Problems arising from theta correspondences

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Finding roots of polynomials is one of the oldest problems in mathematics and number theory. A formula for roots of a quadratic polynomial goes back fifteen hundred years to Brahmagupta, an Indian mathematician. Cubic and quartic polynomial equations were solved by Italian renaissance mathematicians. Higher degree polynomials resisted until modern times, when it become clear that it is not possible to write a simple formula for their roots. Instead, mathematicians realized, polynomials can be understood by looking at permutations of their roots. Permutations of roots form a mathematical object called a group. The group is a measure of the difficulty of a polynomial. Groups were studied extensively in the 20th century and continue to be studied today. Our knowledge of groups can be in turn translated into understanding of polynomials. This is the main object of this project. In particular, we can understand large polynomials whose corresponding groups form a family called G2. These are large and non-trivial groups, for example, the smallest has 12096 elements.The PI will study groups of transformations arising from triality. In mathematics, triality refers to an interaction among three vector spaces. Perhaps the most interesting case is when the three spaces have dimension 8. Then the principle of triality gives rise to several exceptional mathematical structures: the exceptional projective plane, where the usual axioms of Euclid hold while some expected properties do not, and exceptional groups of transformations called G2 and D4. The group G2 was discovered by german mathematicians about one hundred years ago, but has recently attracted much attention due to its role in physics and string theory, in particular.The main object of this work is to classify representations of the group G2 over local fields, as predicted by Langlands conjectures. To that end we study exceptional theta correspondences and dual pairs where one member of the dual pair is G2. The key new ingredient is a conservation principle for G2, somewhat analogous to the conservation principle for classical groups, however, with the group D4 playing the role of the Kudla-Rallis doubling trick. Thus much of the work is devoted to study of D4 and its representations.