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- Abel Prize for Langlands’ Unified Theory
His work ultimately boiled down to proving part of the Shimura—Taniyama—Weil conjecture, now known as the modularity theorem. Essentially a special case of the Langlands program, this theorem asserts the existence of a correspondence between elliptic curves and modular forms.
Hill / Jordan / Hurley | Representation in Congress | | A Unified Theory
Elliptic curves are fundamentally number-theoretic objects. One can obtain information about the points on the curve by reducing modulo , for each prime.
Wiles had the insight that elliptic curves could be studied via Galois representations, cementing the relationship with the Langlands program. Despite signs of early talent he, along with most of his classmates, had no plans to attend university. A teacher in his final year of school convinced him otherwise with an hour-long impassioned plea in front of his class. He proceeded to the University of British Columbia and then to Yale for graduate study, where he was left largely to his own devices. On the strength of his work on elliptic operators and Lie groups, he was appointed Instructor at Princeton upon graduation in An early conversation with Selberg would prove fruitful.
Selberg had been developing the theory of real analytic Eisenstein series for discrete subgroups of with quotients of finite volume. Classically, an Eisenstein series is a well-behaved function of a complex variable, holomorphic on the upper half-plane, that displays a certain symmetry a functional equation when acted upon by the group. The archetypal Eisenstein series is , where the sum is over all integers and that are not both zero, and where is an integer.
The Eisenstein series considered by Selberg are not necessarily holomorphic, but are square-integrable eigenfunctions of the hyperbolic Laplace operator. In applied contexts, eigenfunctions and eigenvalues of the Laplace operator have physical significance. The frequencies produced by beating a drum are eigenvalues and the wave forms are eigenfunctions. Selberg showed that Eisenstein series represent the continuous spectrum of and can be analytically continued to a meromorphic function on. To unveil the discrete part of the spectrum, Selberg developed his trace formula.
For a basic example of a trace formula, consider a diagonalisable matrix. By examining this time period, Grose is able to move beyond problems of observational equivalence hampering past scholars. The results are based on participant observation in 17 congressional districts and quantitative analyses of the entire U. Congress from the late s through the s.
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Abel Prize for Langlands’ Unified Theory
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