Can you recognise a musician only from listening to her play an instrument? In general, the answer is probably no. Nonetheless, we might be able to learn enough from her pieces to narrow down the possibilities. One of the main aims of Spectral Geometry is to "answer" a mathematical version of the latter question: is it possible to determine (or recover) the Riemannian metric on a manifold by the spectrum of the Laplace operator associated to it. In this analogy the musician becomes a Riemannian metric and the role of the musical instrument is played by the manifold. A Riemannian metric on a manifold allows us to measure the length of and angle between tangent vectors and can be used to make sense of lengths of curves on an abstract manifold. It also gives rise in natural way to an operator to an unbounded operator (the Laplace operator) on a Hilbert space. A milder version of our question is then: to what extent does the spectrum of the Laplace operator determine the Riemannian metric.
Our course will start with a quick review of multilinear algebra, followed by an introduction to the theory smooth manifolds. We then delve into Riemannian geometry and discuss some concepts arising from a Riemannian metric such as different notions of curvature. The Laplace operator associated to Riemannian metric is then introduced and some basic properties of its spectrum on compact manifolds are discussed. After the computation of the spectrum in some simple cases, we will discuss how the spectrum is related to the different kinds of curvature previously introduced. We will also look at some negative answers to the above question; i.e., discuss examples of essentially different Riemannian manifolds whose Laplace operators have identical spectrum. Further concepts will be decided upon by availability of time and the interests of the participants.
Prerequisites for this course are a solid knowledge of linear algebra and single and multivariable analysis as taught in the first two semesters of the bachelor degree in mathematics. Basic knowledge of some concepts from functional analysis such as linear operators on Hilbert spaces, and measure theory are also strongly recommended.
Until further notice, the lecture will take place online. The zoom-link will be shared on moodle some time before every lecture. We will upload the material (the lecture notes and exercises) for every week one week earlier on moodle. This gives you some time to go through the material and prepare your questions. In the lecture time, I will give a review of the notes and answer your questions. I will assume that you have acquired some understanding of the material before the lecture and therefore can use the online lectures mainly as an opportunity to gain a better understanding of the material in the notes.
Date and Time:
Lecture: Mondays 1415-1545, and Thursdays 1015-1145
Exercise Session: Tuesdays 1215-1345 (Rubens Longhi)
Lecturer: Mehran Seyedhosseini