Algebraic geometry is the study, using algebraic tools, of geometric objects defined as the solution sets to systems of polynomial equations in several variables. This course is the second in a two-quarter introductory sequence that develops the basic theory of this classical mathematical field. Whereas the fall-quarter course focused more on the subject’s algebraic underpinnings, this quarter will concentrate on geometric interpretations and applications. Topics to be discussed include Bézout’s Theorem, rational varieties, cubic curves and surfaces (including the remarkable 27-line theorem), and the connection between varieties and manifolds. The theoretical discussion will be supported by a large number of examples and exercises. The course should appeal to those with an interest in gaining a deeper understanding of the mathematical interplay among algebra, geometry, and topology.

I’m definitely attending the Winter Quarter!

]]>Algebraic geometry is the study, using algebraic tools, of geometric objects defined as the solution sets to systems of polynomial equations in several variables. This course is the second in a two-quarter introductory sequence that develops the basic theory of this classical mathematical field. Whereas the fall-quarter course focused more on the subject’s algebraic underpinnings, this quarter will concentrate on geometric interpretations and applications. Topics to be discussed include Bézout’s Theorem, rational varieties, cubic curves and surfaces (including the remarkable 27-line theorem), and the connection between varieties and manifolds. The theoretical discussion will be supported by a large number of examples and exercises. The course should appeal to those with an interest in gaining a deeper understanding of the mathematical interplay among algebra, geometry, and topology.

Alright math nerds, it’s that time again! Be sure to register for Mike Miller’s excellent follow-on course for Algebraic Geometry.

Don’t forget to use the coupon code EARLY to save 10% with an early registration–time is limited!

]]>The injury was to the professor’s hand, but I’m pretty sure it wasn’t due to excessive hand-waiving…

]]>In the sixteenth and seventeenth centuries, gamblers and mathematicians transformed the idea of chance from a mystery into the discipline of probability, setting the stage for a series of breakthroughs that enabled or transformed innumerable fields, from gambling, mathematics, statistics, economics, and finance to physics and computer science. This book tells the story of ten great ideas about chance and the thinkers who developed them, tracing the philosophical implications of these ideas as well as their mathematical impact. Persi Diaconis and Brian Skyrms begin with Gerolamo Cardano, a sixteenth-century physician, mathematician, and professional gambler who helped develop the idea that chance actually can be measured. They describe how later thinkers showed how the judgment of chance also can be measured, how frequency is related to chance, and how chance, judgment, and frequency could be unified. Diaconis and Skyrms explain how Thomas Bayes laid the foundation of modern statistics, and they explore David Hume’s problem of induction, Andrey Kolmogorov’s general mathematical framework for probability, the application of computability to chance, and why chance is essential to modern physics. A final idea―that we are psychologically predisposed to error when judging chance―is taken up through the work of Daniel Kahneman and Amos Tversky. Complete with a brief probability refresher, Ten Great Ideas about Chance is certain to be a hit with anyone who wants to understand the secrets of probability and how they were discovered.]]>

The Workshop on Applied Category Theory 2018 takes place in May 2018. A principal goal of this workshop is to bring early career researchers into the applied category theory community. Towards this goal, we are organising the Adjoint School. The Adjoint School will run from January to April 2018.

There’s still some time left to apply. And if nothing else, this looks like it’s got some interesting resources.

h/t John Carlos Baez

]]>Women in Harvard's math department report a bevy of inequalities—from a discouraging absence of female faculty to a culture of "math bro" condescension.

A story about math that sadly doesn’t feature equality.

Oddly not featured in the story was any reference to the Lawrence H. Summers incident of 2005. Naturally, one can’t pin the issue on him as this lack of diversity has spanned the life of the university, but apparently the math department didn’t get the memo when the university president left.

I’ve often heard that the fish stinks from the head, but apparently it’s the whole fish here.

]]>If you’re aware of things I’ve missed, or which have appeared since, please do let me know in the comments.

- Harpreet Bedi (YouTube) 68 lectures (Note: His website also has some other good lectures on Galois Theory and Algebraic Topology)
- Miles Reed(How to Download Miles Reid’s Algebraic Geometry videos)
- Basic Algebraic Geometry: Varieties, Morphisms, Local Rings, Function Fields and Nonsingularity (NPTEL)
- Algebraic geometry for physicists by Ugo Bruzzo
- Lectures on Algebraic Geometry by L. Goettsche (ICTP)
- Talks given at the AMS Summer Institute in Algebraic Geometry (2015)
- Classical Algebraic Geometry Today (MSRI Workshop 2009)
- Lectures by Harris, Hartshorne, Maclagan, and Beelen at ELGA2011

Some other places with additional (sometimes overlapping resources), particularly for more advanced/less introductory lectures:

- Video Lectures for Algebraic Geometry (MathOverflow)
- Sites to Learn Algebraic Geometry (MathOverflow)
- Video lectures of Algebraic Geometry-Hartshorne-Shafarevich (MathOverflow)

When: Wednesday, October 25, 2017, from 4:30 PM – 6:30 PM PDT Where: UCLA California NanoSystems Institute (CNSI), 570 Westwood Plaza, Los Angeles, CA 90095 While there is no mathematical formula for writing television comedy, for the writers of The Simpsons, Futurama, and The Big Bang Theory, mathematical formulas (along with classic equations and cutting-edge theorems) can sometimes be an integral part of those shows. In a lively and nerdy discussion, five of these writers (who have advanced degrees in math, physics, and computer science) will share their love of numbers and talent for producing laughter. Mathematician Sarah Greenwald, who teaches and writes about math in popular culture, will moderate the panel. The event will begin with a lecture by bestselling author Simon Singh (The Simpsons and Their Mathematical Secrets), who will examine some of the mathematical nuggets hidden in The Simpsons (from Euler’s identity to Mersenne primes) and discuss how Futurama has also managed to include obscure number theory and complex ideas about geometry. Tickets: Tickets are $15 each and seating is limited, so reserve your seat soon. Tickets can be purchased here via Eventbrite (ticket required for entry to the event). A limited number of free tickets will be reserved for UCLA students. We ask that students come to IPAM between 9:00am and 3:00pm on Friday, October 20, to present your BruinCard and pick up your ticket (one ticket per BruinCard, nontransferable). If any tickets remain, we will continue distributing free tickets to students on Monday, Oct. 23, starting at 9:00am until we run out. Both your ticket and BruinCard must be presented at the door for entry. Doors open at 4:00. Please plan to arrive early to check in and find a seat. We expect a large audience.

Okay math nerds, this looks like an interesting lecture if you’re in Los Angeles next Wednesday. I remember reading and mostly liking Singh’s book *The Simpsons and Their Mathematical Secrets* a few years back.

The hard core math crowd may be disappointed in the level, but it could be an interesting group to get out and be social with.

My review of *The Simpsons and Their Mathematical Secrets* from Goodreads:

]]>I’m both a math junkie and fan of the Simpsons. Singh’s book is generally excellent and well written and covers a broad range of mathematical areas. I’m a major fan of his book Big Bang: The Origin of the Universe, but find myself wanting much more from this effort. Much of my problem stems from my very deep knowledge of math and its history as well as having read most of the vignettes covered here in other general popular texts multiple times. Fortunately most readers won’t suffer from this and will hopefully find some interesting tidbits both about the Simpsons and math here to whet their appetites.

There were several spots at which I felt that Singh stretched a bit too far in attempting to tie the Simpsons to “mathematics” and possibly worse, several spots where he took deliberate detours into tangential subjects that had absolutely no relation to the Simpsons, but these are ultimately good for the broader public reading what may be the only math-related book they pick up this decade.

This could be considered a modern-day version of E.T. Bell‘s Men of Mathematics but with an overly healthy dose of side-entertainment via the Simpsons and Futurama to help the medicine go down.

Chapter 3: Rings, Section 3 – Chapter 4: Arithmetic in F[x], Sections 1 & 2

Reviewing over some algebra for my algebraic geometry class

]]>I don’t think she’s used the specific words in the book yet, but O’Neil is fundamentally writing about social justice and transparency. To a great extent both governments and increasingly large corporations are using these Weapons of Math Destruction inappropriately. Often it may be the case that the algorithms are so opaque as to be incomprehensible by their creators/users, but, as I suspect in many cases, they’re being used to actively create social injustice by benefiting some classes and decimating others. The evolving case of Facebook’s involvement in potentially shifting the outcome of the 2016 Presidential election especially via “dark posts” is an interesting case in point with regard to these examples.

In some sense these algorithms are like viruses running rampant in a large population without the availability of antibiotics to tamp down or modify their effects. Without feedback mechanisms and the ability to see what is going on as it happens the scale issue she touches on can quickly cause even greater harm over short periods of time.

I like that one of the first examples she uses for modeling is that of preparing food for a family. It’s simple, accessible, and generic enough that the majority of people can relate directly to it. It has lots of transparency (even more than her sabermetrics example from baseball). Sadly, however, there is a large swath of the American population that is poor, uneducated, and living in horrific food deserts that they may not grasp the subtleties of even this simple model. As I was reading, it occurred to me that there is a reasonable political football that gets pushed around from time to time in many countries that relates to food and food subsidies. In the United States it’s known as the Supplemental Nutrition Assistance Program (aka SNAP) and it’s regularly changing, though fortunately for many it has some nutritionists who help to provide a feedback mechanism for it. I suspect it would make a great example of the type of Weapon of Mass Destruction she’s discussing in this book. Those who are interested in a quick overview of it and some of the consequences can find a short audio introduction to it via the Eat This Podcast episode *How much does a nutritious diet cost? Depends what you mean by “nutritious”* or *Crime and nourishment Some costs and consequences of the Supplemental Nutrition Assistance Program* which discusses an interesting crime related sub-consequence of something as simple as when SNAP benefits are distributed.

I suspect that O’Neil won’t go as far as to bring religion into her thesis, so I’ll do it for her, but I’ll do so from a more general moral philosophical standpoint which underpins much of the Judeo-Christian heritage so prevalent in our society. One of my pet peeves of moralizing (often Republican) conservatives (who often both wear their religion on their sleeves as well as beat others with it–here’s a good recent case in point) is that they never seem to follow the Golden Rule which is stated in multiple ways in the Bible including:

He will reply, ‘Truly I tell you, whatever you did not do for one of the least of these, you did not do for me.

In a country that (says it) values meritocracy, much of the establishment doesn’t seem to put much, if any value, into these basic principles as they would like to indicate that they do.

I’ve previously highlighted the application of mathematical game theory before briefly in relation to the Golden Rule, but from a meritocracy perspective, why can’t it operate at all levels? By this I’ll make tangential reference to Cesar Hidalgo‘s thesis in his book *Why Information Grows* in which he looks not at just individuals (person-bytes), but larger structures like firms/companies (firmbytes), governments, and even nations. Why can’t these larger structures have their own meritocracy? When America “competes” against other countries, why shouldn’t it be doing so in a meritocracy of nations? To do this requires that we as individuals (as well as corporations, city, state, and even national governments) need to help each other out to do what we can’t do alone. One often hears the aphorism that “a chain is only as strong as it’s weakest link”, why then would we actively go out of our way to create weak links within our own society, particularly as many in government decry the cultures and actions of other nations which we view as trying to defeat us? To me the statistical mechanics of the situation require that we help each other to advance the status quo of humanity. Evolution and the Red Queeen Hypothesis dictates that humanity won’t regress back to the mean, it may be regressing itself toward extinction otherwise.

Chapter One – Bomb Parts: What is a Model

You can often see troubles when grandparents visit a grandchild they haven’t seen for a while.

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Upon meeting her a year later, they can suffer a few awkward hours because their models are out of date.

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Racism, at the individual level, can be seen as a predictive model whirring away in billions of human minds around the world. It is built from faulty, incomplete, or generalized data. Whether it comes from experience or hearsay, the data indicates that certain types of people have behaved badly. That generates a binary prediction that all people of that race will behave that same way.

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Needless to say, racists don’t spend a lot of time hunting down reliable data to train their twisted models.

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the workings of a recidivism model are tucked away in algorithms, intelligible only to a tiny elite.

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A 2013 study by the New York Civil Liberties Union found that while black and Latino males between the ages of fourteen and twenty-four made up only 4.7 percent of the city’s population, they accounted for 40.6 percent of the stop-and-frisk checks by police.

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So if early “involvement” with the police signals recidivism, poor people and racial minorities look far riskier.

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The questionnaire does avoid asking about race, which is illegal. But with the wealth of detail each prisoner provides, that single illegal question is almost superfluous.

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judge would sustain it. This is the basis of our legal system. We are judged by what we do, not by who we are.

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(And they’ll be free to create them when they start buying their own food.) I should add that my model is highly unlikely to scale. I don’t see Walmart or the US Agriculture Department or any other titan embracing my app and imposing it on hundreds of millions of people, like some of the WMDs we’ll be discussing.

You have to love the obligatory parental aphorism about making your own rules when you have your own house.

Yet the US SNAP program does just this. It could be an interesting example of this type of WMD.

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three kinds of models.

namely: baseball, food, recidivism

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The first question: Even if the participant is aware of being modeled, or what the model is used for, is the model opaque, or even invisible?

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many companies go out of their way to hide the results of their models or even their existence. One common justification is that the algorithm constitutes a “secret sauce” crucial to their business. It’s intellectual property, and it must be defended,

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the second question: Does the model work against the subject’s interest? In short, is it unfair? Does it damage or destroy lives?

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While many may benefit from it, it leads to suffering for others.

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The third question is whether a model has the capacity to grow exponentially. As a statistician would put it, can it scale?

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scale is what turns WMDs from local nuisances into tsunami forces, ones that define and delimit our lives.

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So to sum up, these are the three elements of a WMD: Opacity, Scale, and Damage. All of them will be present, to one degree or another, in the examples we’ll be covering

Think about this for a bit. Are there other potential characteristics?

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You could argue, for example, that the recidivism scores are not totally opaque, since they spit out scores that prisoners, in some cases, can see. Yet they’re brimming with mystery, since the prisoners cannot see how their answers produce their score. The scoring algorithm is hidden.

This is similar to anti-class action laws and arbitration clauses that prevent classes from realizing they’re being discriminated against in the workplace or within healthcare. On behalf of insurance companies primarily, many lawmakers work to cap awards from litigation as well as to prevent class action suits which show much larger inequities that corporations would prefer to keep quiet. Some of the recent incidences like the cases of Ellen Pao, Susan J. Fowler, or even Harvey Weinstein are helping to remedy these types of things despite individuals being pressured to stay quiet so as not to bring others to the forefront and show a broader pattern of bad actions on the part of companies or individuals. (This topic could be an extended article or even book of its own.)

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the point is not whether some people benefit. It’s that so many suffer.

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And here’s one more thing about algorithms: they can leap from one field to the next, and they often do. Research in epidemiology can hold insights for box office predictions; spam filters are being retooled to identify the AIDS virus. This is true of WMDs as well. So if mathematical models in prisons appear to succeed at their job—which really boils down to efficient management of people—they could spread into the rest of the economy along with the other WMDs, leaving us as collateral damage.

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Yellow–general highlights and highlights which don’t fit under another category below

Orange–Vocabulary word; interesting and/or rare word

Green–Reference to read

Blue–Interesting Quote

Gray–Typography Problem

Red–Example to work through

I’m reading this as part of Bryan Alexander’s online book club.

]]>Chapter 3: Rings, Sections 1 and 2

Reviewing over some algebra for my algebraic geometry class

]]>Chapter 5: Congruence in and Congruence-Class arithmetic, Sections 1 and 2

Reviewing over some algebra for my algebraic geometry class tonight. I always did love the pedagogic design of this textbook. The way he builds up algebraic structures is really lovely.

]]>Our new book club reading is Cathy O’Neil’s Weapons of Math Destruction. In this post I’ll lay out a reading agenda, along with ways to participate. The way people read along in this book club is through the web, essentially. It’s a distributed experience.

It occurs to me while reading the set up for this distributed online book club that posting on your own site and syndicating elsewhere (POSSE) while pulling back responses in an IndieWeb fashion is an awesome idea for this type of online activity. Now if only the social silos supported salmention!

I’m definitely in for this general schedule and someone has already gifted me a copy of the book. Given the level of comments I suspect will come about, I’m putting aside the fact that this book wasn’t written for me as an audience and will read along with the crowd. I’m much more curious how Bryan’s audience will see and react to it. But I’m also interested in the functionality and semantics of an online book club run in such a distributed way.

]]>This mathematician died last week. He won the Fields Medal in 2002 for proving the Milnor conjecture in a branch of algebra known as algebraic K-theory. He continued to work on this subject until he helped prove the more general Bloch-Kato conjecture in 2010. Proving these results — which are too technical to easily describe to nonmathematicians! — required him to develop a dream of Grothendieck: the theory of motives. Very roughly, this is a way of taking the space of solutions of a collection of polynomial equations and chopping it apart into building blocks. But the process of 'chopping up', and also these building blocks, called 'motives', are very abstract — nothing simple or obvious.

There’s some interesting personality and history in this short post of John’s.

]]>The Institute for Advanced Study is deeply saddened by the passing of Vladimir Voevodsky, Professor in the School of Mathematics. Voevodsky, a truly extraordinary and original mathematician, made many contributions to the field of mathematics, earning him numerous honors and awards, including the Fields Medal. Celebrated for tackling the most difficult problems in abstract algebraic geometry, Voevodsky focused on the homotopy theory of schemes, algebraic K-theory, and interrelations between algebraic geometry, and algebraic topology. He made one of the most outstanding advances in algebraic geometry in the past few decades by developing new cohomology theories for algebraic varieties. Among the consequences of his work are the solutions of the Milnor and Bloch-Kato Conjectures. More recently he became interested in type-theoretic formalizations of mathematics and automated proof verification. He was working on new foundations of mathematics based on homotopy-theoretic semantics of Martin-Löf type theories. His new "Univalence Axiom" has had a dramatic impact in both mathematics and computer science.

Sad to hear of Dr. Voevodsky’s passing just as I was starting into my studies of algebraic geometry…

]]>**Algebraic Geometry-Lecture 1 notes [.pdf file with embedded and linked audio]**

I’ve previously written some notes about how to best access and use these types of notes in the past. Of particular note, one must download the .pdf file and open in a recent version of Adobe Acrobat to take advantage of the linked/embedded audio file. (Trust me, it’s worth doing as it will be like you were there with the 20 of us who showed up last night!)

For those who prefer just the audio files separately, they can be listened to here, or downloaded.

Again, the recommended text is *Elementary Algebraic Geometry* by Klaus Hulek (AMS, 2003) ISBN: 0-8218-2952-1.

For those new to Dr. Miller’s classes, I’ve written up some hints/tips about them in the past as well.

]]>If you write clearly, then your readers may understand your mathematics and conclude that it isn't profound. Worse, a referee may find your errors. Here are some tips for avoiding these awful possibilities.

I want to come back and read this referenced article by Milne. The comments on this are pretty interesting as well.

]]>It’s “a definitive study for all time, like writing the final book,” says one researcher who’s mapping out new classes of geometric structures.]]>

This is a genuine introduction to algebraic geometry. The author makes no assumption that readers know more than can be expected of a good undergraduate. He introduces fundamental concepts in a way that enables students to move on to a more advanced book or course that relies more heavily on commutative algebra. The language is purposefully kept on an elementary level, avoiding sheaf theory and cohomology theory. The introduction of new algebraic concepts is always motivated by a discussion of the corresponding geometric ideas. The main point of the book is to illustrate the interplay between abstract theory and specific examples. The book contains numerous problems that illustrate the general theory. The text is suitable for advanced undergraduates and beginning graduate students. It contains sufficient material for a one-semester course. The reader should be familiar with the basic concepts of modern algebra. A course in one complex variable would be helpful, but is not necessary. It is also an excellent text for those working in neighboring fields (algebraic topology, algebra, Lie groups, etc.) who need to know the basics of algebraic geometry.

Dr. Miller emailed me yesterday to confirm that the textbook for his Fall UCLA Extension course *Elementary Algebraic Geometry* by Klaus Hulek (AMS, 2003) ISBN: 0-8218-2952-1.

Sadly, I totally blew the prediction of which text he’d use. I was so far off that this book wasn’t even on my list to review! I must be slipping…

]]>Homotopy type theory is a new branch of mathematics that combines aspects of several different fields in a surprising way. It is based on a recently discovered connection between homotopy theory and type theory. It touches on topics as seemingly distant as the homotopy groups of spheres, the algorithms for type checking, and the definition of weak ∞-groupoids. Homotopy type theory offers a new “univalent” foundation of mathematics, in which a central role is played by Voevodsky’s univalence axiom and higher inductive types. The present book is intended as a first systematic exposition of the basics of univalent foundations, and a collection of examples of this new style of reasoning — but without requiring the reader to know or learn any formal logic, or to use any computer proof assistant. We believe that univalent foundations will eventually become a viable alternative to set theory as the “implicit foundation” for the unformalized mathematics done by most mathematicians.]]>

Algebraic geometry is the study, using algebraic tools, of geometric objects defined as the solution sets to systems of polynomial equations in several variables. This introductory course, the first in a two-quarter sequence, develops the basic theory of the subject, beginning with seminal theorems—the Hilbert Basis Theorem and Hilbert’s Nullstellensatz—that establish the dual relationship between so-called varieties—both affine and projective—and certain ideals of the polynomial ring in some number of variables. Topics covered in this first quarter include: algebraic sets, projective spaces, Zariski topology, coordinate rings, the Grassmannian, irreducibility and dimension, morphisms, sheaves, and prevarieties. The theoretical discussion will be supported by a large number of examples and exercises. The course should appeal to those with an interest in gaining a deeper understanding of the mathematical interplay among algebra, geometry, and topology. Prerequisites: Some exposure to advanced mathematical methods, particularly those pertaining to ring theory, fields extensions, and point-set topology.

Dr. Michael Miller has announced the topic for his Fall math class at UCLA Extension: Algebraic Geometry!!

Yes math fans, as previously hinted at in prior conversations, we’ll be taking a deep dive into the overlap of algebra and geometry. Be sure to line up expeditiously as registration for the class won’t happen until July 31, 2017.

While it’s not yet confirmed, some sources have indicated that this may be the first part of a two quarter sequence on the topic. As soon as we have more details, we’ll post them here first. As of this writing, there is no officially announced textbook for the course, but we’ve got some initial guesses and the best are as follows (roughly in decreasing order):

*Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra*(Undergraduate Texts in Mathematics) 4th ed. by David A. Cox, John Little, and Donal O’Shea*Algebraic Geometry: An Introduction*(Universitext) by Daniel Perrin*An Invitation to Algebraic Geometry*(Universitext) by Karen E. Smith, Lauri Kahanpää, Pekka Kekäläinen, William Traves*Algebraic Geometry*(Dover Books on Mathematics) by Solomon Lefschetz (Less likely based on level and age, but Dr. Miller does love inexpensive Dover editions)

For those who are new to Dr. Miller’s awesome lectures, I’ve written some hints and tips on what to expect.

Most of his classes range from about 20-30 people, many of them lifelong regulars. (Yes, there are dozens of people like me who will take almost everything he teaches–he’s that good. This class, my 22nd, will be the start of my second decade of math with him.)

Mathematical Sciences Building, 520 Portola Plaza, Los Angeles, CA 90095

]]>A Course in Game Theory presents the main ideas of game theory at a level suitable for graduate students and advanced undergraduates, emphasizing the theory's foundations and interpretations of its basic concepts. The authors provide precise definitions and full proofs of results, sacrificing generalities and limiting the scope of the material in order to do so. The text is organized in four parts: strategic games, extensive games with perfect information, extensive games with imperfect information, and coalitional games. It includes over 100 exercises.

Tangentially suggested after reading *In Game Theory, No Clear Path to Equilibrium* by Erica Klarreich (Quanta Magazine)

Free, personal copy is downloadable in .pdf format with registration here.

]]>(.pdf download) Subjectivity and correlation, though formally related, are conceptually distinct and independent issues. We start by discussing subjectivity. A mixed strategy in a game involves the selection of a pure strategy by means of a random device. It has usually been assumed that the random device is a coin flip, the spin of a roulette wheel, or something similar; in brief, an ‘objective’ device, one for which everybody agrees on the numerical values of the probabilities involved. Rather oddly, in spite of the long history of the theory of subjective probability, nobody seems to have examined the consequences of basing mixed strategies on ‘subjective’ random devices, i.e. devices on the probabilities of whose outcomes people may disagree (such as horse races, elections, etc.).

Suggested by *In Game Theory, No Clear Path to Equilibrium* by Erica Klarreich (Quanta Magazine)

For a constant ϵ, we prove a poly(N) lower bound on the (randomized) communication complexity of ϵ-Nash equilibrium in two-player NxN games. For n-player binary-action games we prove an exp(n) lower bound for the (randomized) communication complexity of (ϵ,ϵ)-weak approximate Nash equilibrium, which is a profile of mixed actions such that at least (1−ϵ)-fraction of the players are ϵ-best replying.

Suggested by *In Game Theory, No Clear Path to Equilibrium* by Erica Klarreich (Quanta Magazine)

Kaisa Matomäki has proved that properties of prime numbers over long intervals hold over short intervals as well. The techniques she uses have transformed the study of these elusive numbers.]]>

John Nash’s notion of equilibrium is ubiquitous in economic theory, but a new study shows that it is often impossible to reach efficiently.

There’s a couple of interesting sounding papers in here that I want to dig up and read. There are some great results that sound like they are crying out for better generalization and classification. Perhaps some overlap with information theory and complexity?

To some extent I also find myself wondering about repeated play as a possible random walk versus larger “jumps” in potential game play and the effects this may have on the “evolution” of a solution by play instead of a simpler closed mathematical solution.

]]>Cotton twill hat features a full color embroidered Johns Hopkins lacrosse design showcasing our shielded blue jay. Unstructured low profile fit. Just the right wash; renowned perfect fit. Fabric strap closure with brass slide buckle. 100% cotton twill. Adjustable. Black. By Legacy.

I’d love to have a Johns Hopkins hat just like this with “Math” instead of “Lacrosse”. Surely the department has them made occasionally?

]]>Songs about communication, telephones, conversation, satellites, love, auto-tune and even one about a typewriter! They all relate at least tangentially to the topic at hand. To up the ante, everyone should realize that digital music would be impossible without Shannon’s seminal work.

Let me know in the comments or by replying to one of the syndicated copies listed below if there are any great tunes that the list is missing.

Enjoy the list and the book!

]]>I am totally stunned to learn that Maryam Mirzakhani died today, aged 40, after a severe recurrence of the cancer she had been fighting for several years. I had planned to email her some wishes for a speedy recovery after learning about the relapse yesterday; I still can’t fully believe that she didn’t make it.

A nice obituary about a fantastic mathematician from a fellow Fields Prize winner.

]]>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.”

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.

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.

]]>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.]]>

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.

"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.

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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