Rings of Algebraic Differential Operators in Mathematical Physics and Geometry

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BerestThe proposed research lies at the intersection of several areas ofmathematics 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 partialdifferential equations. One such problem (known as the problem of lacunas)deals with propagation of waves without diffusion, another with heatkernels on Riemannian manifolds with finite asymptotics. Despite beinganalytic by their nature, these problems exhibit a remarkable feature:they have algebraic solutions. The earlier work of the PI and hiscollaborators links these classical problems of analysis to variousquestions of algebraic geometry and noncommutative algebra, concerningrings of differential operators on algebraic varieties and their specificrepresentations (projective D-modules). One of the primary goals of thisproposal is to systematically study the properties of such modules, theirendomorphism rings and moduli spaces. The results obtained so far (mostlyin 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 onideas of homotopical algebra.The mathematical problems addressed in this project arise from the practicalquestion: what happens when a short signal (light or sound, say) is omittedfrom a point A, travels through a medium and arrives at point B? There aremany possibilities: there could be focusing, diffraction, persistence of faintechoes, or still a short `clean-cut' signal at B. This last possibility is ofparticular interest, but it may occur only under very special conditions.To describe these conditions in precise mathematical terms is one of the goalsof the proposal. The results sought in this direction are fundamental for ourunderstanding of wave phenomena and may have applications in related physicalsciences, 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.