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Fourier series, tiling, and point configurations: a journey from analysis to combinatorics to number theory and back

Mathematics Colloquium

Fourier series, tiling, and point configurations: a journey from analysis to combinatorics to number theory and back
Series: Mathematics Colloquium
Location: ONLINE
Presenter: Alex Iosevich, University of Rochester

The question of whether L^2(\Omega)\Omega a bounded domain in {\Bbb R}^d possesses an orthogonal basis of exponentials, or even an approximate version thereof, i.e a basis of the form {\{e^{2 \pi i x \cdot a}\}}_{a \in A} , superficially appears to be a "technical" question in the realm of classical analysis. In this talk, we are going to see that this question, both in the Euclidean and discrete settings, is connected on a variety of levels with beautiful problems in discrete geometry, geometric measure theory, number theory, and beyond. We are going to illustrate this point using several results, some proved decades ago, and some much more recent. The emphasis throughout the talk will be on the inherent simplicity of the underlying ideas and the utter absurdity of separating mathematics into narrow categories.