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Introducing Franz Luef - Creating a Mathematical Dictionary

bridges vol. 24, December 2009 / News from the Network: Austrian Researchers Abroad
By Astrid Roemer

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Franz Luef

"It was fun." Rarely does one hear this statement in reference to mathematics classes in school. It makes one wonder if there is something like a mathematics gene after all, giving some selected few the innate ability to see the underlying mathematical structure of the world surrounding us. If there is such a gene, Franz Luef, a Marie Curie Fellow at UC Berkeley who focuses on Time-Frequency Analysis, is certain to have it.

From an early age, Luef was drawn to mathematics, exhibiting a natural talent which developed from a playful approach to the subject to a deeper need to understand. "During high school I had some questions which I would have loved to have had an answer to, for example what is π? So, I sat down and tried to compute it. I came up with some formulas which I later found in textbooks." Encouraged by his early independent successes, he started to read about mathematics, figuring that if he was good at mathematics he would also be good at various other subjects like physics or mechanics.



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Despite his initial attraction to mathematics, physics was his favorite subject - theoretical physics in particular, "because there is this big theory of quantum mechanics. When I first heard about it, it was such a mystery." Instead of choosing between mathematics and physics, he decided to get an M.A. in both subjects at the University of Vienna, only later

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The 21st century way of getting from Point A to Point B.

deciding to pursue a career as a mathematician. "My seminal change of focus came after attending a lecture on topology," Luef explains. It is somehow telling that his change of direction resulted from a lecture on this topic, as topology basically deals with the various ways and properties of how one can get from point A to point B. Take your way to work in the morning as an example: You leave your apartment, point A, then you can choose between different ways of transportation (e.g., public transportation, bicycling, walking) to get to your office, point B. Depending on the chosen mode of transportation, the route, the distance covered, and the time it takes to reach your office will vary respectively. Even though topology certainly is a fascinating field of research in its own right, it was not the content of the lecture itself but the explanation of the scientific approach that made Luef come to the conclusion that, in comparison to physicists, mathematicians "seemed to have a clearer understanding of what's going on."

Assembling the pieces

Going from A(ustria) to B(erkeley) to work with Prof. Marc Rieffel is exactly what Luef intended in 2005 after finishing his Ph.D. dissertation: "Gabor analysis meets non-commutative geometry." In order to do so, he applied for two fellowships - and was awarded both. One was the Austrian Max Kade Fellowship , which he declined in favor of the European Marie Curie International Outgoing Fellowship . Clearly, as both these grants are extremely prestigious and any application has to be peer-reviewed, Luef's fellow mathematicians were of the opinion that his research was original, innovative, and groundbreaking.

Asked about his field of research, Luef gives a simple, yet (to anyone without a Ph.D. in mathematics) cryptic answer: "The combination of time-frequency analysis with noncommutative geometry." Remember when you were a kid and built little houses with Lego blocks? To build the aforementioned structure, we have to put different Lego building blocks together to get an idea what Luef's research is all about:

Building Block #1: Time-Frequency Analysis

A set of mathematical techniques used to characterize and/or manipulate signals with changing frequency, e.g., speech, music, heart rate. Note that signals are described as waves.

 

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Wave function with increasing frequency.

Whenever you listen to a song on your MP3 player, you are using a time-frequency analysis application. The MP3 format employs a form of "lossy" data compression, i.e., applying an algorithm to the audio wave function of the original recording in order to reduce the data size (thus the "loss") of the audio file while still maintaining the impression of a faithful reproduction of the original audio. Put very simply, the algorithm cuts off all the peaks of the audio wave function that are inaudible to the human ear in the first place, so any data below 20 Hz and above 20 kHZ will be eliminated.

Building Block #2: Noncommutative Geometry (NCG)

The name stems from the Latin word commutare, which basically means exchangeable in terms of sequence. In mathematical terms, commutative means that the sequence on one side equals the reversed sequence on the other side, whereas noncommutative means that the sequence on one side does not equal the reversed sequence on the other side. Combined with  classic geometry, NCG deals with noncommutative spaces, i.e., xy does not always equal yx.


An easy example of a commutative process is addition: clearly a commutative action since a+b = b+a, e.g., 1+2 = 2+1. Another everyday example of a commutative action would be putting on your shoes; no matter if you put on your left or your right shoe first, the result is always the same. Division, however, is a noncommutative action: a/b ≠ b/a, e.g., 3/1 ≠ 1/3. A noncommutative action in everyday life would be washing and drying your clothes - changing the sequence of actions leads to significantly different results.

Whereas classic geometry (Ancient Greek: geo = earth, metria = measures), which is concerned with all types of measures (size, shape, relative position of objects, and properties of space), is one of the oldest branches of mathematics, NCG is a fairly recent development in mathematics. NCG was created by the mathematician Alain Connes in the 1980s and deals with noncommutative spaces, i.e., in NCG xy does not always equal yx. On the plane of mortal human existence, called everyday life, it makes no difference if you take 4 steps per 1 meter or 1 step per 4 meters, you would always cover the same distance, ending up in the same spot both times with, in the latter case, probably an entry in the Guinness Book of World Records. But in NCG you might find yourself in a different place altogether, depending on which combination of steps and length units (1x4 or 4x1) you have taken. Understandably, this concept may prove challenging to grasp for the uninitiated with no technical training in mathematics or physics so, for the sake of argument, let us simply take it at face value.

Building Block #3: Heisenberg's uncertainty principle

This is a concept in quantum mechanics which states that certain pairs of physical properties cannot both be known to arbitrary precision, e.g., the more precisely one knows the position of a particle, the less precisely its momentum can be known. Note that, in quantum mechanics, a particle is described by a wave.

 

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The flying Heisenberg uncertainty principle.

Look at a hummingbird and you will find that you cannot discern a single wing beat. All that is observable to the human eye is a blur, since a hummingbird flaps its wings with a frequency up to 70 times per second. Only with a high-speed camera is it possible to make a hummingbird's wing beat clearly visible. Unfortunately, such equipment does not, nor will it ever, exist for quantum mechanical systems. This was the fundamental insight of Werner Heisenberg, as stated in his uncertainty principle. The subatomic particles will remain in a more or less sharp blur.

The bigger picture

Now put all the building blocks together: signals described by waves and a toolkit for manipulating signals (Time-Frequency Analysis), a toolkit for locating and describing places non-mathematicians would never have thought of (Noncommutative Geometry), and a principle stating that certain pairs of physical properties of a particle (described by a wave) can never be pinned down with arbitrary precision at the same time (Heisenberg's uncertainty principle). Where do you end up? With the basic principles of wireless communication and ways of improving it. This is one of the projects Luef is currently working on. To illustrate this rather abstract work description with a more real-life example: Whenever you use your mobile phone or Skype to talk to someone, the audio signal of your speech is translated into a digital signal composed of zeros and ones, which is transmitted and afterwards reassembled into an audio signal, always accompanied by what is perceived as noise (this is where Heisenberg's uncertainty principle comes in). The improvement of the audio signal reconstruction is the goal of Luef's research. Still, we are talking here about classic basic research, several steps removed from the actual application. Luef leaves that part to the engineers.

The innovative part of his research is the way he comes up with functions better suited for the reassembly process, namely by building a bridge between two distinct fields of mathematics - Time-Frequency Analysis and Noncommutative Geometry. "What I try to do is to create a dictionary that allows you to translate a notion from one side to the other side." As every

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The Pythagorean theorem - a mathematical translation.

mathematical field has its own tools, its own way of solving problems, transferring a problem from one field into another allows Luef to tackle it with a different toolkit. Sometimes it happens that tools from another field are more apt for successfully tackling the challenge at hand. This is a common mathematical technique, which every one of us has frequently used in school. The best example would be the Pythagorean theorem in geometry, which states that in a right triangle the square of the hypotenuse (the side opposite the right angle), is equal to the sum of the squares of the other two sides. You can prove this theorem either using techniques from geometry (constructing two squares - one would be the merged squares of a and b and the other would be constructed from c - placing them on top of each other, and seeing that their areas are identical) or by translating it into algebraic terms so you end up with the familiar equation  a2 + b2 = c2 which is easier to deal with in classrooms as it only requires pen and paper instead of strings and sticks.

When asked how many people in the world actually understand his research in detail, Luef answered: "Maybe ten," followed by the explanation that it is the combination of these two fields that usually confuses his colleagues, because knowing and understanding time-frequency analysis certainly does not imply that you understand noncommutative geometry, one of the more abstract fields of mathematics, and vice versa. One has to bear in mind that humankind's knowledge of mathematics has increased exponentially since the Greeks invented/discovered it. One can, of course, argue about whether mathematics is an invention at all or whether it pre-existed as the underlying structure of the universe and we are simply in the process of uncovering it bit by bit. The vast mathematical body of knowledge, as a whole, can nowadays no longer be grasped by a single individual's mind as it has branched out and became increasingly diversified and specialized over the centuries. The last time a single human being could know and understand all the mathematics of their time was probably some time in the late 19th century - a reassuring fact for those of us whose mathematical education ended with our high school diploma.

The social mathematician

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Mathematicians - eccentric and brilliant? Could be. Recluses? Not necessarily.

Imagine that there are only ten people in the whole world who understand in detail what you are working on. This sounds like a very lonely line of work, bringing to mind the stereotype of a mathematician - the eccentric, brilliant recluse. But this stereotype is the exception, says Luef. "In general, mathematicians love to talk about their work with other mathematicians." Communication is just as vital in mathematics as in every other science. By talking to others one gets new ideas and gets acquainted with the work of colleagues which might even be relevant to furthering one's own research. According to Luef, "People would be surprised how social mathematics is."

Luef remains modest and rather vague when talking about his plans for the future after the end of his Marie Curie Fellowship at Berkeley in 2011. "I would like to have a tenure track position at some university" - "some" referring in geographical terms to Europe or North America. Regarding his professional future, he behaves like a particle according to the Heisenberg uncertainty principle. As we know his exact momentary position, we have no clue about his momentum.  And with the passage of time as another unknown variable, we can only state for certain that Luef "will be somewhere, but we don't know where it will be."
 

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This article is based on an interview conducted by the author, Astrid Roemer, with Dr. Franz Luef, International Outgoing Marie Curie Fellow in the Department of Mathematics at UC Berkeley.


Sources:

Franz Luef
http://homepage.univie.ac.at/franz.luef/index.php

Alain Connes
http://www.alainconnes.org/en/

Thierry Masson - An informal introduction to the ideas and concepts of noncommutative geometry
http://arxiv.org/abs/math-ph/0612012

Masoud Khalkhali - Very basic noncommutative geometry
http://arxiv.org/abs/math/0408416

Alain Connes & Matilde Marcolli - A walk in the noncommutative garden
http://arxiv.org/abs/math/0601054

Philip J. Davis, Reuben Hersh. The Mathematical Experience. Birkhäuser, 1995.


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