The well-known Pythagorean theorem which relates the lengths of sides of a right triangle to the length of the hypotenuse dates as far back in time as the Babylonians. Later in school when one gets a flavour of trigonometry, this result is neatly packaged in the equation $cos^2 \theta + sin^2 \theta = 1$. With further mathematical sophistication in terms of analytic geometry, one starts talking about $cos \theta, sin \theta$ as representing projections of a unit vector onto the x-axis and y-axis. In this talk, we will discuss the manifestations of the Pythagorean theorem and its converse in higher dimensions. In $n$ dimensions, we will see how this can be viewed as the problem of characterizing diagonal entries of a $n \times n$ projection matrix. Somewhat surprisingly, the set of vectors representing diagonal entries of rank $k$ projections in $M_n(\mathbb{C})$ forms a convex set with a simple description of its extreme points. In more generality, these results reveal connections between various parts of mathematics like combinatorics, representation theory, symplectic geometry which we will briefly touch upon.

Venue

Seminar Room, School of Mathematical Sciences

Speaker

Dr. Soumyashant Nayak

Affiliation

University of Pennsylvania

Title

The Many Forms of the Pythagorean Theorem