Simple math shows how widespread vaccination can disrupt the exponential spread of disease and prevent epidemics.

This is a very clear and lucid article with some very basic math that shows the value of vaccines. I highly recommend it to everyone.

]]>Mayonnaise: 20 parts oil: 1 part liquid: 1 part yolk

Hollandaise: 5 parts butter: 1 part liquid: 1 part yolk

Vinaigrette: 3 parts oil: 1 part vinegar

Rule of thumb: You probably don’t need as much yolk as you thought you did.

I like that he provides the simple ratios with some general advice up front and then includes some ideas about variations before throwing in a smattering of specific recipes that one could use. For my own part, most of these chapters could be cut down to two pages and then perhaps even then cut the book down to a single sheet for actual use in the kitchen.

Part 4: Fat-Based Sauces

But what greatly helps the oil and water to remain separate is, among other things, a molecule in the yolk called lecithin, which, McGee explains, is part water soluble and part fat soluble.

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The traditional ratio, not by weight, is excellent and works beautifully: Hollandaise = 1 pound butter: 6 yolks. This ratio seems to have originated with Escoffier. Some cookbooks call for considerably less butter per yok, as little as 3 and some even closer to 2 to 1, but then you’re creeping into sabayon territory; whats more, I believe it’s a cook’s moral obligation to add more butter given the chance.

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more butter given the chance! Reminiscent of the Paula Deen phrase: “Mo’e butta is mo’e betta.”

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]]>

How do nonsensical counting-out rhymes like these enter the lexicon?

I’d read this a year or two ago for a specific purpose and revisited it again today just for entertainment. There’s some interesting history hiding in this sort of exercise.

I also considered these rhymes as simple counting games, but the’re not really used to count up as if they were ordinals. Most people couldn’t even come close to saying how many things they’d have counted if they sang such a song. I also find that while watching children sing these while “counting” they typically do so with a choice for each syllable, but this often fails in the very young so that they can make their own “mental” choice known while still making things seem random. For older kids, with a little forethought and some basic division one can make something seemingly random and turn it into a specific choice as well.

So what are these really and what purpose did they originally serve?

]]>Okay so right now I go to coffee shops to solve math problems alone, it's peaceful, I like it But someone mentioned they do cute tea parties with their girl squad & I said wow I want something like that but we all bring math textbooks & solve problems next to each other (1/2)

It’s not specifically femme yet does involve tea, but I’ve noticed something informal like this at the Starbucks just two blocks West of CalTech in Pasadena.

Separately but related, “adults” looking for a varied advanced math outlet in the Los Angeles area are welcome to join Dr. Mike Miller’s classes at UCLA ExtensionÂ on Tuesday nights from 7-10pm. We’re working on Algebraic Geometry this quarter. For those who might need notes to play catch up, I’ve got copies, with full audio recordings, that I’m happy to share.

]]>There is a proof for Brouwer's Fixed Point Theorem that uses a bridge - or portal - between geometry and algebra. Analogous to the relationship between geometry and algebra, there is a mathematical âportalâ from a looser version of geometry -- topology -- to a more âsophisticatedâ version of algebra. This portal can take problems that are very difficult to solve topologically, and recast them in an algebraic light, where the answers may become easier. Written and Hosted by Tai-Danae Bradley; Produced by Rusty Ward; Graphics by Ray Lux; Assistant Editing and Sound Design by Mike Petrow and Meah Denee Barrington; Made by Kornhaber Brown (www.kornhaberbrown.com)

I had originally started following Tai-Danae Bradley on Instagram having found her account via the #math tag. Turns out she’s burning up the world explaining some incredibly deep and complex mathematics in relatively simple terms. If you’re into math and not following her work already, get with the program. She’s awesome!

Personal Website: http://www.math3ma.com/

Twitter: @math3ma

Instagram: @math3ma

YouTube series: PBS Infinite Series

While this particular video leaves out a masters degree’s worth of detail, it does show some incredibly powerful mathematics by analogy. The overall presentation and descriptions are quite solid for leaving out as much as they do. This may be some of the best math-based science communication I’ve seen in quite a while.

I must say that I have to love and laugh at the depth and breadth of the comments on the video too. At best, this particular video, which seems to me to be geared toward high school or early college viewers and math generalists, aims to introduce come general topics and outline an incredibly complex proof in under 9 minutes. People are taking it to task for omitting “too much”! To completely understand and encapsulate the entirety of the topics at hand one would need coursework including a year’s worth of algebra, a year’s worth of topology including some algebraic topology, and a minimum of a few months worth of category theory. Even with all of these, to fill in all the particular details, I could easily see a professor spending an hour at the chalkboard filling in the remainder without any significant handwaving. The beauty of what she’s done is to give a very motivating high level perspective on the topic to get people more interested in these areas and what can be done with them. For the spirit of the piece, one might take her to task a bit for not giving more credit to the role Category Theory is playing in the picture, but then anyone interested is going to spend some time on her blog to fill in a lot of those holes. I’d challenge any of the comments out there to attempt to do what she’s done in under 9 minutes and do it better.

]]>Lecture one of six in an introductory set of lectures on category theory.

Take Away from the lecture: Morphisms are just as important as the objects that they morph. Many different types of mathematical constructions are best described using morphisms instead of elements. (This isn’t how things are typically taught however.)

Would have been nice to have some more discussion of the required background for those new to the broader concept. There were a tremendous number of examples from many areas of higher math that many viewers wouldn’t have previously had. I think it’s important for them to know that if they don’t understand a particular example, they can move on without much loss as long as they can attempt to apply the ideas to an area of math they are familiar with. Having at least a background in linear algebra and/or group theory are a reasonable start here.

In some of the intro examples it would have been nice to have seen at least one more fully fleshed out to better demonstrate the point before heading on to the multiple others which encourage the viewer to prove some of the others on their own.

Thanks for these Steven, I hope you keep making more! There’s such a dearth of good advanced math lectures on the web, I hope these encourage others to make some of their own as well.ï»ż

]]>The main purpose of this blog is to share updates about the open-access, open-source textbook Understanding Linear Algebra. Though work is continuing on this project, the HTML version of the text is now freely available, the forthcoming PDF version will also be free, and low-cost print options will be provided. The PreTeXt source code will be posted on GitHub as well.

h/t Robert Talbert

]]>My awesome colleague @davidaustinm is unveiling his new, open-source linear algebra text at the JMM, but you can access it NOW at his (new!) blog, the aptly named "More Linear Algebra": https://t.co/AAreqGk8DW

— Robert Talbert (@RobertTalbert) January 9, 2018

I'd like to embark on yet another mini-series here on the blog. The topic this time? Limits and colimits in category theory! But even if you're not familiar with category theory, I do hope you'll keep reading. Today's post is just an informal, non-technical introduction. And regardless of your categorical background, you've certainly come across many examples of limits and colimits, perhaps without knowing it! They appear everywhere - in topology, set theory, group theory, ring theory, linear algebra, differential geometry, number theory, algebraic geometry. The list goes on. But before diving in, I'd like to start off by answering a few basic questions.

A great little introduction to category theory! Can’t wait to see what the future installments bring.

Interestingly I came across this on Instagram. It may be one of the first times I’ve seen math at this level explained in pictorial form via Instagram.

]]>The theory developed here (that you will not find in any other course :) has much in common with (and complements) statistical mechanics and field theory courses; partition functions and transfer operators are applied to computation of observables and spectra of chaotic systems. Nonlinear dynamics 1: Geometry of chaos (see syllabus) Topology of flows - how to enumerate orbits, Smale horseshoes Dynamics, quantitative - periodic orbits, local stability Role of symmetries in dynamics Nonlinear dynamics 2: Chaos rules (see syllabus) Transfer operators - statistical distributions in dynamics Spectroscopy of chaotic systems Dynamical zeta functions Dynamical theory of turbulence The course, which covers the same material and the same exercises as the Georgia Tech course PHYS 7224, is in part an advanced seminar in nonlinear dynamics, aimed at PhD students, postdoctoral fellows and advanced undergraduates in physics, mathematics, chemistry and engineering.

An interesting looking online course that appears to be on a white-labeled Coursera platform.

I’ve come acrossÂ Predrag Cvitanovic’s work on Group Theory and Lie Groups before, so this portends some interesting work. I’ll have to see if I can carve out some time to sample some of it.

]]>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 subjectâs algebraic underpinnings, this quarter will concentrate on geometric interpretations and applications. Topics to be discussed include BĂ©zoutâs 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 subjectâs algebraic underpinnings, this quarter will concentrate on geometric interpretations and applications. Topics to be discussed include BĂ©zoutâs 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 Humeâs problem of induction, Andrey Kolmogorovâs general mathematical framework for probability, the application of computability to chance, and why chance is essential to modern physics. A final ideaâthat we are psychologically predisposed to error when judging chanceâis 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 inequalitiesâfrom 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 â 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 Eulerâs 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*Â 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 havenât 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 donât 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 cityâs population, they accounted for 40.6 percent of the stop-and-frisk checks by police.

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So if early âinvolvementâ 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 theyâll be free to create them when they start buying their own food.) I should add that my model is highly unlikely to scale. I donât 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 weâll 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 âsecret sauceâ crucial to their business. Itâs intellectual property, and it must be defended,

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the second question: Does the model work against the subjectâs 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 weâll 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 theyâre 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. Itâs that so many suffer.

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And hereâs 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 jobâwhich really boils down to efficient management of peopleâthey 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 OâNeilâs Weapons of Math Destruction. In this post Iâll lay out a reading agenda, along with ways to participate. The way people read along in this book club is through the web, essentially. Itâs 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 â which are too technical to easily describe to nonmathematicians! â 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 â 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-LĂ¶f 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 isÂ *Elementary Algebraic Geometry*Â 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.

]]>Itâs âa definitive study for all time, like writing the final book,â says one researcher whoâs 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 âcourse *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 â-groupoids. Homotopy type theory offers a new âunivalentâ foundation of mathematics, in which a central role is played by Voevodskyâs 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 â 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 âimplicit foundationâ 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 theoremsâthe Hilbert Basis Theorem and Hilbertâs Nullstellensatzâthat establish the dual relationship between so-called varietiesâboth affine and projectiveâand 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 KahanpĂ€Ă€, Pekka KekĂ€lĂ€inen, 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.)

]]>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 readingÂ *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 âobjectiveâ 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 âsubjectiveâ 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 Ï”, we prove a poly(N) lower bound on the (randomized) communication complexity of Ï”-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 (Ï”,Ï”)-weak approximate Nash equilibrium, which is a profile of mixed actions such that at least (1âÏ”)-fraction of the players are Ï”-best replying.

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

Kaisa MatomĂ€ki 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 Nashâs 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 canât fully believe that she didnât make it.

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

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