Research in Several Complex Variables and Applications

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Proposal Number DMS-0203072PI: Laszlo LempertABSTRACTThis project has two components. One concerns complex analysison infinite dimensional manifolds, and the proposed problems are mostly cohomological in nature. They center around thecomputation of Dolbeault and sheaf cohomology groups. Two typesof manifolds will be investigated: loop spaces of finitedimensional compact complex manifolds and Stein manifolds. One(longer term) goal will be to prove a Hirzebruch--Riemann--Rochtype index theorem on loop spaces, another to discover anappropriate class of sheaves for which cohomology can be provedto vanish on infinite dimensional Stein manifolds (generalizationof the notion of coherent sheaves). I will also work on solvingthe inhomogeneous Cauchy--Riemann equations in pseudoconvexdomains in Hilbert spaces. The finite dimensional component isabout extending the concept of holomorphic motion from the complex plane to motions of higher dimensional complex manifoldsand their subsets. This will hopefully provide a useful tool forhigher dimensional complex dynamics.Mathematics in general elucidates the formal structuresthat underlie the exploration of our world. An example ofsuch a structure is a function, which formalizes the notionof "dependence" among variable quantities. For instancetemperature at some location on Earth depends on thelatitude and the longitude of the location, but also onthe altitude, and the time of the measurement: we have a functionof four variables. In another context, the temperature of saya mole of a gas depends on the velocity of each molecule of thegas, thus on trillions and trillions of variables. Functionsof such a huge number of variables are best understood byidealizing to infinitely many variables. These are the mainobjects of the proposed research: functions of infinitelymany variables. The variables are complex rather than realnumbers. Introducing complex variables typically streamlinesproblems, and having solved the complexified problem one canoften go back and obtain, by restriction, a solution to theoriginal real problem. The problems to be considered are partlymotivated by various mathematical disciplines, and also bytheoretical physics.