When a German retiree proved a famous long-standing mathematical conjecture, the response was underwhelming.

As he was brushing his teeth on the morning of July 17, 2014, Thomas Royen, a little-known retired German statistician, suddenly lit upon the proof of a famous conjecture at the intersection of geometry, probability theory and statistics that had eluded top experts for decades.

Known as the Gaussian correlation inequality (GCI), the conjecture originated in the 1950s, was posed in its most elegant form in 1972 and has held mathematicians in its thrall ever since. “I know of people who worked on it for 40 years,” said Donald Richards, a statistician at Pennsylvania State University. “I myself worked on it for 30 years.”

Royen hadn’t given the Gaussian correlation inequality much thought before the “raw idea” for how to prove it came to him over the bathroom sink. Formerly an employee of a pharmaceutical company, he had moved on to a small technical university in Bingen, Germany, in 1985 in order to have more time to improve the statistical formulas that he and other industry statisticians used to make sense of drug-trial data. In July 2014, still at work on his formulas as a 67-year-old retiree, Royen found that the GCI could be extended into a statement about statistical distributions he had long specialized in. On the morning of the 17th, he saw how to calculate a key derivative for this extended GCI that unlocked the proof. “The evening of this day, my first draft of the proof was written,” he said.

Not knowing LaTeX, the word processer of choice in mathematics, he typed up his calculations in Microsoft Word, and the following month he posted his paper to the academic preprint site arxiv.org. He also sent it to Richards, who had briefly circulated his own failed attempt at a proof of the GCI a year and a half earlier. “I got this article by email from him,” Richards said. “And when I looked at it I knew instantly that it was solved.”

Upon seeing the proof, “I really kicked myself,” Richards said. Over the decades, he and other experts had been attacking the GCI with increasingly sophisticated mathematical methods, certain that bold new ideas in convex geometry, probability theory or analysis would be needed to prove it. Some mathematicians, after years of toiling in vain, had come to suspect the inequality was actually false. In the end, though, Royen’s proof was short and simple, filling just a few pages and using only classic techniques. Richards was shocked that he and everyone else had missed it. “But on the other hand I have to also tell you that when I saw it, it was with relief,” he said. “I remember thinking to myself that I was glad to have seen it before I died.” He laughed. “Really, I was so glad I saw it.”

Richards notified a few colleagues and even helped Royen retype his paper in LaTeX to make it appear more professional. But other experts whom Richards and Royen contacted seemed dismissive of his dramatic claim. False proofs of the GCI had been floated repeatedly over the decades, including two that had appeared on arxiv.org since 2010. Bo’az Klartag of the Weizmann Institute of Science and Tel Aviv University recalls receiving the batch of three purported proofs, including Royen’s, in an email from a colleague in 2015. When he checked one of them and found a mistake, he set the others aside for lack of time. For this reason and others, Royen’s achievement went unrecognized.

Proofs of obscure provenance are sometimes overlooked at first, but usually not for long: A major paper like Royen’s would normally get submitted and published somewhere like the *Annals of Statistics*, experts said, and then everybody would hear about it. But Royen, not having a career to advance, chose to skip the slow and often demanding peer-review process typical of top journals. He opted instead for quick publication in the *Far East Journal of Theoretical Statistics*, a periodical based in Allahabad, India, that was largely unknown to experts and which, on its website, rather suspiciously listed Royen as an editor. (He had agreed to join the editorial board the year before.)

With this red flag emblazoned on it, the proof continued to be ignored. Finally, in December 2015, the Polish mathematician Rafał Latała and his student Dariusz Matlak put out a paper advertising Royen’s proof, reorganizing it in a way some people found easier to follow. Word is now getting around. Tilmann Gneiting, a statistician at the Heidelberg Institute for Theoretical Studies, just 65 miles from Bingen, said he was shocked to learn in July 2016, two years after the fact, that the GCI had been proved. The statistician Alan Izenman, of Temple University in Philadelphia, still hadn’t heard about the proof when asked for comment last month.

No one is quite sure how, in the 21st century, news of Royen’s proof managed to travel so slowly. “It was clearly a lack of communication in an age where it’s very easy to communicate,” Klartag said.

“But anyway, at least we found it,” he added — and “it’s beautiful.”

In its most famous form, formulated in 1972, the GCI links probability and geometry: It places a lower bound on a player’s odds in a game of darts, including hypothetical dart games in higher dimensions.

Imagine two convex polygons, such as a rectangle and a circle, centered on a point that serves as the target. Darts thrown at the target will land in a bell curve or “Gaussian distribution” of positions around the center point. The Gaussian correlation inequality says that the probability that a dart will land inside both the rectangle and the circle is always as high as or higher than the individual probability of its landing inside the rectangle multiplied by the individual probability of its landing in the circle. In plainer terms, because the two shapes overlap, striking one increases your chances of also striking the other. The same inequality was thought to hold for any two convex symmetrical shapes with any number of dimensions centered on a point.

Special cases of the GCI have been proved — in 1977, for instance, Loren Pitt of the University of Virginia established it as true for two-dimensional convex shapes — but the general case eluded all mathematicians who tried to prove it. Pitt had been trying since 1973, when he first heard about the inequality over lunch with colleagues at a meeting in Albuquerque, New Mexico. “Being an arrogant young mathematician … I was shocked that grown men who were putting themselves off as respectable math and science people didn’t know the answer to this,” he said. He locked himself in his motel room and was sure he would prove or disprove the conjecture before coming out. “Fifty years or so later I still didn’t know the answer,” he said.

Despite hundreds of pages of calculations leading nowhere, Pitt and other mathematicians felt certain — and took his 2-D proof as evidence — that the convex geometry framing of the GCI would lead to the general proof. “I had developed a conceptual way of thinking about this that perhaps I was overly wedded to,” Pitt said. “And what Royen did was kind of diametrically opposed to what I had in mind.”

Royen’s proof harkened back to his roots in the pharmaceutical industry, and to the obscure origin of the Gaussian correlation inequality itself. Before it was a statement about convex symmetrical shapes, the GCI was conjectured in 1959 by the American statistician Olive Dunn as a formula for calculating “simultaneous confidence intervals,” or ranges that multiple variables are all estimated to fall in.

Suppose you want to estimate the weight and height ranges that 95 percent of a given population fall in, based on a sample of measurements. If you plot people’s weights and heights on an *x*–*y* plot, the weights will form a Gaussian bell-curve distribution along the *x*-axis, and heights will form a bell curve along the *y*-axis. Together, the weights and heights follow a two-dimensional bell curve. You can then ask, what are the weight and height ranges — call them –*w* < *x < **w and –**h* < *y < **h — such that 95 percent of the population will fall inside the rectangle formed by these ranges? *

If weight and height were independent, you could just calculate the individual odds of a given weight falling inside –*w* < *x < **w and a given height falling inside –**h* < *y < **h, then multiply them to get the odds that both conditions are satisfied. But weight and height are correlated. As with darts and overlapping shapes, if someone’s weight lands in the normal range, that person is more likely to have a normal height. Dunn, generalizing an inequality posed three years earlier, conjectured the following: The probability that both Gaussian random variables will simultaneously fall inside the rectangular region is always greater than or equal to the product of the individual probabilities of each variable falling in its own specified range. (This can be generalized to any number of variables.) If the variables are independent, then the joint probability equals the product of the individual probabilities. But any correlation between the variables causes the joint probability to increase. *

Royen found that he could generalize the GCI to apply not just to Gaussian distributions of random variables but to more general statistical spreads related to the squares of Gaussian distributions, called gamma distributions, which are used in certain statistical tests. “In mathematics, it occurs frequently that a seemingly difficult special problem can be solved by answering a more general question,” he said.

Royen represented the amount of correlation between variables in his generalized GCI by a factor we might call *C*, and he defined a new function whose value depends on *C*. When *C* = 0 (corresponding to independent variables like weight and eye color), the function equals the product of the separate probabilities. When you crank up the correlation to the maximum, *C* = 1, the function equals the joint probability. To prove that the latter is bigger than the former and the GCI is true, Royen needed to show that his function always increases as *C* increases. And it does so if its derivative, or rate of change, with respect to *C* is always positive.

His familiarity with gamma distributions sparked his bathroom-sink epiphany. He knew he could apply a classic trick to transform his function into a simpler function. Suddenly, he recognized that the derivative of this transformed function was equivalent to the transform of the derivative of the original function. He could easily show that the latter derivative was always positive, proving the GCI. “He had formulas that enabled him to pull off his magic,” Pitt said. “And I didn’t have the formulas.”

Any graduate student in statistics could follow the arguments, experts say. Royen said he hopes the “surprisingly simple proof … might encourage young students to use their own creativity to find new mathematical theorems,” since “a very high theoretical level is not always required.”

Some researchers, however, still want a geometric proof of the GCI, which would help explain strange new facts in convex geometry that are only de facto implied by Royen’s analytic proof. In particular, Pitt said, the GCI defines an interesting relationship between vectors on the surfaces of overlapping convex shapes, which could blossom into a new subdomain of convex geometry. “At least now we know it’s true,” he said of the vector relationship. But “if someone could see their way through this geometry we’d understand a class of problems in a way that we just don’t today.”

Beyond the GCI’s geometric implications, Richards said a variation on the inequality could help statisticians better predict the ranges in which variables like stock prices fluctuate over time. In probability theory, the GCI proof now permits exact calculations of rates that arise in “small-ball” probabilities, which are related to the random paths of particles moving in a fluid. Richards says he has conjectured a few inequalities that extend the GCI, and which he might now try to prove using Royen’s approach.

Royen’s main interest is in improving the practical computation of the formulas used in many statistical tests — for instance, for determining whether a drug causes fatigue based on measurements of several variables, such as patients’ reaction time and body sway. He said that his extended GCI does indeed sharpen these tools of his old trade, and that some of his other recent work related to the GCI has offered further improvements. As for the proof’s muted reception, Royen wasn’t particularly disappointed or surprised. “I am used to being frequently ignored by scientists from [top-tier] German universities,” he wrote in an email. “I am not so talented for ‘networking’ and many contacts. I do not need these things for the quality of my life.”

The “feeling of deep joy and gratitude” that comes from finding an important proof has been reward enough. “It is like a kind of grace,” he said. “We can work for a long time on a problem and suddenly an angel — [which] stands here poetically for the mysteries of our neurons — brings a good idea.”

Source: A Long-Sought Proof, Found and Almost Lost

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Some personal thoughts and opinions on what ``good quality mathematics'' is, and whether one should try to define this term rigorously. As a case study, the story of Szemer\'edi's theorem is presented.

This looks like a cool little paper.

Update 3/17/17

And indeed it was. The opening has lovely long (though possibly incomplete) list of aspects of good mathematics toward which mathematicians should strive. The second section contains an interesting example which looks at the history of a theorem and it’s effect on several different areas. To me most of the value is in thinking about the first several pages. I highly recommend this to all young budding mathematicians.

In particular, as a society, we need to be careful of early students in elementary and high school as well as college as the pedagogy of mathematics at these lower levels tends to weed out potential mathematicians of many of these stripes. Students often get discouraged from pursuing mathematics because it’s “too hard” often because they don’t have the right resources or support. These students, may in fact be those who add to the well-roundedness of the subject which help to push it forward.

I believe that this diverse and multifaceted nature of “good mathematics” is very healthy for mathematics as a whole, as it it allows us to pursue many different approaches to the subject, and exploit many different types of mathematical talent, towards our common goal of greater mathematical progress and understanding. While each one of the above attributes is generally accepted to be a desirable trait to have in mathematics, it can become detrimental to a field to pursue only one or two of them at the expense of all the others.

As I look at his list of scenarios, it also reminds me of how areas within the humanities can become quickly stymied. The trouble in some of those areas of study is that they’re not as rigorously underpinned, as systematic, or as brutally clear as mathematics can be, so the fact that they’ve become stuck may not be noticed until a dreadfully much later date. These facts also make it much easier and clearer in some of these fields to notice the true stars.

As a reminder for later, I’ll include these scenarios about research fields:

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- A field which becomes increasingly ornate and baroque, in which individual

results are generalised and refined for their own sake, but the subject as a

whole drifts aimlessly without any definite direction or sense of progress;- A field which becomes filled with many astounding conjectures, but with no

hope of rigorous progress on any of them;- A field which now consists primarily of using ad hoc methods to solve a collection

of unrelated problems, which have no unifying theme, connections, or purpose;- A field which has become overly dry and theoretical, continually recasting and

unifying previous results in increasingly technical formal frameworks, but not

generating any exciting new breakthroughs as a consequence; or- A field which reveres classical results, and continually presents shorter, simpler,

and more elegant proofs of these results, but which does not generate any truly

original and new results beyond the classical literature.

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Georg Cantor showed that some infinities are bigger than others. Did he assault mathematical wisdom or corroborate it?

In 1883, the brilliant German mathematician Georg Cantor produced the first rigorous, systematic, mathematical theory of the infinite. It was a work of genius, quite unlike anything that had gone before. And it had some remarkable consequences. Cantor showed that some infinities are bigger than others; that we can devise precise mathematical tools for measuring these different infinite sizes; and that we can perform calculations with them. This was seen an assault not only on intuition, but also on received mathematical wisdom. In due course, I shall sketch some of the main features of Cantor’s work, including his most important result, commonly known as ‘Cantor’s theorem’. But first I want to give a brief historical glimpse of why this work was perceived as being so iconoclastic. Ultimately, my aim is to show that this perception was in fact wrong. My contention will be that Cantor’s work, far from being an assault on received mathematical wisdom, actually served to corroborate it.

The standard conception of the infinite is that which is endless, unlimited, unsurveyable, immeasurable. Ever since people have been able to reflect, they have treated the infinite with a curious combination of perplexity, suspicion, fascination and respect. On the one hand, they have wondered whether we can even make sense of the infinite: mustn’t it, by its very nature, elude our finite grasp? On the other hand, they have been reluctant, indeed unable, to ignore it altogether.

In the fourth century BCE, Aristotle responded to this dilemma by drawing a distinction. He believed that there is one kind of infinity that really can’t be made sense of, and another that is a familiar and fundamental feature of reality. To the former he gave the label ‘actual’. To the latter he gave the label ‘potential’. An ‘actual’ infinity is one that is located at some point *in* time. A ‘potential’ infinity is one that is spread *over* time. Thus an infinitely big physical object, if there were such a thing, would be an example of an actual infinity. Its infinite bulk would be there all at once. An endlessly ticking clock, on the other hand, would be an example of a potential infinity. Its infinite ticking would be forever incomplete: however long the clock had been ticking, there would always be more ticks to come. Aristotle thought that there was something deeply problematic, if not incoherent, about an actual infinity. But he thought that potential infinities were there to be acknowledged in any process that will never end, such as the process of counting, or the process of dividing an object into smaller and smaller parts, or the passage of time itself.

Aristotle’s distinction proved to be enormously influential. Its importance to subsequent discussion of the infinite is hard to exaggerate. For more than 2,000 years, it more or less had the status of orthodoxy. But later thinkers, unlike Aristotle himself, construed the references to time in the actual/potential distinction as a metaphor for something more abstract. Having a location ‘in time’, or being there ‘all at once’, came to assume broader meanings than they had done in Aristotle. Eventually, exception to an actual infinity became exception to the very idea that the infinite could be a legitimate object of mathematical study in its own right. Cue Cantor.

Precisely what Cantor did was to demonstrate, with unimpeachable rigour, that the infinite *can* be a legitimate object of mathematical study in its own right. In particular, Cantor showed that we can acknowledge infinitely big sets – such as the set of numbers 1, 2, 3, 4, 5, 6, etc – and investigate the mathematical properties of these sets. To the extent that this involves considering the members of such sets all together, their infinity can be thought of as being there ‘all at once’.

At the core of Cantor’s work is the very idea of comparing sets in size with one another. Now, you can often tell that two sets are the same size just by counting their members. For example, suppose you’re at a meeting, and suppose you count first the number of men in the room, and then the number of women in the room, and it turns out that there are 12 of each. Then you know that the set of men in the room is the same size as the set of women there. But you can also sometimes tell that two sets are the same size *without* counting. Thus, suppose you’re at a meeting where you don’t know how many people are present, but you notice that people are sitting around the table in such a way that men and women alternate. Then you can tell that the set of men in the room is the same size as the set of women there, even though you don’t know how many there are of either. There are even cases where you can tell that two sets are the same size without being in a *position* to count. Thus you know that the set of older twins that have ever been born is the same size as the set of younger twins that have ever been born. The basic principle here is that, whenever it’s possible to *pair off* all the members of one set with all the members of another, as it is in the case of the alternating men and women and as it is in the case of the twins, then the two sets are the same size.

This is where it really does get ultra-curious – that there are distinctions of size to be drawn even in the infinite case

Does this principle extend to infinite sets? Cantor didn’t see why not. But here things start to get a little weird. Reconsider the set of numbers 1, 2, 3, 4, 5, 6, etc. The members of this set can clearly be paired off with the members of the set of *even* numbers 2, 4, 6, 8, 10, 12, etc. For 1 can be paired with 2; 2 can be paired with 4; 3 can be paired with 6; and so on. So if we extend the principle touted above to infinite sets, then we’re forced to conclude that the set of all the numbers is the same size as the set of those that are even, even though the first of these sets includes everything in the second set *plus* all the odd numbers as well.

Some people react by saying that it just doesn’t make sense to invoke comparisons of size where infinite sets are concerned. But that wasn’t Cantor’s reaction. He took such anomalies in his stride. He accepted that the sets in question – the set of all the numbers and the set of even numbers – are indeed the same size. And, although that’s curious enough, it’s not *ultra*-curious. After all, perhaps we can show that *all* infinite sets are the same size. If so, that wouldn’t be especially counterintuitive: sets would be either finite, in which case there would be a further question as to exactly how big they are, or infinite, in which case there wouldn’t be. But no! Cantor’s remarkable discovery – and this is where it really *does* get ultra-curious – is that there are distinctions of size to be drawn even in the infinite case. Some infinite sets are bigger than others. A pairing of the sort that we’ve been considering *isn’t* always available, even when the two sets concerned are infinite.

To see why not, let’s again focus on the numbers. Not only are there infinitely many of these, there are infinitely many *sets* of these. Here are just a few examples:

the set of even numbers that we have just been considering

the set of squares

the set of numbers that are less than 100

the set of numbers that are greater than 100

the set of numbers that are exactly divisible by 13

the set whose only three members are 6, 17, and 243

But it is impossible to pair off all these *sets* of numbers with *individual* numbers. Cantor had an ingenious argument to show that, whenever sets of numbers *are* paired off with individual numbers, at least one such set will inevitably be left out: so there are more sets of numbers than there are individual numbers. Cantor’s argument trades on the fact that, given such a pairing, some numbers will themselves belong to whatever set they’re paired with, and some won’t. Imagine, for instance, that there is a pairing in which the six sets just mentioned are paired with the first six numbers thus:

1 — the set of even numbers

2 — the set of squares

3 — the set of numbers that are less than 100

4 — the set of numbers that are greater than 100

5 — the set of numbers that are divisible by 13

6 — the set whose only three members are 6, 17 and 243

Then 1 *doesn’t* belong to the set with which it is paired, because it isn’t itself even. By contrast, 3 *does* belong to the set with which it is paired, since it is itself less than 100. Likewise 6 belongs to the set with which it is paired, since it is one of the three members of that set. Let’s call numbers that *don’t* belong to the set with which they’re paired ‘excluded’ and those that *do* belong to the set with which they’re paired ‘included’. Thus 1, 2, 4 and 5 are all excluded, but 3 and 6 are both included. Now the excluded numbers themselves form a set. And *this* is the set that can’t have been paired with any number: this is the set that must have been left out. Why? Well, suppose it *has* been paired with some number, say 821. In other words suppose, as we run down the list started above, we eventually find the following pairing:

821 — the set of excluded numbers

Then a contradiction arises concerning whether 821 is itself excluded or not. If it is, then it belongs to the set with which it is paired (the set of excluded numbers), so it is *in*cluded. If it is included, on the other hand, then it *doesn’t* belong to the set with which it is paired (the set of excluded numbers), so it is *ex*cluded. There is no satisfactory answer to the question of whether 821 is excluded or included.

We must therefore accept that there are more sets of numbers than there are individual numbers. And in fact, with one crucial qualification that we shall come back to, this argument can be applied to anything whatsoever: there are more sets of bananas than there are bananas, more sets of stars than there are stars, more sets of points in space than there are points in space, more sets of sets of bananas than there are sets of bananas, and so on. In general – subject to the crucial qualification that I’ve said we’ll come back to – there are always more sets of things of any given kind than there are individual things of that kind. This is Cantor’s theorem.

But what about sets of *sets*? Are there more of those than there are sets? Surely that’s impossible. How can there be more sets of *anything* than there are sets altogether?

This is a paradox. It is closely related to Russell’s paradox, named after the British philosopher and mathematician Bertrand Russell who discovered it at the beginning of the 20th century. Russell’s paradox turns on the fact that, although a set doesn’t typically belong to itself, some sets, it would appear, do. The set of bananas, for example, doesn’t: it’s a set, not a banana. But the set of things mentioned in this article, it would appear, does: I’ve just mentioned it. Russell’s paradox concerns the set of sets of the former kind: the set of sets that don’t belong to themselves. Does *that* set belong to itself? As with the question whether 821 is excluded or included, there is no satisfactory answer.

Cantor was aware of such paradoxes. But again he was unfazed. He developed a robust and relatively intuitive conception of sets whereby the paradoxes simply don’t arise. On this conception, the members of a set must exist ‘before’ the set itself: the set’s existence is parasitic on theirs. So first there are bananas, *then* there is the set of bananas. First there are sets of bananas, *then* there is the set of sets of bananas. More generally, first there are things that are not sets (bananas, stars, etc); then there are sets of these things; then there are sets of *these* things; and so on, without end. On this conception, then, *no* set belongs to itself. For a set can’t exist ‘before’ itself. (If we want to talk about the set of things mentioned in this article, we first need to be more precise both about what ‘things’ we have in mind and about what counts as ‘mentioning’ one of them. *Once* we’ve done this, we’ll be able to acknowledge such a set, but it won’t belong to itself.) Every set, moreover, is succeeded by further new sets to which it itself belongs, sets that didn’t already exist when it itself came into being. So there is no set of all sets.

This circumvents Russell’s paradox, because the set of sets that don’t belong to themselves, if there were such a thing, *would be* the set of all sets (because no set belongs to itself). However, there is no such thing on this conception. So the question whether that set belongs to itself or not never gets a chance to arise.

Their collective infinity, as opposed to the infinity of any one of them, is potential, not actual

The paradox that there are more sets of sets than there are individual sets is likewise circumvented. Cantor’s theorem applies *only where sets are being compared in size:* this is the crucial qualification to which I referred earlier. Thus, although we can say that there are more sets of bananas than there are bananas, this is because the *set* of sets of bananas is bigger than the *set* of bananas. By contrast, we can’t say that there are more sets of sets than there are sets. That would mean that the *set* of sets of sets is bigger than the *set* of sets. But this makes no sense on Cantor’s conception. Neither the set of sets of sets nor the set of sets exist. So the question of whether one of these sets is bigger than the other, likewise, never gets a chance to arise.

The conception of sets involved here is, as I’ve already said, relatively intuitive. But isn’t it also *strikingly Aristotelian*? There is a temporal metaphor sustaining it. Sets are depicted as coming into existence ‘after’ their members, in such a way that there are ‘always’ more to come. Their *collective* infinity, as opposed to the infinity of any one of them, is potential, not actual: its existence is spread ‘over time’ rather than being located at any one point ‘in time’. Moreover, it is this collective infinity that arguably has the best claim to the title. For recall the concepts that I listed earlier as characterising the standard conception of the infinite: endlessness, unlimitedness, unsurveyability, immeasurability. These concepts more properly apply to the full range of sets than to any one of them. This in turn is because of the very success that Cantor enjoyed in subjecting individual sets to rigorous mathematical scrutiny. He showed, for example, that the set of numbers is *limited in size*. It is limited in size because it doesn’t have as many members as the set of sets of numbers. He also showed (although I didn’t go into the details of this) that its size can be given a precise mathematical *measure*. Isn’t there a sense, therefore, in which he established that the set of numbers is ‘really’ finite and that what is ‘really’ infinite is something of an altogether different kind? Didn’t his work serve, in the end, to corroborate the Aristotelian orthodoxy that ‘real’ infinity can never be actual, but must always be potential?

You might object on the following grounds: to call the set of numbers ‘really’ finite would not only be at variance with standard mathematical terminology, it would also be at variance, contrary to what I seem to be suggesting, with what most people would say. And I agree. Most people, if they’re happy to talk in these terms at all, would say that the set of numbers is infinite. But then again, most people are unaware of Cantor’s work. They would also no doubt say that it’s impossible for one infinite set to be bigger than another. My point isn’t a point about what most people would say. It’s a point about how they *understand* what they would say, and about how that understanding is best able, for any given purpose, to absorb the shock of Cantor’s results. Nothing here is forced on us. Certainly we can say that some infinite sets are bigger than others, as mathematics nowadays routinely does. But we *can* also say that the set of numbers is only finite, as (I have suggested) there would be some rationale for doing. For that matter, we *could* go right back to the drawing board and say that there’s no such thing as the set of numbers in the first place, perhaps on the grounds that the very idea of gathering infinitely many things into a single set is already too great a concession to an ‘actual-infinity-friendly’ conception of the infinite. Sooner or later, on Cantor’s conception, we’re going to have to say *something* along those lines: at the very least, we’re going to have to say that there’s no such thing as the set of sets. Why not be pre-emptive?

None of these remarks are intended as an attack on anything that mathematicians either say or do. I am simply urging greater caution when it comes to interpreting what they say and do, and in particular when it comes to saying how this bears on traditional conceptions of the infinite. Aristotle, on Cantor’s showing, was not so wrong after all.

Source: Why some infinities are bigger than others | Aeon Essays

Lofty goals here, but I’m not quite sure he’s really make the case he set out to in these few words. The comments on the article are somewhat interesting, but seem to devolve into the usual pablum seen on such posts. Nothing stuck out to me as a comment by a solid mathematician, which might have been interesting.

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The world’s foremost expert on pricing strategy shows how this mysterious process works and how to maximize value through pricing to company and customer.]]>

In all walks of life, we constantly make decisions about whether something is worth our money or our time, or try to convince others to part with their money or their time. Price is the place where value and money meet. From the global release of the latest electronic gadget to the bewildering gyrations of oil futures to markdowns at the bargain store, price is the most powerful and pervasive economic force in our day-to-day lives and one of the least understood.

The recipe for successful pricing often sounds like an exotic cocktail, with equal parts psychology, economics, strategy, tools and incentives stirred up together, usually with just enough math to sour the taste. That leads managers to water down the drink with hunches and rules of thumb, or leave out the parts with which they don’t feel comfortable. While this makes for a sweeter drink, it often lacks the punch to have an impact on the customer or on the business.

It doesn’t have to be that way, though, as Hermann Simon illustrates through dozens of stories collected over four decades in the trenches and behind the scenes. A world-renowned speaker on pricing and a trusted advisor to Fortune 500 executives, Simon’s lifelong journey has taken him from rural farmers’ markets, to a distinguished academic career, to a long second career as an entrepreneur and management consultant to companies large and small throughout the world. Along the way, he has learned from Nobel Prize winners and leading management gurus, and helped countless managers and executives use pricing as a way to create new markets, grow their businesses and gain a sustained competitive advantage. He also learned some tough personal lessons about value, how people perceive it, and how people profit from it.

In this engaging and practical narrative, Simon leaves nothing out of the pricing cocktail, but still makes it go down smoothly and leaves you wanting to learn more and do more―as a consumer or as a business person. You will never look at pricing the same way again.

A unique introduction to the theory of linear operators on Hilbert space. The author presents the basic facts of functional analysis in a form suitable for engineers, scientists, and applied mathematicians. Although the Definition-Theorem-Proof format of mathematics is used, careful attention is given to motivation of the material covered and many illustrative examples are presented.Syndicated copies to:

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Pachter, a computational biologist, returns to CalTech to study the role and function of RNA.

Pachter, a computational biologist and Caltech alumnus, returns to the Institute to study the role and function of RNA.

*Lior Pachter (BS ’94) is Caltech’s new Bren Professor of Computational Biology. Recently, he was elected a fellow of the International Society for Computational Biology, one of the highest honors in the field. We sat down with him to discuss the emerging field of applying computational methods to biology problems, the transition from mathematics to biology, and his return to Pasadena.*

Computational biology is the art of developing and applying computational methods to answer questions in biology, such as studying how proteins fold, identifying genes that are associated with diseases, or inferring human population histories from genetic data. I have interests in both the development of computational methods and in answering specific biology questions, primarily related to the function of RNA, a molecule central to the function of cells. RNA molecules transmit information through their roles as products of DNA transcription and as the precursors to translation to protein; they also act as enzymes catalyzing biochemical reactions. I am interested in understanding these functions of RNA through tools that involve the combination of computational methods with sequencing methods that together allow for high-resolution probing of RNA activity and structure in cells.

During my PhD studies at MIT, I took a course in computational biology. In the course of working on a final project for the class, I got connected to the Human Genome Project—a large-scale endeavor to identify the full DNA sequence of a human genome—and I found the biology and associated math questions very interesting. This led me to change my intended direction of research from algebraic combinatorics to computational biology, and my interests expanded from math to statistics, computer science, and genomics.

It’s not very common. However, many prominent genomics biologists have backgrounds in mathematics, computer science, or statistics. For example, one of my mentors in graduate school was Eric Lander, the director of the Broad Institute of MIT and Harvard, who received a PhD in mathematics and then transitioned to working in biology. His transition, like mine years later, was sparked by the possibilities and challenges of utilizing genome sequencing to understand biology.

While genome sequencing has obviously been useful in revealing the sequences that are involved in coding various aspects of the molecular biology of the cell, it has had a secondary impact that is less obvious at first glance. The low cost and high throughput (the ability to process large volumes of material) of genome sequencing allowed for a more “big-data” approach to biology, so that experiments that previously could only be applied to individual genes could suddenly be applied in parallel to all of the genes in the genome. The design and analysis of such experiments demand much more sophisticated mathematics and statistics than had previously been needed in biology.

A result of the scale of these new experiments is the emergence of very large data sets in biology whose interpretation demands the application of state-of-the-art computer science methods. The problems require interdisciplinary dexterity and involve not only management of large data sets but also the development of novel abstract frameworks for understanding their structure. For example, there’s a new technique called RNA-seq, developed by biologists including Barbara Wold [Caltech’s Bren Professor of Molecular Biology], which involves measuring transcription—the process of copying segments of DNA into RNA—in cells. The RNA-seq technique consists of transforming RNA molecules into DNA sequences that allow the researchers to identify and count the original RNA molecules. The development of this technique required not only novel biochemistry and molecular biology, but also new definitions and ideas for how to think about transcriptomes, which are the sets of all the RNA molecules in a cell. I work on improvements to the assay, as well as the development of the associated statistics, computer science, and mathematics.

I was born in Israel and moved to South Africa when I was two. I lived there until moving to Palo Alto, California, at 15. After high school, I studied mathematics at Caltech and pursued my PhD in applied mathematics at MIT. I spent time at Berkeley as a postdoc before becoming professor of mathematics, molecular and cell biology, and computer science, and I held the Raymond and Beverly Sackler Chair in Computational Biology. I joined the Caltech faculty in early 2017.

It’s a great pleasure. As an undergrad, I made very strong connections with very special people who just had a pure love of science. I’ve always missed the unique culture and atmosphere at Caltech and, returning now as a professor, I can feel the spirit of the Institute—an intense love of science emanating from individuals that is unlike anywhere else. It’s a homecoming of sorts.

Source: A Conversation with Lior Pachter (BS ’94) | Caltech

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"Mathematicians have big egos, so they haven’t told anyone that math is easy.”

Math is a notoriously hard subjectfor many kids and adults. There is a gender gap, a race gap, and just generally bad performance in many countries.

John Mighton, a Canadian playwright, author, and math tutor who struggled with math himself, has designed a teaching program that has some of the worst-performing math students performing well and actually enjoying math. There’s mounting evidence that the method works for all kids of all abilities.

His program, JUMP (Junior Undiscovered Math Prodigies) Math, is being used by 15,000 kids in eight US states (it is aligned with the Common Core), more than 150,000 in Canada, and about 12,000 in Spain. The US Department of Education found it promising enough to give a $2.75 million grant in 2012 to Tracy Solomon and Rosemary Tannock, cognitive scientists at the Hospital for Sick Children and the University of Toronto, to conduct a randomized control trial with 1,100 kids and 40 classrooms. The results, out later this year, hope to confirm previous work the two did in 2010, which showed that students from 18 classrooms using JUMP progressed twice as fast on a number of standardized math tests as those receiving standard instruction in 11 other classrooms.

“It would be difficult to attribute these gains to anything but the instruction because we took great pains to make sure the teachers and the students were treated identically except for the instruction they received,” Solomon said.

Mighton has identified two major problems in how we teach math. First, we overload kids’ brains, moving too quickly from the concrete to the abstract. That puts too much stress on working memory. Second, we divide classes by ability, or “stream”, creating hierarchies which disable the weakest learners while not benefitting the top ones.

Mighton argues that over the past decade, the US and Canada have moved to a “discovery” or “inquiry” based approach to math by which kids are meant to figure out a lot of concepts on their own. The example he offers in this Scientific American article is this:

“Discovery-based lessons tend to focus less on problems that can be solved by following a general rule, procedure or formula (such as “find the perimeter of a rectangle five meters long and four meters wide”) and more on complex problems based on real-world examples that can be tackled in more than one way and have more than one solution (“using six square tiles, make a model of a patio that has the least possible perimeter”)”

Solomon said this approach—also called problem-based learning—means the teachers’ role is not to provide direct instruction, but to let kids collaborate to find solutions to complex, realistic problems which have multiple approaches and answers. But too many children don’t have the building blocks from which to discover the answers. They get frustrated, and then fixed in the belief that they are not “math people.”

A key problem with this method is it requires kids to have too much happening in their brains at one time. “This is very difficult for teachers,” said Solomon, and “it’s very difficult for kids.”

Mighton thinks—and offers brain research (pdf) to support it—that kids succeed more with math when it is broken down into small components which are explained carefully and then practiced continually.

To explain the concept to me, he took a basic question—what is 72 divided by 3? He showed me multiple ways to do it, including saying three friends wants to share seven dimes and two pennies. When I pause, even for a second, Mighton apologizes and says he clearly hasn’t explained it well, and takes another stab at it a different way.

Critics would argue that all good teachers approach problems like this, from multiple angles. But many teachers struggle with their own math anxiety, and research shows that they then pass on this anxiety to their students. (That happens with parents too, unfortunately.)

And Nikki Aduba, who helped test Mighton’s method in schools in the London borough of Lambeth, said Mighton has broken down the steps so carefully that nearly everyone could catch on. Many teachers, she said, welcomed this approach. “Many thought, okay to get from A to B there are these three steps, but it turns out there are really five or six,” she said.

When Solomon conducted the pilot program on JUMP, she said it was the small, incremental steps which made the math accessible to all students and allowed some of them to experience success in math for the first time. “Because they can master the increments, they are getting the checks and building the mindset that their efforts can amount to something. That experience motivates them to continue,” she said. By continuing, they practice more math, get more skills, and become the math people they thought they couldn’t be.

Mighton says the small steps are critical. “I am not going to move until everyone can do this,” he said. “Math is like a ladder—if you miss a step, it’s hard to go on. There are a set of sequences.” He has dubbed his method “micro discovery” or “guided discovery.”

There is other evidence for its success. When the Manhattan Charter School piloted the program in in 2013-14 with its fourth graders, it experienced the highest increase in math scores in all of New York City. Now every class in the school is using it.

The program was used in Lambeth, one of the poorest areas of London, with more than 450 of its worst-performing students. At the time they started, 14% were performing at grade level: when the kids took their grade six exams (called Key Stage 2 exams in the UK), 60% passed. Aduba said it worked “brilliantly,” especially for kids who had been struggling.

“The key thing about the JUMP program is it starts small and progresses in very small steps to a very sophisticated level in a relatively short period of time,” she said. “It restored confidence in kids who thought ‘I can’t do maths.’ Suddenly, to be able to do stuff, it boosted their confidence.”

The bigger problem Mighton sees is hierarchies. Teachers tend to assume that in most classrooms there’s a bell-shaped curve—a wide distribution of abilities—and teach accordingly. It means that 20% of the class underperforms, 60% are in the middle, and 20% outperform, leading to a two- or three-grade range of abilities within one classroom.

“When people talk about improving education they want to move the mean higher. They don’t talk about tightening the distribution,” Mighton said.

The reason this matters is that, as research shows (pdf), kids compare themselves to each other early on and decide whether or not they are “math people.” Children who decide they are not math people are at risk of developing something Stanford psychologist Carol Dweck calls a “fixed” mindset: They think their talents are innate and cannot be improved upon. Thirty years of mindset research shows that kids with a fixed mindset take fewer risks and underperform those who think their efforts matter.

Dweck has examined JUMP and says it encourages a “growth” mindset: the belief that your abilities can improve with your efforts. “The kids move at an exciting pace; it feels like it should be hard but it’s not hard, they have this feeling of progress, that [they] can be good at this,” she said at a math conference.

Mighton says the problem with the bell curve is that everyone worries about the kids at the top getting bored. “Our data shows that if you teach to the whole class, the whole class does better,” he says. And, by moving together and having so many children experience success in math, they experience what Durkheim calls “collective effervescence,” the joy of knowing they can do it, rather than the joy of just getting a high mark.

As school districts move away from the most commercially savvy educational publishers to programs based on proper evidence—a shift that has been taking place over the past decade, albeit slowly—programs like JUMP will likely have more success. Until he won the Schwab entrepreneur of the year award in 2015, Mighton—who has been working on JUMP for 15 years—has had no marketing team and has invested all of his budget into testing and refining the materials. (JUMP is a nonprofit, and all its materials are available on its website.) Pearson, by way of contrast, is a £5.3 billion ($6.6 billion) company with tentacles in every corner of the education market.

While many people try to paint their methods as new, Mighton is the first to admit that what he is teaching is age-old. He believes math has been overhyped as hard, and all that students *and* teachers need is to have things broken down properly. Many have dubbed these simple steps as “drill and kill”. But he says the steps can be made fun, like puzzles.

Mathematicians“have big egos, so they haven’t told anyone that math is easy,” he said at the World Economic Forum in Davos last month. “Logicians proved more than 100 years ago it can be broken into simple steps.”

Source: JUMP Math, a teaching method that’s proving there’s no such thing as a bad math student — Quartz

]]>The equations of gauge theory lie at the heart of our understanding of particle physics. The Standard Model, which describes the electromagnetic, weak, and strong forces, is based on the Yang-Mills equations. Starting with the work of Donaldson in the 1980s, gauge theory has also been successfully applied in other areas of pure mathematics, such as low dimensional topology, symplectic geometry, and algebraic geometry.

More recently, Witten proposed a gauge-theoretic interpretation of Khovanov homology, a knot invariant whose origins lie in representation theory. Khovanov homology is a “categorification” of the celebrated Jones polynomial, in the sense that its Euler characteristic recovers this polynomial. At the moment, Khovanov homology is only defined for knots in the three-sphere, but Witten’s proposal holds the promise of generalizations to other three-manifolds, and perhaps of producing new invariants of four-manifolds.

This workshop will bring together researchers from several different fields (theoretical physics, mathematical gauge theory, topology, analysis / PDE, representation theory, symplectic geometry, and algebraic geometry), and thus help facilitate connections between these areas. The common focus will be to understand Khovanov homology and related invariants through the lens of gauge theory.

This workshop will include a poster session; a request for posters will be sent to registered participants in advance of the workshop.

Edward Witten will be giving two public lectures as part of the Green Family Lecture series:

March 6, 2017

*From Gauge Theory to Khovanov Homology Via Floer Theory*

The goal of the lecture is to describe a gauge theory approach to Khovanov homology of knots, in particular, to motivate the relevant gauge theory equations in a way that does not require too much physics background. I will give a gauge theory perspective on the construction of singly-graded Khovanov homology by Abouzaid and Smith.

March 8, 2017

*An Introduction to the SYK Model*

The Sachdev-Ye model was originally a model of quantum spin liquids that was introduced in the mid-1990′s. In recent years, it has been reinterpreted by Kitaev as a model of quantum chaos and black holes. This lecture will be primarily a gentle introduction to the SYK model, though I will also describe a few more recent results.

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Open for submission now

Deadline for manuscript submissions: 31 August 2017

A special issue of *Entropy* (ISSN 1099-4300).
## Special Issue Editor

## Special Issue Information

Deadline for manuscript submissions: **31 August 2017**

Dear Colleagues,

Whereas Bayesian inference has now achieved mainstream acceptance and is widely used throughout the sciences, associated ideas such as the principle of maximum entropy (implicit in the work of Gibbs, and developed further by Ed Jaynes and others) have not. There are strong arguments that the principle (and variations, such as maximum relative entropy) is of fundamental importance, but the literature also contains many misguided attempts at applying it, leading to much confusion.

This Special Issue will focus on Bayesian inference and MaxEnt. Some open questions that spring to mind are: Which proposed ways of using entropy (and its maximisation) in inference are legitimate, which are not, and why? Where can we obtain constraints on probability assignments, the input needed by the MaxEnt procedure?

More generally, papers exploring any interesting connections between probabilistic inference and information theory will be considered. Papers presenting high quality applications, or discussing computational methods in these areas, are also welcome.

Dr. Brendon J. Brewer

*Guest Editor*

**Submission**

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. Papers will be published continuously (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are refereed through a peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. *Entropy* is an international peer-reviewed Open Access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1500 CHF (Swiss Francs).

No papers have been published in this special issue yet.

Source: Entropy | Special Issue : Maximum Entropy and Bayesian Methods

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Back in December, you hopefully thoroughly immersed yourself in *The Map of Physics*, an animated video–a visual aid for the modern age–that mapped out the field of physics, explaining all the connections between classical physics, quantum physics, and relativity.

You can’t do physics without math. Hence we now have *The Map of Mathematics*. Created by physicist Dominic Walliman, this new video explains “how pure mathematics and applied mathematics relate to each other and all of the sub-topics they are made from.” Watch the new video above. You can buy a poster of the map here. And you can download a version for educational use here.

*Follow Open Culture on* *Facebook,** **Twitter, Instagram, Google Plus,** and Flipboard and share intelligent media with your friends. Or better yet,* *sign up for our daily email* *and get a daily dose of Open Culture in your inbox. And if you want to make sure that our posts* *definitely* *appear in your Facebook newsfeed,* *just follow these simple steps**.*

We discuss properties of the "beamsplitter addition" operation, which provides a non-standard scaled convolution of random variables supported on the non-negative integers. We give a simple expression for the action of beamsplitter addition using generating functions. We use this to give a self-contained and purely classical proof of a heat equation and de Bruijn identity, satisfied when one of the variables is geometric.Syndicated copies to:

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Instagram filter used: Clarendon

Photo taken at: UCLA Math Sciences Building

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There follows a discussion of flipping coins and the fact that frequencies have more random variation when the sample size is small, but he never stops to see if this is enough to explain the observation.

My intuition told me it did not, so I went and got some brain cancer data.

Jordan Ellenberg is called out a bit by Rick Durrett for one of his claims in the best seller *How Not To Be Wrong: The Power of Mathematical Thinking*.

I remember reading that section of the book and mostly breezing through that argument primarily as a simple example with a limited, but direct point. Durrett decided to delve into the applied math a bit further.

These are some of the subtle issues one eventually comes across when experts read others’ works which were primarily written for much broader audiences.

I also can’t help thinking that one paints a target on one’s back with a book title like that…

BTW, the quote of the day has to be:

Syndicated copies to:… so I went and got some brain cancer data.

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NIMBioS will host an Tutorial on Uncertainty Quantification for Biological Models

**Meeting dates:** June 26-28, 2017

**Location**: NIMBioS at the University of Tennessee, Knoxville

**Organizers:**

Marisa Eisenberg, School of Public Health, Univ. of Michigan

Ben Fitzpatrick, Mathematics, Loyola Marymount Univ.

James Hyman, Mathematics, Tulane Univ.

Ralph Smith, Mathematics, North Carolina State Univ.

Clayton Webster, Computational and Applied Mathematics (CAM), Oak Ridge National Laboratory; Mathematics, Univ. of Tennessee

**Objectives:**

Mathematical modeling and computer simulations are widely used to predict the behavior of complex biological phenomena. However, increased computational resources have allowed scientists to ask a deeper question, namely, “how do the uncertainties ubiquitous in all modeling efforts affect the output of such predictive simulations?” Examples include both epistemic (lack of knowledge) and aleatoric (intrinsic variability) uncertainties and encompass uncertainty coming from inaccurate physical measurements, bias in mathematical descriptions, as well as errors coming from numerical approximations of computational simulations. Because it is essential for dealing with realistic experimental data and assessing the reliability of predictions based on numerical simulations, research in uncertainty quantification (UQ) ultimately aims to address these challenges.

Uncertainty quantification (UQ) uses quantitative methods to characterize and reduce uncertainties in mathematical models, and techniques from sampling, numerical approximations, and sensitivity analysis can help to apportion the uncertainty from models to different variables. Critical to achieving validated predictive computations, both forward and inverse UQ analysis have become critical modeling components for a wide range of scientific applications. Techniques from these fields are rapidly evolving to keep pace with the increasing emphasis on models that require quantified uncertainties for large-scale applications. This tutorial will focus on the application of these methods and techniques to mathematical models in the life sciences and will provide researchers with the basic concepts, theory, and algorithms necessary to quantify input and response uncertainties and perform sensitivity analysis for simulation models. Concepts to be covered may include: probability and statistics, parameter selection techniques, frequentist and Bayesian model calibration, propagation of uncertainties, quantification of model discrepancy, adaptive surrogate model construction, high-dimensional approximation, random sampling and sparse grids, as well as local and global sensitivity analysis.

This tutorial is intended for graduate students, postdocs and researchers in mathematics, statistics, computer science and biology. A basic knowledge of probability, linear algebra, and differential equations is assumed.

**Application deadline:** March 1, 2017

**To apply, you must complete an application on our online registration system:**

- Click here to access the system
- Login or register
- Complete your user profile (if you haven’t already)
- Find this tutorial event under
*Current Events Open for Application*and click on*Apply*

Participation in NIMBioS tutorials is by application only. Individuals with a strong interest in the topic are encouraged to apply, and successful applicants will be notified within two weeks after the application deadline. If needed, financial support for travel, meals, and lodging is available for tutorial attendees.

**Summary Report**. TBA

**Live Stream.** The Tutorial will be streamed live. Note that NIMBioS Tutorials involve open discussion and not necessarily a succession of talks. In addition, the schedule as posted may change during the Workshop. To view the live stream, visit http://www.nimbios.org/videos/livestream. A live chat of the event will take place via Twitter using the hashtag #uncertaintyTT. The Twitter feed will be displayed to the right of the live stream. We encourage you to post questions/comments and engage in discussion with respect to our Social Media Guidelines.

Source: NIMBioS Tutorial: Uncertainty Quantification for Biological Models

]]>Prof. Walter B. Rudin presents the lecture, "Set Theory: An Offspring of Analysis." Prof. Jay Beder introduces Prof. Dattatraya J. Patil who introduces Prof....]]>

MyScript MathPad is a mathematic expression demonstration that lets you handwrite your equations or mathematical expressions on your screen and have them rendered into their digital equivalent for easy sharing. Render complex mathematical expressions easily using your handwriting with no constraints. The result can be shared as an image or as a LaTeX* or MathML* string for integration in your documents.

This looks like something I could integrate into my workflow.

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A mathematical model could lead to a new approach to the study of what is possible, and how it follows from what already exists.

Innovation is one of the driving forces in our world. The constant creation of new ideas and their transformation into technologies and products forms a powerful cornerstone for 21st century society. Indeed, many universities and institutes, along with regions such as Silicon Valley, cultivate this process.

And yet the process of innovation is something of a mystery. A wide range of researchers have studied it, ranging from economists and anthropologists to evolutionary biologists and engineers. Their goal is to understand how innovation happens and the factors that drive it so that they can optimize conditions for future innovation.

This approach has had limited success, however. The rate at which innovations appear and disappear has been carefully measured. It follows a set of well-characterized patterns that scientists observe in many different circumstances. And yet, nobody has been able to explain how this pattern arises or why it governs innovation.

Today, all that changes thanks to the work of Vittorio Loreto at Sapienza University of Rome in Italy and a few pals, who have created the first mathematical model that accurately reproduces the patterns that innovations follow. The work opens the way to a new approach to the study of innovation, of what is possible and how this follows from what already exists.

The notion that innovation arises from the interplay between the actual and the possible was first formalized by the complexity theorist Stuart Kauffmann. In 2002, Kauffmann introduced the idea of the “adjacent possible” as a way of thinking about biological evolution.

I know he discusses some of this in At Home in the Universe.

The adjacent possible is all those things—ideas, words, songs, molecules, genomes, technologies and so on—that are one step away from what actually exists. It connects the actual realization of a particular phenomenon and the space of unexplored possibilities.

But this idea is hard to model for an important reason. The space of unexplored possibilities includes all kinds of things that are easily imagined and expected but it also includes things that are entirely unexpected and hard to imagine. And while the former is tricky to model, the latter has appeared close to impossible.

What’s more, each innovation changes the landscape of future possibilities. So at every instant, the space of unexplored possibilities—the adjacent possible—is changing.

“Though the creative power of the adjacent possible is widely appreciated at an anecdotal level, its importance in the scientific literature is, in our opinion, underestimated,” say Loreto and co.

Nevertheless, even with all this complexity, innovation seems to follow predictable and easily measured patterns that have become known as “laws” because of their ubiquity. One of these is Heaps’ law, which states that the number of new things increases at a rate that is sublinear. In other words, it is governed by a power law of the form V(n) = knβ where β is between 0 and 1.

Words are often thought of as a kind of innovation, and language is constantly evolving as new words appear and old words die out.

This evolution follows Heaps’ law. Given a corpus of words of size n, the number of distinct words V(n) is proportional to n raised to the β power. In collections of real words, β turns out to be between 0.4 and 0.6.

Another well-known statistical pattern in innovation is Zipf’s law, which describes how the frequency of an innovation is related to its popularity. For example, in a corpus of words, the most frequent word occurs about twice as often as the second most frequent word, three times as frequently as the third most frequent word, and so on. In English, the most frequent word is “the” which accounts for about 7 percent of all words, followed by “of” which accounts for about 3.5 percent of all words, followed by “and,” and so on.

This frequency distribution is Zipf’s law and it crops up in a wide range of circumstances, such as the way edits appear on Wikipedia, how we listen to new songs online, and so on.

These patterns are empirical laws—we know of them because we can measure them. But just why the patterns take this form is unclear. And while mathematicians can model innovation by simply plugging the observed numbers into equations, they would much rather have a model which produces these numbers from first principles.

Enter Loreto and his pals (one of which is the Cornell University mathematician Steve Strogatz). These guys create a model that explains these patterns for the first time.

They begin with a well-known mathematical sand box called Polya’s Urn. It starts with an urn filled with balls of different colors. A ball is withdrawn at random, inspected and placed back in the urn with a number of other balls of the same color, thereby increasing the likelihood that this color will be selected in future.

This is a model that mathematicians use to explore rich-get-richer effects and the emergence of power laws. So it is a good starting point for a model of innovation. However, it does not naturally produce the sublinear growth that Heaps’ law predicts.

That’s because the Polya urn model allows for all the expected consequences of innovation (of discovering a certain color) but does not account for all the unexpected consequences of how an innovation influences the adjacent possible.

The upshot of the whole thing:

So Loreto, Strogatz, and co have modified Polya’s urn model to account for the possibility that discovering a new color in the urn can trigger entirely unexpected consequences. They call this model “Polya’s urn with innovation triggering.”

The exercise starts with an urn filled with colored balls. A ball is withdrawn at random, examined, and replaced in the urn.

If this color has been seen before, a number of other balls of the same color are also placed in the urn. But if the color is new—it has never been seen before in this exercise—then a number of balls of entirely new colors are added to the urn.

Loreto and co then calculate how the number of new colors picked from the urn, and their frequency distribution, changes over time. The result is that the model reproduces Heaps’ and Zipf’s Laws as they appear in the real world—a mathematical first. “The model of Polya’s urn with innovation triggering, presents for the first time a satisfactory first-principle based way of reproducing empirical observations,” say Loreto and co.

The team has also shown that its model predicts how innovations appear in the real world. The model accurately predicts how edit events occur on Wikipedia pages, the emergence of tags in social annotation systems, the sequence of words in texts, and how humans discover new songs in online music catalogues.

Interestingly, these systems involve two different forms of discovery. On the one hand, there are things that already exist but are new to the individual who finds them, such as online songs; and on the other are things that never existed before and are entirely new to the world, such as edits on Wikipedia.

Loreto and co call the former novelties—they are new to an individual—and the latter innovations—they are new to the world.

Curiously, the same model accounts for both phenomenon. It seems that the pattern behind the way we discover novelties—new songs, books, etc.—is the same as the pattern behind the way innovations emerge from the adjacent possible.

That raises some interesting questions, not least of which is why this should be. But it also opens an entirely new way to think about innovation and the triggering events that lead to new things. “These results provide a starting point for a deeper understanding of the adjacent possible and the different nature of triggering events that are likely to be important in the investigation of biological, linguistic, cultural, and technological evolution,” say Loreto and co.

We’ll look forward to seeing how the study of innovation evolves into the adjacent possible as a result of this work.

Ref: arxiv.org/abs/1701.00994: Dynamics on Expanding Spaces: Modeling the Emergence of Novelties

Source: Mathematical Model Reveals the Patterns of How Innovations Arise

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AMS Open Math Notes is a repository of freely downloadable mathematical works in progress hosted by the American Mathematical Society as a service to researchers, teachers and students. These draft works include course notes, textbooks, and research expositions in progress. They have not been published elsewhere, and, as works in progress, are subject to significant revision. Visitors are encouraged to download and use these materials as teaching and research aids, and to send constructive comments and suggestions to the authors.

h/t to Terry Tao for the notice.

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Retired UCI math professor Steven Roman has just started making a series of Group Theory lectures on YouTube.

Retired UCI math professor Steven Roman has just started making a series of Group Theory lectures on YouTube. No prior experience in group theory is necessary. He’s the author of the recent *Fundamentals of Group Theory: An Advanced Approach*. [1]

He hopes to eventually also offer lectures on ring theory, fields, vector spaces, and module theory in the near future.

[1]

S. Roman, *Fundamentals of Group Theory: An Advanced Approach*, 2012th ed. Birkhäuser, 2011.

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This short introduction to category theory is for readers with relatively little mathematical background. At its heart is the concept of a universal property, important throughout mathematics. After a chapter introducing the basic definitions, separate chapters present three ways of expressing universal properties: via adjoint functors, representable functors, and limits. A final chapter ties the three together. For each new categorical concept, a generous supply of examples is provided, taken from different parts of mathematics. At points where the leap in abstraction is particularly great (such as the Yoneda lemma), the reader will find careful and extensive explanations.

Tom Leinster has released a digital e-book copy of his textbook *Basic Category Theory* on arXiv. [1]

h/t to John Carlos Baez for the notice:

My friend Tom Leinster has written a great introduction to that wonderful branch of math called category theory! It’s free:

https://arxiv.org/abs/1612.09375

It starts with the basics and it leads up to a trio of related concepts, which are all ways of talking about universal properties.

Huh? What’s a ‘universal property’?

In category theory, we try to describe things by saying what they do, not what they’re made of. The reason is that you can often make things out of different ingredients that still do the same thing! And then, even though they will not be strictly the same, they will be isomorphic: the same in what they do.

A universal property amounts to a precise description of what an object does.

Universal properties show up in three closely connected ways in category theory, and Tom’s book explains these in detail:

through representable functors (which are how you actually hand someone a universal property),

through limits (which are ways of building a new object out of a bunch of old ones),

through adjoint functors (which give ways to ‘freely’ build an object in one category starting from an object in another).

If you want to see this vague wordy mush here transformed into precise, crystalline beauty, read Tom’s book! It’s not easy to learn this stuff – but it’s good for your brain. It literally rewires your neurons.

Here’s what he wrote, over on the category theory mailing list:

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Dear all,

My introductory textbook “Basic Category Theory” was published by Cambridge University Press in 2014. By arrangement with them, it’s now also free online:

https://arxiv.org/abs/1612.09375

It’s also freely editable, under a Creative Commons licence. For instance, if you want to teach a class from it but some of the examples aren’t suitable, you can delete them or add your own. Or if you don’t like the notation (and when have two category theorists ever agreed on that?), you can easily change the Latex macros. Just go the arXiv, download, and edit to your heart’s content.

There are lots of good introductions to category theory out there. The particular features of this one are:

• It’s short.

• It doesn’t assume much.

• It sticks to the basics.

[1]

T. Leinster, *Basic Category Theory*, 1st ed. Cambridge University Press, 2014.

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Tweet a positive 9-digit (or smaller) integer at @PrimesAsAService. It will reply via Twitter to tell you if the number prime or not.

Some of the usable commands one can tweet to the bot for answers follow. (Hint: Click on the buttons with the tweet text to auto-generate the relevant Tweet.)

- To factor a number into prime factors, tweet:

@primesasservice # factor

and replace the # with your desired number - To get the greatest common factor of two numbers, tweet:

@primesasservice #1 #2 gcf

and replace #1 and #2 with your desired numbers - To get a random prime number, tweet:

@primesasservice random - To find out if two numbers are coprime, tweet:

@primesasservice #1 #2 coprime

replace #1 and #2 with your desired numbers

If you ask about a prime number with a twin prime, it should provide the twin.

Pro tip: You should be able to drag and drop any of the buttons above to your bookmark bar for easy access/use in the future.

Happy prime tweeting!

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Instagram filter used: Clarendon

Photo taken at: UCLA Bookstore

I just saw Emily Riehl‘s new book *Category Theory in Context* on the shelves for the first time. It’s a lovely little volume beautifully made and wonderfully typeset. While she does host a free downloadable copy on her website, the book and the typesetting is just so pretty, I don’t know how one *wouldn’t* purchase the physical version.

I’ll also point out that this is one of the very first in Dover’s new series Aurora: Dover Modern Math Originals. Dover has one of the greatest reprint collections of math texts out there, I wish them the best in publishing new works with the same quality and great prices as they always have! We need more publishers like this.

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I do know, however, that there were a few who couldn’t make part of the Fall course, but who had some foundation in the subject and wanted to join us for the more advanced portion in the second half. Toward that end, below are the details for the course:

## Introduction to Complex Analysis: Part II | MATH X 451.41 – 350370

## Course Description

Complex analysis is one of the most beautiful and practical disciplines of mathematics, with applications in engineering, physics, and astronomy, to say nothing of other branches of mathematics. This course, the second in a two-part sequence, builds on last quarter’s development of the differentiation and integration of complex functions to extend the principles to more sophisticated and elegant applications of the theory. Topics to be discussed include conformal mappings, Laurent series and meromorphic functions, Riemann surfaces, Riemann Mapping Theorem, analytical continuation, and Picard’s Theorem. The course should appeal to those whose work involves the application of mathematics to engineering problems, and to those interested in how complex analysis helps explain the structure and behavior of the more familiar real number system and real-variable calculus.

Winter 2017

Days: Tuesdays

Time: 7:00PM to 10:00PM

Dates: Jan 10, 2017 to Mar 28, 2017

Contact Hours: 33.00

Location: UCLA, Math Sciences Building

Course Fee(s): $453.00

Available for Credit: 3 units

Instructors: Michael Miller

No refund after January 24, 2017.

Class will not meet on one Tuesday to be announced.Recommended Textbook:

Complex Analysis with Applicationsby Richard A. Silverman, Dover Publications; ISBN 0-486-64762-5

For many who will register, this certainly won’t be their first course with Dr. Miller–yes, he’s that good! But for the newcomers, I’ve written some thoughts and tips to help them more easily and quickly settle in and adjust: Dr. Michael Miller Math Class Hints and Tips | UCLA Extension

If you’d like additional details as well as lots of alternate textbooks, see the announcement for the first course in the series.

If you missed the first quarter and are interested in the second quarter but want a bit of review or some of the notes, let me know in the comments below.

I look forward to seeing everyone in the Winter quarter!

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This is the signal for the second.

How can you * not *follow this twitter account?!

Now I’m waiting for a Shannon bot and a Weiner bot. Maybe a John McCarthy bot would be apropos too?!

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Peter Woit has just made the final draft (dated 10/25/16) of his new textbook *Quantum Theory, Groups and Representations: An Introduction* freely available for download from his website. It covers quantum theory with a heavy emphasis on groups and representation theory and “contains significant amounts of material not well-explained elsewhere.”

He expects to finish up the diagrams and publish it next year some time, potentially through Springer.

Syndicated copies to:

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I enjoyed his treatment of inversion, but it seems like there’s a better way of laying the idea out, particularly for applications. Straightforward coverage of nested intervals and rectangles, limit points, convergent sequences, Cauchy convergence criterion. Given the level, I would have preferred some additional review of basic analysis and topology; he seems to do the bare minimum here.

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Syndicated copies to:Advanced Data Analysis from an Elementary Point of View

by Cosma Rohilla ShaliziThis is a draft textbook on data analysis methods, intended for a one-semester course for advance undergraduate students who have already taken classes in probability, mathematical statistics, and linear regression. It began as the lecture notes for 36-402 at Carnegie Mellon University.

By making this draft generally available, I am not promising to provide any assistance or even clarification whatsoever. Comments are, however, welcome.

The book is under contract to Cambridge University Press; it should be turned over to the press before the end of 2015. A copy of the next-to-final version will remain freely accessible here permanently.

Table of contents:

I. Regression and Its Generalizations

- Regression Basics
- The Truth about Linear Regression
- Model Evaluation
- Smoothing in Regression
- Simulation
- The Bootstrap
- Weighting and Variance
- Splines
- Additive Models
- Testing Regression Specifications
- Logistic Regression
- Generalized Linear Models and Generalized Additive Models
- Classification and Regression Trees

II. Distributions and Latent Structure- Density Estimation
- Relative Distributions and Smooth Tests of Goodness-of-Fit
- Principal Components Analysis
- Factor Models
- Nonlinear Dimensionality Reduction
- Mixture Models
- Graphical Models

III. Dependent Data- Time Series
- Spatial and Network Data
- Simulation-Based Inference

IV. Causal Inference- Graphical Causal Models
- Identifying Causal Effects
- Causal Inference from Experiments
- Estimating Causal Effects
- Discovering Causal StructureAppendices

- Data-Analysis Problem Sets
- Reminders from Linear Algebra
- Big O and Little o Notation
- Taylor Expansions
- Multivariate Distributions
- Algebra with Expectations and Variances
- Propagation of Error, and Standard Errors for Derived Quantities
- Optimization
- chi-squared and the Likelihood Ratio Test
- Proof of the Gauss-Markov Theorem
- Rudimentary Graph Theory
- Information Theory
- Hypothesis Testing
- Writing R Functions
- Random Variable Generation
Planned changes:

- Unified treatment of information-theoretic topics (relative entropy / Kullback-Leibler divergence, entropy, mutual information and independence, hypothesis-testing interpretations) in an appendix, with references from chapters on density estimation, on EM, and on independence testing
- More detailed treatment of calibration and calibration-checking (part II)
- Missing data and imputation (part II)
- Move d-separation material from “causal models” chapter to graphical models chapter as no specifically causal content (parts II and IV)?
- Expand treatment of partial identification for causal inference, including partial identification of effects by looking at all data-compatible DAGs (part IV)
- Figure out how to cut at least 50 pages
- Make sure notation is consistent throughout: insist that vectors are always matrices, or use more geometric notation?
- Move simulation to an appendix
- Move variance/weights chapter to right before logistic regression
- Move some appendices online (i.e., after references)?
(Text last updated 30 March 2016; this page last updated 6 November 2015)

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Learn about quantum computation and quantum information in this advanced graduate level course from MIT.

Syndicated copies to:## About this course

Already know something about quantum mechanics, quantum bits and quantum logic gates, but want to design new quantum algorithms, and explore multi-party quantum protocols? This is the course for you!

In this advanced graduate physics course on quantum computation and quantum information, we will cover:

- The formalism of quantum errors (density matrices, operator sum representations)
- Quantum error correction codes (stabilizers, graph states)
- Fault-tolerant quantum computation (normalizers, Clifford group operations, the Gottesman-Knill Theorem)
- Models of quantum computation (teleportation, cluster, measurement-based)
- Quantum Fourier transform-based algorithms (factoring, simulation)
- Quantum communication (noiseless and noisy coding)
- Quantum protocols (games, communication complexity)
Research problem ideas are presented along the journey.

## What you’ll learn

- Formalisms for describing errors in quantum states and systems
- Quantum error correction theory
- Fault-tolerant quantum procedure constructions
- Models of quantum computation beyond gates
- Structures of exponentially-fast quantum algorithms
- Multi-party quantum communication protocols
## Meet the instructor

Isaac Chuang Professor of Electrical Engineering and Computer Science, and Professor of Physics MIT

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With these notes, you should be able to toggle the settings in the file to read and listen to the notes almost as if you were attending the class live. I’ve done my best to write everything exactly as it was written on the board and only occasionally added small bits of additional text.

If you haven’t registered yet, you can watch the notes as if you were actually in the class and still join us next Tuesday night without missing a beat. There are over 25 people in the class not counting several I know who had to miss the first session.

Hope to see you then!

Syndicated copies to:## Viewing and Playing a Pencast PDF

Pencast PDF is a new format of notes and audio that can play in Adobe Reader X or above.

You can open a Pencast PDF as you would other PDF files in Adobe Reader X. The main difference is that a Pencast PDF can contain ink that has associated audio—called “active ink”. Click active ink to play its audio. This is just like playing a Pencast from Livescribe Online or in Livescribe Desktop. When you first view a notebook page, active ink appears in green type. When you click active ink, it turns gray and the audio starts playing. As audio playback continues, the gray ink turns green in synchronization with the audio. Non-active ink (ink without audio) is black and does not change appearance.

## Audio Control Bar

Pencast PDFs have an audio control bar for playing, pausing, and stopping audio playback. The control bar also has jump controls, bookmarks (stars), and an audio timeline control.

## Active Ink View Button

There is also an active ink view button. Click this button to toggle the “unwritten” color of active ink from gray to invisible. In the default (gray) setting, the gray words turn green as the audio plays. In the invisible setting, green words seem to write themselves on blank paper as the audio plays.

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A nice beginning overview of where they’re going and philosophy of the book. Makes the subject sound beautiful and wondrous, though they do use the word ‘miraculous’ which is overstepping a bit in almost any math book whose history is over a century old.

Their opening motivation for why complex instead of just real:

However, everything changes drastically if we make a natural, but misleadingly simple-looking assumption on that it is differentiable in the complex sense. This condition is called

holomorphicity, and it shapes most of the theory discussed in this book.

We shall start our study with some general characteristic properties of holomorphic functions, which are subsumed by three rather miraculous facts:

Contour integration: If is holomorphic in , then for appropriate closed paths inRegularity: If is holomorphic, then is indefinitely differentiable.Analytic continuation: If and are holomorphic functions in which are equal in an arbitrarily small disc in , then everywhere in .

This far into both books, I think I’m enjoying the elegance of Stein/Shakarchi better than Ahlfors.

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A very beautiful classical theory on field extensions of a certain type (Galois extensions) initiated by Galois in the 19th century. Explains, in particular, why it is not possible to solve an equation of degree 5 or more in the same way as we solve quadratic or cubic equations. You will learn to compute Galois groups and (before that) study the properties of various field extensions. We first shall survey the basic notions and properties of field extensions: algebraic, transcendental, finite field extensions, degree of an extension, algebraic closure, decomposition field of a polynomial. Then we shall do a bit of commutative algebra (finite algebras over a field, base change via tensor product) and apply this to study the notion of separability in some detail. After that we shall discuss Galois extensions and Galois correspondence and give many examples (cyclotomic extensions, finite fields, Kummer extensions, Artin-Schreier extensions, etc.). We shall address the question of solvability of equations by radicals (Abel theorem). We shall also try to explain the relation to representations and to topological coverings. Finally, we shall briefly discuss extensions of rings (integral elemets, norms, traces, etc.) and explain how to use the reduction modulo primes to compute Galois groups.

I’ve been watching MOOCs for several years and this is one of the few I’ve come across that covers some more advanced mathematical topics. I’m curious to see how it turns out and what type of interest/results it returns.

It’s being offered by National Research University – Higher School of Economics (HSE) in Russia.

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Logical probability theory was developed as a quantitative measure based on Boole's logic of subsets. But information theory was developed into a mature theory by Claude Shannon with no such connection to logic. But a recent development in logic changes this situation. In category theory, the notion of a subset is dual to the notion of a quotient set or partition, and recently the logic of partitions has been developed in a parallel relationship to the Boolean logic of subsets (subset logic is usually mis-specified as the special case of propositional logic). What then is the quantitative measure based on partition logic in the same sense that logical probability theory is based on subset logic? It is a measure of information that is named "logical entropy" in view of that logical basis. This paper develops the notion of logical entropy and the basic notions of the resulting logical information theory. Then an extensive comparison is made with the corresponding notions based on Shannon entropy.

Ellerman is visiting at UC Riverside at the moment. Given the information theory and category theory overlap, I’m curious if he’s working with John Carlos Baez, or what Baez is aware of this.

Based on a cursory look of his website(s), I’m going to have to start following more of this work.

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The book is a collection of papers written by a selection of eminent authors from around the world in honour of Gregory Chaitin s 60th birthday. This is a unique volume including technical contributions, philosophical papers and essays. Hardcover: 468 pages; Publisher: World Scientific Publishing Company (October 18, 2007); ISBN: 9789812770820]]>

Download a pre-publication version of the book which will be published by Cambridge University Press. The book arises from notes of courses taught at the second year graduate level at the University of Minnesota and is suitable to accompany study at that level.

Syndicated copies to:

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## Course Description

Complex analysis is one of the most beautiful and useful disciplines of mathematics, with applications in engineering, physics, and astronomy, as well as other branches of mathematics. This introductory course reviews the basic algebra and geometry of complex numbers; develops the theory of complex differential and integral calculus; and concludes by discussing a number of elegant theorems, including many–the fundamental theorem of algebra is one example–that are consequences of Cauchy’s integral formula. Other topics include De Moivre’s theorem, Euler’s formula, Riemann surfaces, Cauchy-Riemann equations, harmonic functions, residues, and meromorphic functions. The course should appeal to those whose work involves the application of mathematics to engineering problems as well as individuals who are interested in how complex analysis helps explain the structure and behavior of the more familiar real number system and real-variable calculus.

## Prerequisites

Basic calculus or familiarity with differentiation and integration of real-valued functions.

## Details

MATH X 451.37 – 268651 Introduction to Complex Analysis

Fall 2016

Time 7:00PM to 10:00PM

Dates Tuesdays, Sep 20, 2016 to Dec 06, 2016

Contact Hours 33.00

Location: UCLA, Math Sciences Building

Standard credit (3.9 units) $453.00

Instructor: Michael Miller

Register Now at UCLA

For many who will register, this certainly won’t be their first course with Dr. Miller — yes, he’s that good! But for the newcomers, I’ve written some thoughts and tips to help them more easily and quickly settle in and adjust:

Dr. Michael Miller Math Class Hints and Tips | UCLA Extension

I often recommend people to join in Mike’s classes and more often hear the refrain: “I’ve been away from math too long”, or “I don’t have the prerequisites to even begin to think about taking that course.” For people in those categories, you’re in luck! If you’ve even had a soupcon of calculus, you’ll be able to keep up here. In fact, it was a similar class exactly a decade ago by Mike Miller that got me back into mathematics. (Happy 10th math anniversary to me!)

I look forward to seeing everyone in the Fall!

Dr. Miller is back from summer vacation and emailed me this morning to say that he’s chosen the textbook for the class. We’ll be using *Complex Analysis with Applications* by Richard A. Silverman. [1]

(Note that there’s another introductory complex analysis textbook from Silverman that’s offered through Dover, so be sure to choose the correct one.)

As always in Dr. Miller’s classes, the text is just *recommended* (read: not required) and in-class notes are more than adequate. To quote him directly, “We will be using as a basic guide, but, as always, supplemented by additional material and alternate ways of looking at things.”

The bonus surprise of his email: He’s doing two quarters of Complex Analysis! So we’ll be doing both the Fall and Winter Quarters to really get some depth in the subject!

If you’re like me, you’ll probably take a look at some of the other common (and some more advanced) textbooks in the area. Since I’ve already compiled a list, I’ll share it:

*Complex Analysis*by Joseph Bak and Donald J. Newman [2]*Complex Analysis*by Theodore Gamelin [3]*Complex Variables and Applications*by James Brown and Ruel Churchill [4]*Fundamentals of Complex Analysis with Applications to Engineering, Science, and Mathematics*by Edward Saff and Arthur D. Snider (Pearson, 2014, 3rd edition) [5]

*Complex Analysis*by Lars Ahlfors [6]*Complex Analysis*by Serge Lang [7]*Functions of One Complex Variable*(Graduate Texts in Mathematics by John B. Conway (Springer, 1978) [8]*Complex Analysis*(Princeton Lectures in Analysis, No. 2) by Elias M. Stein and Rami Shakarchi (Princeton University Press, 2003) [9]

[1]

R. A. Silverman, *Complex Analysis with Applications*, 1st ed. Dover Publications, Inc., 2010, pp. 304–304 [Online]. Available: http://amzn.to/2c7KaQy

[2]

J. Bak and D. J. Newman, *Complex Analysis*, 3rd ed. Springer, 2010, pp. 328–328 [Online]. Available: http://amzn.to/2bLPW89

[3]

T. Gamelin, *Complex Analysis*. Springer, 2003, pp. 478–478 [Online]. Available: http://amzn.to/2bGNQct

[4]

J. Brown and R. V. Churchill, *Complex Variables and Applications*, 8th ed. McGraw-Hill, 2008, pp. 468–468 [Online]. Available: http://amzn.to/2bLQWcu

[5]

E. B. Saff and A. D. Snider, *Fundamentals of Complex Analysis with Applications to Engineering, Science, and Mathematics*, 3rd ed. Pearson, 2003, pp. 563–563 [Online]. Available: http://amzn.to/2f3Nyj6

[6]

L. V. Ahlfors, *Complex Analysis*, 3rd ed. McGraw-Hill, 1979, pp. 336–336 [Online]. Available: http://amzn.to/2bMXrxm

[7]

S. Lang, *Complex Analysis*, 4th ed. Springer, 2003, pp. 489–489 [Online]. Available: http://amzn.to/2c7OaR0

[8]

J. B. Conway, *Functions of One Complex Variable*, 2nd ed. Springer, 1978, pp. 330–330 [Online]. Available: http://amzn.to/2cggbF1

[9]

El. M. Stein and R. Shakarchi, *Complex Analysis*. Princeton University Press, 2003, pp. 400–400 [Online]. Available: http://amzn.to/2bGOG9c

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