There are perhaps only three types of universally crucial quantities in this world. The first is energy E, or equivalently mass m according to Einstein's energy-mass transition formula: E=M c². The second is momentum  or angular momentum (or spin) J. And the last,  yet perhaps the most mysterious and fascinating, is entropy S.

With this well claimed, you start to question me:  "Jackie, are you teaching a course in physics or mathematics?"

It doesn't matter, as long as you do bear a great mission in your mind and heart, and an extraordinary goal to accomplish. If still you are attempting to comfort your inner psychology, then please let us call it mathematical physics, but in this splendid era of information and digitial technology.

In classical mechanics, unlike energy (or Hamiltonian) and momentum, entropy does not belong to a single particular state in the phase space. Instead, it is immersed within an ensemble (or often a large collection) of all possible states (or quantized (eigen) states as in quantum mechanics), and characterizes the overall degree of freedom of such a system, or pessimistically speaking, chaos. Thus entropy is a powerful tool for studying many-body systems, such as a balloon of gas molecules, a lattice of  crystals, a beam of light full of photons, and, even in the much larger scale, the earth as a system, and a black hole in the universe.

Now, your nerves just cannot stop jumping and fluttering - "So Jackie, at the end of the day, what you are saying is that you are indeed planning to teach physics !!!" Hold on, please. In 1910, how many even greatest minds on this planet  once trusted Einstein's equation: "Are you insane, Einstein?! how can touchable and visible mass be turned into abstract and intangible energy?" That is also my point --- why cannot mechanical or tangible physics be useful for abstract and sometimes seemingly psychological theory of information and patterns? So, please, calm down, and let me finish. [This as well natually reminds you the celebrated duality between tangible "particles" and abstract "waves" in quantum mechanics.]

It is certainly not I who have proposed this approach for the first time in this world. In fact, the real credit should go to the great applied mathematician Claude Shannon, the founder of modern information and communication theory. In this semester, we plan to read his original and seminal paper that was written in the wake of World War II (also an intense war of coding and code breaking), in which the originally very "mechanical" notion of Entropy was ingeniously introduced as a fundamental tool or variable for rigorously studying the mathematical theory behind telegraph, telephony, human languages, etc., or simply, the theory for information and communication.

On the other hand, according to one of the most influential figures of modern Pattern Theory, Ulf Grenander, a pattern is often formed from an ensemble of basic elements (or "atoms") which are "assembled" together under certain local (or short-range) interaction/communication (regularity) "energies." Therefore it seems very natural that an abstract Pattern Theory has to be a many-body theory, which, just like the duality between particles and waves in physics, is necessarily connected to the more tangible theory of Thermodynamics and Statistical Mechanics. This, of course, is only the benign speculation (or faith) of a young applied mathematician.

"What is information, and what is pattern? " is a serious question to be investigated by everyone of us in this information era, not only computer scientists or mathematicians,  just as Neo asked Murpheus in the Hollyhood hit "The Matrix" - "What is the Matrix?"

The Matrix is a world with patterns and information all designed and created by human or machine intelligence for being felt "real", while we are living in a complex "real" world,  and constantly, and often subconsciously, searching for the meaning, information, and patterns of this world. Are they really different? Perhaps not, just as what Einstein's equation is saying. Or perhaps you will feel as much shocked someday as Neo did - there indeed exists no solid boundary between physics/the physical world and abstract information and patterns ! 

Yes, physics is about information, and information is a new interpretation or novel view of physics. Recent advancements in string theory and black holes are indeed indicating that perhaps this entire universe is simply a hologram, or The Matrix !

Such story-telling style is certainly attractive and amusing for this friendly mission statement. But what  is more exciting  and impressive is that  today we do not have to become as vague as story-telling. Numerous tools  such as the Entropy are allowing us to explore all the fascinating facets of the information and pattern theory quantitatively, robustly,  reproducibly, and accurately. It is the perfect time for young applied mathematicians to devote part of their movie time to this new course. Yes, we do not have to WATCH Neo in an IMAX theatre. We ARE Neo's, in this real life.

And, that is THE  mission of this course. Many theories and  applications in  Mathematical Image and Vision Analysis (Miva) will be further explored in this course, including Gibbs image model by Geman and Geman (brothers)(1984), visual pattern learning based on the Maximum Entropy Principle by Zhu and Mumford (1997), the entropy based searching of best wavelet bases by Coifman and Wickerhauser (1992), and the mathematical theory of perception by Mumford (ICM 2002).

And, not mentioning all the great topics in Statistical Mechanics that are prevailing in modern applied mathematics: Monte-Carlo similuation, Gibbs/Markov random fields, simulated annealing, renormalization group method, mean-field and large deviation theory, and phase transitions...

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