The Fourier transform is one of the most celebrated tools in mathematics. In this talk, I will introduce the Fourier transform in $\mathbb{R}^d$ and discuss its many properties and dualities. In particular, I will expand upon a fundamental notion from harmonic analysis called the uncertainty principle—a function and its Fourier transform cannot be simultaneously localized.
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Let F(x,y) be an irreducible polynomial over Q, and let C be the plane curve cut out by F(x,y) = 0. We say that a number field K/Q is generated by F if there exists an algebraic point P on C such that K = Q(P). Recently, Mazur and Rudin suggested the idea of studying the geometry of C by instead investigating the set of all number fields generated by C. This relationship can also be examined in the opposite direction, by fixing a curve or a family of curves and asking what can be said about the fields generated by that curve or family.
Random matrices have a variety of applications in many areas of math and science. I will introduce and motivate random matrices through the lens of population dynamics and briefly discuss other applications. The goal of the talk is to give a description of some cornerstone results from random matrices such as the probability distribution of the eigenvalues of a large random matrix, i.e. the Wigner semicircle law, and the probability distribution of the largest eigenvalue of a large random matrix, i.e. the Tracy-Widom distribution.
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