications include differential equations, quantum mechanics and other fields of mathematical physics, representation theory, and number theory. Broader impact. The audience of the spring school, four conferences, four lecture courses, as well as the readership of the book, included a large number of students, postdocs, and other young mathematicians that got an ample opportunity to study and practice a broad array of methods and ideas of microlocal analysis and its applications.
