+91-674-249-4082

Submitted by sutanu on 31 May, 2017 - 08:10

Date/Time:

Saturday, June 24, 2017 - 15:30 to 16:30

Venue:

Seminar Room, School of Mathematical Sciences

Speaker:

Dr. Soumyashant Nayak

Affiliation:

University of Pennsylvania

Title:

The Many Forms of the Pythagorean Theorem

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.

**School of Mathematical Sciences**

NISER, PO- Bhimpur-Padanpur, Via- Jatni, District- Khurda, Odisha, India, PIN- 752050

Tel: +91-674-249-4081

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