Modular forms, which are among the most beautiful and important objects in number theory, are functions of a complex variable with an infinite group of symmetries and that also often lead via their Fourier expansions to interesting arithmetical functions. They have important applications in many parts of pure mathematics, ranging from Diophantine equations to differential geometry to coding theory, but in recent years also many different kinds of applications in mathematical physics. In the talk, which assumes no prior knowledge, I will try to explain what modular forms (and a more recent variant called "mock modular forms") are, with explicit examples, and then describe two or three of their most surprising recent appearances in physics: in connection with the string theory of black holes; in connection with the various brands of "moonshine" (the "monstrous" version discovered in the 80's and the "Mathieu" and "umbral" versions discovered recently); and in connection with "Nahm's conjecture", which came originally from conformal field theory and has now been proved (by Calegari, Garoufalidis and myself) and discovered to be related to quantum invariants of knots.
Don Zagier (born June 29, 1951, Heidelberg) is an American mathematician whose main area of work is number theory. He received his Ph.D. in Bonn at the age of 20, and became professor at the age of 24 at the University of Bonn. He was professor at the University of Maryland (1979-1990), Kyushu University (1990-1993), University of Utrecht (1990-2001), Collège de France (2000-2014). Since 1995 he is one of the directors of the Max Planck Institute for Mathematics in Bonn. Since October 2014, he is a Distinguished Staff Associate at ICTP (Trieste).
Zagier won the Cole Prize in Number Theory in 1987, the von Staudt Prize in 2001, and the Gauss Lectureship of the German Mathematical Society in 2007. He became a foreign member of the Royal Netherlands Academy of Arts and Sciences in 1997 and a member of the National Academy of Sciences in 2017. He was invited speaker of the ICM 1986 in Berkley.
Zagier is author of many of the most influential results in number theory and arithmetic algebraic geometry of the last 30 years. He is author of 12 books and over 140 research papers. His main achievements go from the modularity of intersection numbers of algebraic cycles in Hilbert modular surfaces and modular curves, to the Gross-Zagier formula on the arithmetic of elliptic curves; from the calculation of the Euler characteristic of moduli spaces of curves to the description of special values of Dirichlet L-series using polylogarithms.
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