Seminars, Colloquia, and Conferences
2016 Norbert Wiener Lectures
Noam Elkies
Professor, Department of Mathematics, Harvard University
CANONICAL FORMS
(April 78, 2016)
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Public Lecture
Canonical forms: the mathematical structure of musical canons
Thursday, April 7, 2016  4:30pm  Varis Lecture Hall, Granoff Music Center 155
Reception: 5:30pm; Murnane Lobby, Granoff Music Center
Abstract:
Musical canons, from simple rounds like Three Blind Mice to the
compendium of canons Bach compiled in his Musical Offering, have
a history almost as long as that of Western music itself, and continue
to fascinate musical composers, performers and listeners. In a canon the
same melody is played or sung in two or more parts at once; this melody must
therefore make musical sense both as a tune and in harmony with a delayed or
otherwise modified copy of itself. How does one go about constructing such a
melody? This challenge has a mathematical flavor. It turns out that some kinds
of canons are so easy to create that they can be improvised in real time, while
other kinds are more demanding, and in some cases only a handful of examples are
known. The talk will be illustrated with both abstract diagrams and specific musical
examples, and may also digress into generalizations of canons (the forms known
collectively as 'invertible counterpoint') and the reasonsbesides showing off
that so many composers incorporate canons into their music.
Colloquium
Poisson summation and packing problems
Friday, April 8, 2016  4:00pm  Science and Technology Center 136
Reception: 3:00pm; Science and Technology Center Atrium
Abstract:
The Poisson summation formula asserts that the sum over the integers of "any"
function \(f\) on \(\mathbb R\) equals the sum over integers of the Fourier transform
of \(f\). We give some examples and applications, including evaluations of familiar
sums (such as Euler's solution
\(1 + \frac 14 + \frac 19 + \frac{1}{16} + ... + \frac{1}{n^2} + ... = \frac{\pi^2}{6} \)
of the Basel problem) and lessfamiliar ones such as
\(\frac 12 + \frac 15 + \frac{1}{10} + \frac{1}{17} + ... + \frac{1}{n^2+1} + ... \)
We then generalize to functions on \(n\)dimensional space \(\mathbb{R}^n\), and report on the recent
application to the solution by Viazovska et al. of the problem of densest packing of \(\mathbb{R}^n\)
by identical spheres in the "magic dimensions" \(n=8\) and \(n=24\).
The Norbert Wiener Lectures were initially funded by an anonymous gift
to the Department of Mathematics. The 2016 lectures are cosponsored by the Department of Music and Tufts STS.
All talks are free and open to the public.
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