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Unearthing the visions of a master: harmonic Maass forms and number theory
 HARVARDMIT CURRENT DEVELOPMENTS IN MATHEMATICS 2008, INTERNATIONAL
, 2008
"... Together with his collaborators, most notably Kathrin Bringmann and Jan Bruinier, the author has been researching harmonic Maass forms. These nonholomorphic modular forms play central roles in many subjects: arithmetic geometry, combinatorics, modular forms, and mathematical physics. Here we outli ..."
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Together with his collaborators, most notably Kathrin Bringmann and Jan Bruinier, the author has been researching harmonic Maass forms. These nonholomorphic modular forms play central roles in many subjects: arithmetic geometry, combinatorics, modular forms, and mathematical physics. Here we outline the general facets of the theory, and we give several applications to number theory: partitions and qseries, modular forms, singular moduli, Borcherds products, extensions of theorems of KohnenZagier and Waldspurger on modular Lfunctions, and the work of Bruinier and Yang on GrossZagier formulae. What is surprising is that this story has an unlikely beginning: the pursuit of the solution to a great mathematical mystery.
Infinite products in number theory and geometry
"... Abstract. We give an introduction to the theory of Borcherds products and to some number theoretic and geometric applications. In particular, we discuss how the theory can be used to study the geometry of Hilbert modular surfaces. 1. ..."
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Abstract. We give an introduction to the theory of Borcherds products and to some number theoretic and geometric applications. In particular, we discuss how the theory can be used to study the geometry of Hilbert modular surfaces. 1.
The Classification of the Finite Simple Groups: An Overview
 MONOGRAFÍAS DE LA REAL ACADEMIA DE CIENCIAS DE ZARAGOZA. 26: 89–104, (2004)
, 2004
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Algebras, BPS States, and Strings
, 1995
"... We clarify the role played by BPS states in the calculation of threshold corrections of D=4, N=2 heterotic string compactifications. We evaluate these corrections for some classes of compactifications and show that they are sums of logarithmic functions over the positive roots of generalized KacMoo ..."
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We clarify the role played by BPS states in the calculation of threshold corrections of D=4, N=2 heterotic string compactifications. We evaluate these corrections for some classes of compactifications and show that they are sums of logarithmic functions over the positive roots of generalized KacMoody algebras. Moreover, a certain limit of the formulae suggests a reformulation of heterotic string in terms of a gauge theory based on hyperbolic algebras such as E10. We define a generalized KacMoody Lie superalgebra associated to the BPS states. Finally we discuss the relation of our results with string duality.
HARMONIC MAASS FORMS, MOCK MODULAR FORMS, AND QUANTUM MODULAR FORMS
"... Abstract. This short course is an introduction to the theory of harmonic Maass forms, mock modular forms, and quantum modular forms. These objects have many applications: black holes, Donaldson invariants, partitions and qseries, modular forms, probability theory, singular moduli, Borcherds product ..."
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Abstract. This short course is an introduction to the theory of harmonic Maass forms, mock modular forms, and quantum modular forms. These objects have many applications: black holes, Donaldson invariants, partitions and qseries, modular forms, probability theory, singular moduli, Borcherds products, central values and derivatives of modular Lfunctions, generalized GrossZagier formulae, to name a few. Here we discuss the essential facts in the theory, and consider some applications in number theory. This mathematics has an unlikely beginning: the mystery of Ramanujan’s enigmatic “last letter ” to Hardy written three months before his untimely death. Section 15 gives examples of projects which arise naturally from the mathematics described here. Modular forms are key objects in modern mathematics. Indeed, modular forms play crucial roles in algebraic number theory, algebraic topology, arithmetic geometry, combinatorics, number theory, representation theory, and mathematical physics. The recent history of the subject includes (to name a few) great successes on the Birch and SwinnertonDyer Conjecture, Mirror Symmetry, Monstrous Moonshine, and the proof of Fermat’s Last Theorem. These celebrated works are dramatic examples of the evolution of mathematics.
1 ELLIPTIC GENERA, REAL ALGEBRAIC VARIETIES AND QUASIJACOBI FORMS
, 904
"... Abstract. This paper surveys the push forward formula for elliptic class and various applications obtained in the papers by L.Borisov and the author. In the remaining part we discuss the ring of quasiJacobi forms which allows to characterize the functions which are the elliptic genera of almost com ..."
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Abstract. This paper surveys the push forward formula for elliptic class and various applications obtained in the papers by L.Borisov and the author. In the remaining part we discuss the ring of quasiJacobi forms which allows to characterize the functions which are the elliptic genera of almost complex manifolds and extension of Ochanine elliptic genus to certain singular real algebraic varieties.
ORDERS AT INFINITY OF MODULAR FORMS WITH HEEGNER DIVISORS
"... Abstract. Borcherds described the exponents a(n) in product expansions f = q h Q ∞ n=1 (1−qn) a(n) of meromorphic modular forms with a Heegner divisor. His description does not directly give any information about h, the order of vanishing at infinity of f. We give padic formulas for h in terms of g ..."
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Abstract. Borcherds described the exponents a(n) in product expansions f = q h Q ∞ n=1 (1−qn) a(n) of meromorphic modular forms with a Heegner divisor. His description does not directly give any information about h, the order of vanishing at infinity of f. We give padic formulas for h in terms of generalized traces given by sums over the zeroes and poles of f. Specializing to the case of the Hilbert class polynomial f = Hd(j(z)) yields padic formulas for class numbers that generalize past results of Bruinier, Kohnen and Ono. We also give new proofs of known results about the irreducible decomposition of the supersingular polynomial Sp(X).
′ λ,N,n: M ♯
"... Abstract: We extend results of Bringmann and Ono that relate certain generalized traces of MaassPoincaré series to Fourier coefficients of modular forms of halfintegral weight. By specializing to cases in which these traces are usual traces of algebraic numbers, we generalize results of Zagier des ..."
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Abstract: We extend results of Bringmann and Ono that relate certain generalized traces of MaassPoincaré series to Fourier coefficients of modular forms of halfintegral weight. By specializing to cases in which these traces are usual traces of algebraic numbers, we generalize results of Zagier describing arithmetic traces associated to modular forms. We define correspondences Zλ,N,m: M ♯