Research in Representation Theory and Automorphic Forms

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AbstractWallachThis project aims to use the methods of analysis, homological algebra and geometry in order to study the structure and applications of representations of real reductive groups. In particular the principal investigator plans to continue his development of what he calls "transfer" between real forms of reductive groups over the complex numbers. A new ingredient involves determining explicit multiplicity formulas (analogous to Blattner's formula) for restriction of representations of semisimple groups to symmetric subgroups (noncompact). The theory also provides a new method for proving the unitarizability of representations that are difficult to analyze analytically. This work uses basic invariant theory and computationally intensive algebraic geometry. For the latter, Matt Clegg and the author will be completing the code for the Groebner array which is now in place. This hardware has also been helpful in solving several problems in combinatorics, representation theory, and quantum computing.The relatively inexpensive access to fast computational tools has changed the landscape of "pure mathematics". In fact, it has blurred the distinction between pure and applied mathematics. Mathematicians can now do experiments involving massive calculations to help predict the form of theorems to be proved by mathematical reasoning. This proposed research will make use of these tools. It will also work to enhance the usefulness of massive calculations whose output can be mathematical objects that are described by files that are billions of bytes long . Although the basic research aims of this project are outside the scope of short term applications the methodology is expected to lead to practical applications. For example, methods to help in deciding which aspects of an immense file that describes a complicated object are most important without having the ability to read the full file.