The proposed research lies at the intersection of several areas of mathematics and mathematical physics, including geometric analysis, representation theory, noncommutative algebra and algebraic geometry. It is motivated by old (yet unsolved) problems in the theory of partial differential equations. One such problem (known as the problem of lacunas) deals with propagation of waves without diffusion, another with heat kernels on Riemannian manifolds with finite asymptotics. Despite being analytic by their nature, these problems exhibit a remarkable feature: they have algebraic solutions. The earlier work of the PI and his collaborators links these classical problems of analysis to various questions of algebraic geometry and noncommutative algebra, concerning rings of differential operators on algebraic varieties and their specific representations (projective D-modules). One of the primary goals of this proposal is to systematically study the properties of such modules, their endomorphism rings and moduli spaces. The results obtained so far (mostly in the case of curves) reveal many interesting connections with noncommutative geometry, representation theory, combinatorics and integrable systems. Another goal of the proposal is thus to investigate some of these connections. In a different direction, the PI intends to develop a new method for computing asymptotics of heat kernels based on ideas of homotopical algebra.
The mathematical problems addressed in this project arise from the practical question: what happens when a short signal (light or sound, say) is omitted from a point A, travels through a medium and arrives at point B? There are many possibilities: there could be focusing, diffraction, persistence of faint echoes, or still a short `clean-cut' signal at B. This last possibility is of particular interest, but it may occur only under very special conditions. To describe these conditions in precise mathematical terms is one of the goals of the proposal. The results sought in this direction are fundamental for our understanding of wave phenomena and may have applications in related physical sciences, including the theory of electromagnetic and acoustic waves, space communication technologies, magneto-hydrodynamics, crystal optics, etc. As a broader impact, the PI expects that the interdisciplinary nature of this work will stimulate communication and collaboration between experts in the various areas involved.
