Minimal Representations and Functoriality

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Abstract: SavinGordan Savin is continuing his work on minimal representations, with applications to and explicit constructions of some cases of Langlands functoriality which can not be obtained by any other method. The subject of automorphic forms is a crossroad of analysis, algebra, and number theory. It is a subject with roots in indispensable mathematical theories such as Fourier Analysis which impact our everyday lives in many ways. Telecommunications, data transmission, and modern instruments of radiology would not be possible without Fourier Analysis. A purpose of this research is to use tools of analysis (calculus) to answer some questions in number theory, such as to get a better understanding of zeroes of polynomials. Almost everybody knows how to solve a quadratic equation. It is also possible, although it is less known, to find zeroes of any polynomial of degree 3 or 4. A higher degree polynomial, in general, cannot be solved, but can be studied using some special functions of complex variable (called modular forms). This circle of ideas, which surprisingly relates mathematical theories which at first glance do not appear related, is about 30 years old, and called the Langlands Program, after Robert P. Langlands of the Institute of Advanced Sciences in Princeton.