Below are my posts, from the most recent to the earliest. The blog is now set up via 'DISQUS" for uploading comments but I also welcome comments sent to me at

dbmumford@gmail.com. I would like to post some these comments with my replies unless you ask that they be kept private. I won't post any comments sent under a pseudonym however: nothing on my blog justifies hiding. Subscriptions to the blog can be made through the

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October 11, 2015

Tags: beautiful-mathematics, brain localization, and more

Two questions about the general nature of math have a certain interest to the larger public: mathematicians like to talk of a 'beautiful result' but what does this mean?; and is there a special part of cortex which is highly active when people do math? Recently Professors Michael Atiyah and Semir Zeki have put these two questions together, collaborating on an astonishing *experimental* investigation of these questions entitled "The experience of mathematical beauty and its neural correlates". Fifteen mathematicians were scanned using fMRI while viewing 60 mathematical formulas and rating them as ugly, neutral or beautiful. All 60 equations are shown in the sidebar, in 6 jpegs: you can magnify them by clicking. Their main result is that activity in the mOFC = medial (near the centerline) orbital (in the inward curl of the cortex above the eyes) frontal cortex *correlates* to some extent with their judgement of beauty (though strangely activity in mOFC relative to baseline diminishes). My aim in this post is to argue for the view that the subjective nature and attendant excitement during mathematical activity, including a sense of its beauty, varies greatly from mathematician to mathematician and that that would make it plausible for quite different parts of the brain to be active during mathematical reflection. I do not claim any scientific basis for this as my only evidence comes from opportunities to talk with colleagues and being struck with the remarkably diverse ways they seem to have of 'doing math'.

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July 20, 2015

Tags: economics, work, technology, employment, and more

Economics is an area that is built on mathematical models that simplify highly complex phenomena. People often forget that, as a result, economic models omit human and historical factors that are fundamentally non-mathematical and outside its scope. Thus the impossibility of building mathematical models of human psychology undermines that basic building block of economics, the "rational economic agent". But I also want to argue that advances in technology are transforming society in ways not dealt with in economic models, by altering the need for most human work, another foundation stone of economics. My thesis in this post is that, in addition to dealing with the Malthusian constraints caused by population growth, the next 50 years will see the growth of a nearly completely automated society that requires only minimal work from the large majority of its citizens. Such a development destroys the basic axioms on which economics is built, not to mention the basic structure of human lives. How in heaven's name will we adjust to such a "gift"?

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June 16, 2015

Tags: dalits, Jefferson, Manu, politics, India, caste, and more

In the Declaration of Independence, Jefferson famously wrote "We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness." Anyone with a brain knows how much these ringing words have been flagrantly ignored, certainly in the US and over pretty much the whole rest of the world. For some years I have been helping Prof. Shankar of the Chennai Mathematical Institute support an orphanage for Dalits (aka untouchables, "Harijans", scheduled castes) near Chennai in India and following to some extent his bulletins on the horrors endured by most Dalits and how they have none of the above rights. I stuck my neck out a few weeks ago writing a letter criticizing the "de-recognizing" of an activist pro-Dalit student group by the Director of the Indian Institute of Technology in Madras. The large reaction to this letter drew my attention to the extent to which this is a huge issue in India, so the purpose of this post is to look at the Indian struggle between the caste system and the ideals embodied in Jefferson's challenge.

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April 1, 2015

Tags: publishing, journals, Klaus Peters, Springer, private equity, and more

The world of professional publishing, of scholarly communication, is in a state of profound transformation. In some fields, for example physics and computer science, researchers have embraced this transformation and are forging new policies and better customs. In my experience, however, mathematicians are one of the most conservative research communities, clinging to old habits in spite of the opportunity to improve their working life. The impetus for this blog at this time is the death of Klaus Peters, a publisher who, more than any other that I have met, saw publishing in mathematics as a service to the professional community and strived tirelessly to find new ways to assist our community. But the changes that have happened in the commercial publishing world deeply disturbed him. I want to make a plea to my colleagues to spend more time considering how we should shape this aspect of our profession and then being open to radical changes: you have nothing to lose but the chains that are binding you to capitalist exploitation and you can gain a freer, simpler world to work in.

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February 9, 2015

Tags: art, Wolfe, Rockmore, beautiful-mathematical-expressions, and more

A couple of years ago, my good friend Dan Rockmore sent me by FedEx a remarkable invitation: write on copper plate "what you think is your most significant and elegant equation", for a limited edition of etchings. Even for Dan whom I've known for unorthodox projects, this seemed off the wall. But OK: Yole Zariski, the wife of my PhD advisor Oscar Zariski, had her artist brother cast some of Oscar's results on a necklace that she loved. Maybe the odd symbols that we put together might be viewed as a contemporary form of magic and, even if not understood, with a McLuhan-esque significance. And now, I learned from Dan, the project has expanded and they are seeking everyone's "beautiful mathematical expressions": login here "

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January 9, 2015

Tags: Pythagoras, discovery, geometry, Mesopotamia, China, India, transmission, proof, and more

The earliest extant documents that show knowledge of the rule relating the lengths of the three sides of a right triangle, that is traditionally named after Pythagoras, are Babylonian tablets dating from the centuries around Hammurabi's time, c. 1800 BCE. I am calling it a rule, not a theorem, following Jens Høyrup's suggestion, because it appears as a rule for connecting these lengths, not a theorem, in most of its early history. In any case, we don't know if Pythagoras proved it or not. After the Babylonians it next appears in extant records in Indian Vedic altar construction manuals, composed and transmitted orally as early as 800 BCE. Due to the wholesale destruction of documents in China in the Qin dynasty (221-206 BCE), the earliest records we have for the rule from China date from the second century BCE. This is a sparse set of sources indeed. But because this rule may be described in math-talk as the first "non-trivial" mathematical theorem to be discovered, there has been extensive debate about when and where it was first found, whether it was discovered independently in several places and how it was found. All this work belongs to what André Weil called "protohistory", an attempt to be scholarly when surviving documents are not only sparse but also possibly unrepresentative of a tradition, and totally absent from other cultures. The full history of Pythagoras's rule is a perfect example of a problem on which one can only speculate. But that's what I want to do in this blog. However, all is not speculation and, for great help in all that a real scholar of the History of Math might study, I want to thank Jens Høyrup for all his help.

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December 14, 2014

Tags: Grothendieck, schemes, Nature-magazine, and more

John Tate and I were asked by Nature magazine to write an obituary for Alexander Grothendieck. Now he is a hero of mine, the person that I met most deserving of the adjective "genius". I got to know him when he visited Harvard and John, Shurik (as he was known) and I ran a seminar on "Existence theorems". His devotion to math, his disdain for formality and convention, his openness and what John and others call his naiveté struck a chord with me.

So John and I agreed and wrote the obituary below. Since the readership of Nature were more or less entirely made up of non-mathematicians, it seemed as though our challenge was to try to make some key parts of Grothendieck's work accessible to such an audience. Obviously the very definition of a scheme is central to nearly all his work, and we also wanted to say something genuine about categories and cohomology. Here's what we came up with:

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November 1, 2014

Tags: quantum-mechanics, quantum-computers, linear algebra, heat-bath, density matrix, and more

Like many pure mathematicians, I have been puzzled over the meaning of Feynman's path integrals and put them in the category of weird ideas I wished I understood. This year, reading Folland's excellent book *Quantum Field Theory -- A Tourist Guide for Mathematicians*, I got a glimmer of what was going on. In a seminar on quantum computers with my good friend John Myers a few years ago, I had played with * finite dimensional* quantum systems, so it was natural to work out Feynman path integrals in the finite dimensional case.What emerged was so clean and even undergraduate-linear-algebra ready, that I want to put this rigorous and simple result in my blog. A similar path was taken by Ben Rudiak-Gould in a recent arxiv submission "The sum-over-histories formulation of quantum mechanics".

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October 30, 2014

Tags: zeta, explicit-formula, zeros, filtering, oscillations, and more

Barry Mazur and William Stein are writing an excellent book entitled *What is Riemann's Hypothesis?* The book leads up to Riemann's 'explicit formula' which, in von Mangoldt's form, is the formula for the discrete distribution supported at the prime powers:
$$ \sum_{\text{primes }p} \sum_{n \ge 1} \log(p) \delta_{p^n} (x)= 1 - \sum_k x^{(\rho_k-1)} - \tfrac{1}{x(x^2-1)} $$
where \( x > 1 \), \(\rho_k \) ranges over the zeros of the zeta function in the critical strip \( 0 < \text{Im}(\rho) < 1\) and the sum over *k* converges weakly as a distribution. This relates primes to the zeta zeros. In this blog, I ask: * can we find the first and maybe more low zeroes hidden in the very smallest primes?*

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October 22, 2014

Tags: high-school-math-education, algebra, money, interest, polynomials, and more

I believe there is a way to present algebra to middle schoolers that breaks the log jam of 'what the hell are *x* and *y*?' To make middle and high school math work it is essential to get students (or most of them anyway) to see how formulas are useful and intuitive ways to see how numbers in their real lives are connected to each other. It is pointless to drill students for three or four years in something most of them will forget as soon as they have taken their SATs. The blog below was addressed to NYTimes readers in the hope that the Times OpEd department might print it and some readers might loose some of their 'fear and loathing' of algebra. Other articles in this vein are in the education page.

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