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Department of Mathematics University of Notre Dame GRADUATE STUDENT SEMINAR Guest Speaker: Eric Wawerczyk University of Notre Dame Date: Monday, October 3, 2016 Time: 4:00 PM Location: 129 Hayes-Healy Hall Lecture Title: Dreams of a Number Theorist: The left side of the right-half- plane Abstract Euler studied the Harmonic series, the sums of reciprocals of squares, cubes, and so on. Chebyshev was able to package these together in the zeta function, defined for a real number greater than 1. Riemann had the idea of defining the function for complex numbers and found that this infinite series converges as long as the real part of the imaginary number is greater than 1. He then uses the inverse mellin transform of the Theta Function (see: modular forms) to derive the "functional equation" which gives a UNIQUE meromorphic extension to the whole complex plane outside of the point s=1 (harmonic series diverges=pole). This unique extension is called the Riemann Zeta Function. For over 150 years Number Theorists have been staring at the left side of the right half plane. We will explore this mysterious left hand side throughout the talk and its relationship to Number Theory. This will serve as our prototypical example of Zeta Functions. Finally, we will discuss generalizations of the zeta functions (Dedekind Zeta Functions, Artin Zeta Functions, Hasse-Weil Zeta Functions) and our daunting task of extending all of them to their own left sides and what mysteries lie within (The Analytic Class Number Formula, The (proven) Weil Conjectures, the Birch and Swinnerton-Dyer conjecture, The Langlands Program, and the (proven) Taniyama-Shimura Conjecture, the Grand Riemann Hypothesis ).

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Robert P. Langlands
The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2018 to Robert P. Langlands of the Institute for Advanced Study, Princeton, USA, “for his visionary program connecting representation theory to number theory.” The Langlands program predicts the existence of a tight The group GL(2) is the simplest example of a non- web of connections between automorphic forms and abelian reductive group. To proceed to the general Galois groups. case, Langlands saw the need for a stable trace formula, now established by Arthur. Together with Ngô’s proof The great achievement of algebraic number theory in of the so-called Fundamental Lemma, conjectured by the first third of the 20th century was class field theory. Langlands, this has led to the endoscopic classification This theory is a vast generalisation of Gauss’s law of of automorphic representations of classical groups, in quadratic reciprocity. It provides an array of powerful terms of those of general linear groups. tools for studying problems governed by abelian Galois groups. The non-abelian case turns out to be Functoriality dramatically unifies a number of important substantially deeper. Langlands, in a famous letter to results, including the modularity of elliptic curves and André Weil in 1967, outlined a far-reaching program that the proof of the Sato-Tate conjecture. It also lends revolutionised the understanding of this problem. weight to many outstanding conjectures, such as the Ramanujan-Peterson and Selberg conjectures, and the Langlands’s recognition that one should relate Hasse-Weil conjecture for zeta functions. representations of Galois groups to automorphic forms involves an unexpected and fundamental insight, now Functoriality for reductive groups over number fields called Langlands functoriality.
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Arxiv:1802.06193V3 [Math.NT] 2 May 2018 Etr ...I Hsppr Efi H Udmna Discrimi Fundamental the fix Fr We Quadratic Paper, Binary I This a Inverse by in the Squares 1.4.]
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