Modular Forms and L-functions

[ambient page updated 15:05, May 17, 2008] ... [ home ] ... [ garrett@math.umn.edu ]
Naturally, various new snippets come into existence before old things are revised or organized. In reverse chronological order:
Notes (reverse chronological order):
This course will illustrate how standard modern mathematics helps us understand complicated phenomena. To do so, we will introduce the phenomena themselves first in (reconstructed forms of) their genuine historical settings, and then reconsider them in more contemporary terms. Part of the point is that modern mathematics helps us do mathematics better.

More specifically, the zeta functions, L-functions, theta series, and modular forms that arose and played an important role in number theory in the 19th century and early 20th (and still do) are more complicated than many more classical mathematical objects and problems. Thus, it is not surprising that assimilation of these things into the body of standard mathematics took many decades. It turns out (as observed by Gelfand, Selberg, Langlands, and others starting in the 1940s, though having antecedents in work of Schur, Frobenius, Lie, Killing, Cartan circa 1890-1910) that spectral decompositions , equivalently, representation theory (of several different genres) can be made to play a decisive role. More colloquially, but quite literally, this is the exploitation of symmetry. We will motivate this viewpoint through many examples, develop the basic techniques, and see how the ideas help us.

And of course many other basic ideas of the last half of the 20th century are useful, as proof mechanisms, as mnemonic devices, and as organizational notions. We will see that it is profitable to expect and look for application of standard ideas such as naive categorical notions (universal mapping properties) from Eilenberg-MacLane, and notions about generalized functions (distributions) from Schwartz and Sobolev. And the relevance of spectral and representation-theoretic structures will be seen to be both pervasive and helpful.

We will regularly return to the primitive original forms of classical questions and see how they can be recast into more intelligible and answerable forms in terms of standard modern mathematics. I hope this will show people that they can acquire sufficient technique and understanding to do new things themselves.

© 1996-2008, Paul Garrett ... [ garrett@math.umn.edu ]
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