Archive for the ‘Science’ Category

Combinatorial Game Theory

August 29, 2007

Combinatorial Game Theory (CGT) is a mathematical discipline that studies two-player, zero-sum games, where the players play sequentially, and there is perfect information and no chance. The aim of CGT is to build mathematical theories that will help players win such games! In practice, the ideas from CGT are of minimal help to the serious player in more complicated games like Chess or Go, but the theories are extremely helpful for simpler but still non-trivial games like Dots-and-Boxes.


The CGT bible is Winning Ways for Your Mathematical Plays, a four-volume set of books (for the 2nd edition from the early 2000’s; the first edition from 1982 was in two volumes), by the eminent mathematicians Elwyn Berlekamp, John Horton Conway, and Richard Guy.

The basic idea of CGT is to develop a calculus, which lets you assign a value to a position in the game. One normally considers games with the condition that a player who cannot move loses. If both players are without a move in a position (whoever moves loses), the position is assigned a value of 0. If the player on the right can make one move that transforms the position to one of value 0, and the player on the left cannot move at all, the position is assigned a value of 1 (if the roles are reversed, the position has value -1).

Now what happens if the player on the left has a move to a position of value 0, and the player on the right has a move to a position of value 1? What is the value of that position? It turns out to be worth 1/2, as can be seen by considering the position obtained by adding such two such positions together with a position of value -1, when one gets back to a position of value 0 where whoever moves loses (I’ll let you work out the details).

There are more different kinds of values than just fractions. Berlekamp, Conway, and Guy consider a multitude of different interesting games, which give rise to ever deeper mathematics, and game positions with strange values along with complex rules for combining those values.

The writing is informal, with constant punning, but the mathematics is serious. Still, one charm of the field is that there is no pre-requisite mathematical material needed to understand these books–bright high school students can launch right in.

After finishing “Winning Ways,” the ambitious student can move on to the two volumes “Games of No Chance” and “More Games of No Chance” (note that nearly all the papers in these books are available online), and then move onto the rest of the papers collected at David Eppstein‘s CGT page.

Also, take a look at Aaron Siegel‘s Combinatorial Game Suite software to help you do CGT computations.

There’s still much more to say, but I’ll leave that for other posts.

On Rationality

August 28, 2007

I want to expand on what I wrote previously in “A Simple But Challenging Game: Part II, this time focusing on Rosenthal’s Centipede Game. To remind you of the rules, in that game there are two players. The players, named Mutt and Jeff, start out with $2 each, and they alternate rounds. On the first round, Mutt can defect by stealing $2 from Jeff, and the game is over. Otherwise, Mutt cooperates by not stealing, and Nature gives Mutt $1. Then Jeff can defect and steal $2 from Mutt, and the game is over, or he can cooperate and Nature gives Jeff $1. This continues until one or the other defects, or each player has $100.

As I previously wrote, in this game, the Nash equilibrium is that Mutt should immediately defect on his first turn. This result is obtained by induction. When both players have $99, it is clearly in Mutt’s interest to steal from Jeff, so that the he will end with $101, and Jeff will end with $97. But that means that when Jeff has $98 and Mutt has $99, Jeff knows what Mutt will do if he cooperates, and can see that he should steal from Mutt, so that he will end with $100 and Mutt will end with $97. But of course that means that when both players have $98, Mutt can see that he should steal from Jeff, and so on, until one reaches the conclusion that Mutt should start the game by stealing from Jeff.

Of course, this Nash equilibrium behavior doesn’t really seem very wise (not to mention ethical), and experiments show that humans will not follow it. Instead they usually will cooperate until the end or near the end of the game, and thus obtain much more money than would “Nashists” who rigorously follow the conclusions of theoretical game theory.

Game theorists often like to characterize the behavior of Nashists as “rational,” which means that they need to explain the “irrational” behavior of humans in the Rosenthal Centipede Game. See for example, this economics web-page, which gives the following “possible explanations of ‘irrational’ behavior”:

There are two types of explanation to account for the divergence. The first assumes that the subject pool contains a certain proportion of altruists who place a positive weight in their utililty function on the payoff of their opponent. Also to the extent that selfish players believe that there is some probability that other players are altruists, they have an incentive to mimic altruistic behaviour by passing.

The second explanation considers the possibility of action errors. Errors in action, or ‘noisy’ play, may result from subjects experimenting with different strategies. Or simply from subjects pressing the wrong key.

Let’s step back for a second and consider what “rational” behavior should mean. A standard definition from economics is that a rational agent will act so as to maximize his expected utility. Let’s accept this definition of “rational.”

The first thing we should note is that “utility” is not usually the same as “pay-off” in a game. As noted in the first explanation above, many people get utility from helping other people get a pay-off. But there are many other differences between pay-offs and utility. You might lose utility from performing actions that seem unethical or unjust, and gain utility from performing actions that seem virtuous or just. You might want to minimize the risk in your pay-off as well as maximize the expected pay-off. You might value pay-offs in a non-linear way, so that the difference between $101 and $100 is very small in terms of utility.

Of course, this difference between pay-off and utility is very annoying theoretically. We’d really like the pay-offs to strictly represent utilities, but unfortunately for experiments, it is only possible to hand out dollars, not some abstract “utils.”

But suppose that the pay-offs in the Rosenthal Centipede Game really did represent utils. Would the game theory result really be “rational” even in that case? Would the only remaining explanation of cooperating behavior be that the players just don’t understand the situation and are making an error?

No. Remember that to be “rational,” an agent should maximize his expected utility. But he can only do that conditioned on some belief about the nature of the person he is playing with. That belief should take the form of a probability distribution for the possible strategies of his opponent. A Nashist rigidly reasons by backward induction that his opponent must always defect at the first opportunity. He even believes this if he plays second, and his opponent cooperates on the first turn! But is this the most accurate belief possible, or the one that will serve to maximize utility? Probably not.

A much more accurate belief could be based on the understanding that even people who understand the backward induction argument can reason beyond it and see that many of their opponents are nevertheless likely to cooperate for a long time, and therefore it pays to cooperate. If you believe that your opponent is likely to cooperate, it is completely “rational” to cooperate. And if this belief that other players are likely to cooperate is backed by solid evidence such as the fact that they started the game by cooperating, then the behavior of the Nashist, based on inaccurate beliefs that cannot be updated, is in fact quite “irrational,” because it does not maximize his utility.

Sophisticated game theorists do in fact understand these points very well, but they muddy the waters by unnecessarily overloading the term “rational” with a second meaning beyond the definition above; they in essence say that “rational” beliefs are those of the Nashist. For example, take a look at this 1995 paper about the centipede game by Nobel Laureate Robert Aumann. Aumann proves that “Common Knowledge of Rationality” (by which he which he essentially means the certain knowledge that all players must always behave as Nashists) will imply backward induction. He specifically adds the following disclaimer at the end of his paper:

We have shown that common knowledge of rationality (CKR) implies backward induction. Does that mean that in perfect information games, only the inductive choices are appropriate or wise? Would we always recommend the inductive choice?

Certainly not. CKR is an ideal (this is not a value judgement; “ideal” is meant as in “ideal gas”) condition that is rarely met in practice; when it is not met, the inductive choice may be not only unreasonable and unwise, but quite simply irrational. In Rosenthal’s (1982) centipede games, for example, even minute departures from CKR may make it incumbent on rational players to “stay in” until quite late in the game (Aumann, 1992); the resulting outcome is very far from that of backward induction. What we have shown is that if there is CKR, then one gets the backward induction outcome; we do not claim that CKR obtains or “should” obtain, and we make no recommendations.

This is all well and good, but why use the horribly misleading name “Common Knowledge of Rationality” for something that would be more properly called “Universal Insistence on Idiocy?”

I hope it is obvious by now why I am skeptical of explanations of various types of human behavior that are based on assuming that all humans are always Nashists, and even more skeptical of recommendations about how we should behave that are based on those same assumptions.

[Acknowledgement: I thank my son Adam for discussions about these issues.]

The Teaching Company

August 24, 2007

My wife and I are both big fans of the college courses produced by the Teaching Company. The courses cover a wide variety of subjects, and come in a range of video and/or audio formats.

I personally find that the audio format usually works somewhat better. The lecturers are very good, but watching a professor lecture on TV is inevitably somewhat dull. On the other hand, when I’m driving or riding in a car or train, I find that an audio lecture fills an ideal amount of mental bandwidth. (Every so often, the lecture gets complicated at the same time as the driving, but one can always rewind).

My favorite course so far was Robert Greenberg’s course on How to Listen to and Understand Great Music. Music courses are naturally a great fit for an audio course!

Otherwise, there are unfortunately not that many science and mathematics courses that go beyond the beginning undergraduate level, although I did enjoy Stephen Nowicki’s course on Biology. If you are interested in history or philosophy, or other subjects in the humanities (like my wife, who is a historian) there are many more interesting options.

The Teaching Company has an unusual pricing policy. The courses are very expensive, except when they go on sale, when they cost roughly one quarter the normal price and are very reasonable. All the courses go on sale on a regular rotation, so unless you are in a tremendous hurry, you should definitely wait until the course you are interested in goes on sale. A lot of the courses are available at libraries too, so you can borrow one to see if you like it first.

Here’s another blog post endorsing the Teaching Company, with some reader comments on their favorite courses.


August 22, 2007

As I mentioned in my previous post about Lenny Guerente’s book “Ageless Quest,” aging research has, within the last 15 years, gone from being a scientific backwater to a mainstream field of scientific research, with new discoveries now regularly featured on the cover of Nature or Science (as in the Nature issue from June 2007 below.)


Although we now are capable of manipulating the aging process, including significantly extending the lifespan of many laboratory animals, it is still a frustrating fact that there is no consensus about the ultimate cause or causes of aging.

One viewpoint, which is probably only held by a significant minority of scientists in the field, is that the aging process is strongly connected to mitochondria, which are the power plants or batteries of our cell, converting nutrients into useful packets of energy in the form of ATP. We’re used to the idea that electronic equipment fails when the batteries go dead, so it’s not such a stretch to take a close look at the mitochondria.

What’s more, mitochondria produce much of the “pollution” in the cell in the form of the free radicals that are a by-product of the oxidative phosphorylation process (the process that turns nutrients into energy). Those free-radicals can damage proteins or DNA, particularly the mitochondrial DNA (this is special DNA, inherited from the mother, that resides in the mitochondria rather than the nucleus) that codes for a few essential mitochondrial proteins.

So one theory says that there is a kind of vicious circle, whereby old mitochondria start emitting more free radicals, which further damages the mitochondria, until the mitochondria are so damaged that they don’t produce sufficient energy and start damaging the rest of the cell. Right now, the consensus view on whether the experimental facts really fit that theory is “Maybe.”


If you want to learn more about mitochondria, I highly recommend “Power, Sex, Suicide: Mitochondria and the Meaning of Life (you’ve just got to love that title), by Nick Lane. Lane’s book is popular science, but it’s a very deep book, and actually proposes theories, including theories of aging, that you won’t see elsewhere in the literature. It’s not an easy book to read, but it’s very worthwhile.

Alternatively, you might enjoy this video of Douglas Wallace lecturing on the role of mitochondria in diseases and aging. Wallace, a professor from UC Irvine, delivers highly entertaining and persuasive lectures.

“On Intelligence” and Numenta

August 21, 2007


“On Intelligence,” written by Jeff Hawkins with Sandra Blakeslee, is a great read, full of provocative ideas about how our brains work.

Hawkins founded Palm Computing and Handspring, but he says that his true lifelong passion has been trying to understand our brains. In 2002, he founded the Redwood Neuroscience Institute (now the Redwood Center for Theoretical Neuroscience at Berkeley).

At Redwood, he developed a theory of the brain, that he expounds in this book. The book is a popular science book; you will not find any equations or explicit algorithms. It reads very smoothly, undoubtably due to the fact that Hawkins was helped in writing the book by Sandra Blakeslee, a science writer for the New York Times.

Hawkins argues that the cortex consists of modules that are all performing the same algorithm. The purpose of that algorithm is to learn to complete and predict the spatio-temporal patterns coming into a module from “lower” modules in the brain’s hierarchy, using feedback from “higher” modules.

I find this hypothesis for how the brain works extremely attractive, and “On Intelligence” argues for the hypothesis almost too well, in that the difficulties of converting the hypothesis into a concrete and useful algorithm will slip by the reader, as if by sleight of hand. The problem, of course, is that it is easy to use words to talk about how a brain might work; the hard part is making a machine do the same thing.

Peter Dayan, a leading computational neuroscientist, has written a much more detailed review. And for a 2006 lecture by Hawkins about his theory, you can watch this video.

After reading the book, I actually spent a few weeks trying to make the hypothesis into a concrete algorithm, without much success. But Hawkins is certainly putting his money where his mouth is: he has founded a company, called Numenta, and Numenta has released software (the “Numenta Platform for Intelligent Computing” or “NuPIC”) that implements part of his theory. NuPIC appears to be based on software originally written by Numenta co-founder and Stanford graduate student Dileep George, but transformed by a team of software developers into a professional-quality product.

However, it is very disappointing that the NUPIC software does not include feedback in its hierarchies, nor does it let you learn and infer temporal patterns (only spatial ones). To be honest, I am not so surprised, because it was these elements (that are obviously central in Hawkins’ theory) that I found it so difficult to integrate into a real algorithm.

Numenta promises that future versions of NuPIC will fill these gaps. I sincerely wish them the best of luck! I should say that I think it is wonderful that an impatient guy like Hawkins pushes the field, using a different approach than the standard academic one.

So in summary, you should definitely read Hawkins’ book, and if you’re intrigued, you should by all means check out the free Numenta software. Whether this line of research will lead to anything significant though, I’m not really sure…

Structure and Interpretation of Computer Programs: Videos and Textbook

August 17, 2007


This set of videos, of Gerald Jay Sussman and Hal Abelson teaching their course on the “Structure and Interpretation of Computer Programs” in July 1986 for Hewlett-Packard employees, is one of the treasures of the internet. The clothes are out of style, but the material presented is still completely relevant.

The SICP web-site has lots of useful additional information, including the complete text of the 2nd edition of their classic textbook.

I am not going to try to write a better review of the Abelon’s and Sussman’s textbook than the Amazon review written by Peter Norvig. Norvig is the head of Research at Google, and a co-author of Artificial Intelligence: a Modern Approach, the leading AI textbook.

But I will add one comment: what I love most about these lectures is the point (see the picture above of Sussman in his wizard hat) that a computer programmer is like a wizard–he creates something real out of ideas. Computers let us be wizards; and I believe we have only scratched the surface of the possible “spells” that we can learn to cast.

If you want a nice implementation of Scheme, the beautiful dialect of Lisp that was invented by Sussman with his student Guy Steele, and used in this textbook, I highly recommend DrScheme.

A Simple but Challenging Game: Part II

August 15, 2007

Here’s Part I again:

My 15-year-old son Adam likes game theory. He invented the following simple game, and asked me about it when I got on the phone with him while I was away at a conference last month (I’ve simplified and formalized the set-up slightly):

There are two players, each of whom is given a real number which is chosen randomly from a uniform distribution between 0.0 and 1.0. The players know their own number but not their opponent’s. One player moves first and has the choice of passing or challenging. If he challenges, both players reveal their number, and the player with the higher number receives a payoff of 1, while the other player receives a payoff of 0. If the first player passes, the second player has a choice of challenging or passing. If he challenges, again both players reveal their numbers and the player with the higher number receives a payoff of 1, while the other player receives a payoff of 0. If the second player also passes, both players receive a payoff of 1/2. They play the game one time, and are interested in maximizing their expected payoff.

What is the right strategy? For example, if you received the number 0.17, would you pass or challenge if you were the first player? What about if you were the second player? What would you do if the number you received was 0.0017?

I’ll tell you more in a later post, but for now why don’t you think about it….

Here’s the promised followup:

It is clear that if any player has the advantage, it’s the second player, because he gets some information from the first player, and can use it to make his decision. Nevertheless, the first player can adopt the strategy of always challenging, and thereby guarantee that he wins half the time. So apparently he should always challenge. This is the answer that was given by “Optionalstopping” in the comments. The same answer was given to me by the evolutionary game theorist Arne Traulsen (who has recently worked on a ground-breaking theory for the emergence of punishment), after I asked him about the game at a lunch conversation.

There is another way to arrive at the same answer. Assume that each player chooses a strategy parameterized by a single value; if he receives a number above that value, he challenges, while if he receives a number below that value he passes. If you work out the Nash equilibrium (basically, that means that both you and your opponent pick the strategy that gives you the best payoff assuming that the opponent is maximizing their payoff), you’ll find (I won’t bore you with the math) that the value for both players is zero–they should always challenge. My son Adam gave an intuitive version of this argument, without the math. Of course it’s true that probabilistic strategies are also possible. I haven’t proven it, but I strongly doubt that introducing probabilistic strategies will change the result that the Nash equilibrium is to always challenge for both players.

So at first I thought the matter was settled. But still, there is something very weird about this result. Would you really challenge if you were the first player and you received a .001? Would you really?

And what if you were the second player, and the first player passed, and you had a .000001? You know that the first player is not “following the rules” of Nash equilibrium. Are you really going to challenge because Nash tells you to? It’s obviously a crazy result! There must be a hole in the arguments.

So what’s wrong with the above arguments?

First let’s start with the Nash equilbrium arguments. By the way, many authors use the term “rational” for players that use strategies dictated by Nash equilibrium arguments, but I think “rational” and “irrational” are excessively loaded terms, so I prefer to instead say that a player that follows a strategy dictated by Nash equilibrium arguments is a “Nashist.”

If you are the second player, and the first player has passed, you can deduce that the first player is not a Nashist. So in order to make a correct play (maximize your expected winnings) you need to choose some probability distribution for what the first player’s strategy is, and then compute whether you will win more or less depending on whether you challenge. There doesn’t seem to be any obvious way to choose the probability distribution for the first player’s strategy, but you can definitely say that he is not certainly a Nashist! A Nashist is stuck (by definition of being a Nashist) believing that all other players are always Nashists, even in the face of clear evidence that they are not (you see why I don’t like to call Nashists “rational”) and would choose the strategy that followed from that obviously wrong belief; he would always challenge as the second player.

A non-Nashist, on the other hand, can come up with a reasonable probability distribution for the first player’s strategy, and come to the conclusion that he should pass if he is the second player and he has a .000001.

OK, so we can see why the second player might want to pass if he receives a .000001. What about the first player: is it wrong to be a Nashist? Should you pass or challenge if you get a .001, or a .000001, or a .000000001? A Nashist would be compelled, by the force of his “idealogy,” to challenge in each case. But you can make a good argument that that’s wrong. Instead, let’s say I am the first player and I have a .000001. I know that if I pass, the second player will be able to deduce that I’m not a Nashist, and will go through the argument given above for the second player. Now “all” I have to do is form my probability distribution for what his probability distribution of my strategy will be, and compute whether I will have more chance of winning depending whether I challenge or not. Again there’s no obvious way to choose these probability distributions, but it seems pretty clear to me to that reasonable probability distributions will give a result that says don’t challenge if you are the first player and you have a sufficiently low number.

Well what about the other argument, that says that the first player should always challenge, since he is at a disadvantage and if he always challenges, he’ll win half the time? It seems paradoxical to think that there is a strategy that can do better than winning half the time for the first player.

Of course, all Nashists will always win exactly half the time, whether they play first or second. If you are playing against someone who you know is a Nashist, it actually doesn’t matter what you do! But suppose instead that you are playing against an ordinary human. You should play forming the best possible probability distribution of what they will do. Many humans will challenge if they have a number above .5 and pass otherwise, whether or not they play first (a very bad strategy by the way). It is perfectly possible, indeed likely, that playing against a population of ordinary humans, there exists a strategy that wins more than half the time for the first player. I can’t prove that strategy exists by arguing in this way (one would need to run experiments to determine the probability distributions of strategies, and then it would be easy to compute), but I’m actually pretty confident that it does exist, and I’m also pretty confident that the strategy involves passing when you are given a .000001 as the first player.

So I don’t believe either argument for always challenging holds up, which is comforting, because always challenging does seem intuitively wrong. Unfortunately, I can’t tell you exactly what the optimal strategy is either, at least until you tell me what the true probability distribution is for player strategies.


By the way, Arne recommended that I pick up Game Theory Evolving, by Herbert Gintis, for my son. It’s a wonderful book, full of interesting games and solved problems in game theory. Adam and I both love it. Gintis gives other examples showing that Nashists (he calls them “self-regarding agents”) can choose bizarre strategies, including “Rosenthal’s Centipede Game:”

The players, Mutt and Jeff, start out with $2 each, and they alternate rounds. On the first round, Mutt can defect by stealing $2 from Jeff, and the game is over. Otherwise, Mutt cooperates by not stealing, and Nature gives Mutt $1. Then Jeff can defect and steal $2 from Mutt, and the game is over, or he can cooperate and Nature gives Jeff $1. This continues until one or the other defects, or each player has $100.

In this game, the Nash equilibrium, obtained by induction by working backwards from the end of the game, when it is clearly “correct” to defect, is that Mutt should immediately defect on his first turn. So that’s what a Nashist would do, but fortunately humans are much more “rational” than Nashists!

Algorithms for Physics

August 14, 2007


Much of my own work is at the intersection of statistical mechanics and algorithms, in particular understanding and developing new algorithms using ideas originating in statistical mechanics. Werner Krauth also works at the intersection of the two fields, but coming from a very different angle: he is a leading expert on the development and application of algorithms to compute and understand the properties of physical systems.

In his recently published book, “Statistical Mechanics: Algorithms and Computations,” targeted at advanced undergraduates or graduate students, he covers a very wide range of interesting algorithms. To give you an idea of the coverage, I’ll just list the chapters: “Monte Carlo methods,” “Hard disks and spheres,” “Density matrices and path integrals,” “Bosons,” “Order and disorder in spin systems, “Entropic forces,”and “Dynamic Monte Carlo methods.”

Krauth’s presentation is leavened by his humor, and he often uses the results obtained using his algorithms to make surprising points about physics that would otherwise be hard to convey.

I am often asked by computer science or electrical engineering scientists and researchers for good introductions to physics, and particularly statistical mechanics, and I’m now happy to be able to recommend this book.


Specifying physics explicitly in terms of algorithms, as Krauth does, gives a very concrete basis for understanding concepts that can otherwise seem terribly abstract. Gerald Sussman and Jack Wisdom make this point in the preface of their already classic book (which is available online) “Structure and Interpretation of Classical Mechanics”:

“Computational algorithms are used to communicate precisely some of the methods used in the analysis of dynamical phenomena. Expressing the methods of variational mechanics in a computer language forces them to be unambiguous and computationally effective. Computation requires us to be precise about the representation of mechanical and geometric notions as computational objects and permits us to represent explicitly the algorithms for manipulating these objects. Also, once formalized as a procedure, a mathematical idea becomes a tool that can be used directly to compute results.”

But while Sussman and Wisdom’s book focuses in great detail on classical mechanics, Krauth’s book covers more broadly subjects in classical mechanics, statistical mechanics, quantum mechanics, and even quantum statistical mechanics. Another difference is that Sussman and Wisdom specify their algorithms in executable Scheme code, while Krauth uses pseudo-code. Of course, both choices have their advantages, just as both of these books are worth your time.

Lectures on Disordered Systems

August 12, 2007

Many physicists study “disordered systems,” such as materials like glasses where the molecules making up the material are arranged randomly in space, in contrast to crystals, where all the particles are arranged in beautiful repeating patterns.

The symmetries of crystals make them much easier to analyze than glasses, and new theoretical methods had to be invented before physicists could make any headway in computing the properties of disordered systems. Those methods have turned out to be closely connected to approaches, such as the “belief propagation” algorithm, that are widely used in computer science, artificial intelligence, and communications theory, with the result that physicists and computer scientists today regularly exchange new ideas and results across their disciplines.

Returning to the physics of disordered systems, physicists began working on the problem in the 1970’s by considering the problem of disordered magnets (also called “spin glasses”). My Ph.D. thesis advisor, Philip W. Anderson summarized the history as follows:

“In 1975 S.F. (now Sir Sam) Edwards and I wrote down the “replica” theory of the phenomenon I had earlier named “spin glass”, followed up in ’77 by a paper of D.J. Thouless, my student Richard Palmer, and myself. A brilliant further breakthrough by G. Toulouse and G. Parisi led to a full solution of the problem, which turned out to entail a new form of statistical mechanics of wide applicability in fields as far apart as computer science, protein folding, neural networks, and evolutionary modelling, to all of which directions my students and/or I contributed.”


In 1992, I presented five lectures on “Quenched Disorder: Understanding Glasses Using a Variational Principle and the Replica Method” at a Santa Fe Institute summer school on complex systems. The lectures were published in a book edited by Lynn Nadel and Daniel Stein, but that book is very hard to find, and I think that these lectures are still relevant, so I’m posting them here. As I say in the introduction, “I will discuss technical subjects, but I will try my best to introduce all the technical material in as gentle and comprehensible a way as possible, assuming no previous exposure to the subject of these lectures at all.”

The first lecture is an introduction to the basics of statistical mechanics. It introduces magnetic systems and particle systems, and describes how to exactly solve non-interacting magnetic systems and particle systems where the particles are connected by springs.

The second lecture introduces the idea of variational approaches. Roughly speaking, the idea of a variational approach is to construct an approximate but exactly soluble system that is as close as possible to the system you are interested in. The grandly titled “Gaussian variational method” is the variational method that tries to find the set of particles and springs that best approximates an interacting particle system. I describe in this second lecture how the Gaussian variational method can be applied to heteropolymers like proteins.

The next three lectures cover the replica method, and combine it with the variational approach. The replica method is highly intricate mathematically. I learned it at the feet of the masters during my two years at the Ecole Normale Superieure (ENS) in Paris. In particular, I was lucky to work with Jean-Philippe Bouchaud, Antoine Georges, and Marc Mezard, who taught me what I knew. I thought it unfortunate that there wasn’t a written tutorial on the replica method, so the result were these lectures. Marc told me that for years afterwards they were given to new students of the replica method at the ENS.

Nowadays, the replica method is a little less popular than it used to be, mostly because it is all about computing averages of quantities over many samples of systems that are disordered according to some probability distribution. While those averages are very useful in physics, they are somewhat less important in computer science, where you usually just want an algorithm to deal with the one disordered system in front of you, rather than an average over all the possible disordered systems.

Santa Fe Institute Lectures