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Atlanta, GA | Posted:
April 26, 2016 *

I work in an area of mathematics called harmonic analysis. This field grew from the fundamental fact that many functions defined over an interval can be decomposed as sums of the simple sine and cosine functions.

I study cases where the above decomposition does not hold - or holds but is not efficient enough - say, because the functions are no longer defined over an interval. The question is whether similar decompositions are possible also in such cases, with the sines and cosines being replaced by other functions with a simple structure.

Usually, the goal is to use functions which mimic the structure of the sines and cosines, in one way or another. By finding good replacements for the trigonometric functions, one obtains a good way to understand the behavior of functions and the interrelationships between them. With this we get an excellent tool to study the mathematical aspect of the way the world around us behaves.

This area is of much interest in natural sciences and engineering, including in sound and image processing, wireless communications and data transmission, methods in quantum mechanics and quantum computing, and the analysis of signals in geophysics and medicine.

Don't be afraid of making mistakes and of asking "stupid questions." The only way to be a scientist is by having the courage to do both.

I joined the faculty in Georgia Tech only recently and was pleasantly surprised by how kind and warm everyone is. I am most excited by the opportunity to be a part of a community that does outstanding science while maintaining the sense of "community."

While going on a walk with your monkey, you encounter a long (though finite) row of poles. The poles are so high that you cannot see the top of any of them. Suddenly, your monkey escapes and jumps to the top of one of the poles. You don't know which one.

The only thing you can do is throw rocks at the poles. If the rock hits at the precise pole your monkey is sitting on, he will jump back to your arms and you could both go home to eat ice cream. However, if you miss, and the rock hits any other pole, then the monkey will jump from the pole he is sitting on to a pole just next to it. So the monkey has two options for where to jump, unless it is at the end of the line of poles.

Find a deterministic tactic that will ensure your success in getting your monkey back and going home to eat your ice cream.

Look for the answer next week in the College of Sciences Facebook page.

When I was an undergraduate student, I very much enjoyed "Proofs from THE BOOK." by Martin Aigner and Günter M. Ziegler. It provides a collection of beautiful mathematical proofs obtained with rather basic tools. Readers would need some basic undergrad knowledge to understand many of these proofs.

The book is dedicated to the famous mathematician Paul Erdos (also referenced by Prasad Tetali).

Here's an excerpt from the preface: "Paul Erdos liked to talk about THE BOOK, in which God maintains the perfect proofs for mathematical theorems... Erdos also said that you need not believe in God but, as a mathematician, you should believe in THE BOOK."

In 1822, Joseph Fourier published his paper regarding the heat equation. The paper includes Fourier's observation that every function can be decomposed into a sum of sines and cosines. (We now know that this is true for many functions but not for every function). This work had a significant impact on the development of mathematics in general and the area of harmonic analysis in particular.

It might be surprising to learn that Fourier wrote a first version of this paper in 1807, and it took him 15 years to succeed in publishing this work. The part of the work regarding the evolution of heat was recognized as significant earlier, but the part regarding the decomposition of functions was considered a disgrace. For this reason, the paper was not published for many years.

Human history has many similar stories of belated recognition, and I think there is a moral in them, although the precise lesson to be learned should be thought of carefully.