Category Archives: Math

Mathematical Rainbows

Source: Quanta, Mar 2020

Ringel’s conjecture is a problem in combinatorics where you connect dots (vertices) with lines (edges) to form graphs. It predicts that there’s a special relationship between a type of large graph with 2n + 1 vertices and a proportionally smaller type of graph with + 1 vertices.

Ringel’s conjecture says that copies of any tree you make can be used to perfectly cover, or tile, the edges of the corresponding complete graph, the same way you could use identical tiles to cover a kitchen floor.

 

Statistical Analysis of Medical Screening

Source: Tofspot.blogspot.com, Mar 2020

Of the 950,000 folks who are not infected. nearly all (95%) get a clean bill of health and of the 50,000 infected souls, nearly all (95%) get a red card. But along the way 2500 infected people go undetected, while 47,000 uninfected get red carded nonetheless.

Perhaps they ate a poppy seed bagel that morning, or the test was run late at night by a dog-tired technician. In any case, you will notice that half of all those getting flagged are not in fact infected.

Consequently, the media reports that the infection rate is 9.5% [(47,500+47,500)/1,000,000] rather than 5%, although what they really mean is that the test-positive rate is 9.5%.

This also affects estimates of the mortality rates. It’s not called the Dreaded Red Squamish for nothing.

But the denominator has been inflated by the false positives, so deaths will be divided by 95,000 rather than the [unknown] 50,000. There will be 47,000 “recoveries” of people who never actually had the disease. OTOH, some of the undetected 2,500 may also shuffle off the coil of mortality, but these will be assigned to collateral conditions (heart disease, asthma, etc.)

Computing, Quantum Mechanics and Mathematics are Entangled

Source: Quanta, Mar 2020

The new proof establishes that quantum computers that calculate with entangled quantum bits or qubits, rather than classical 1s and 0s, can theoretically be used to verify answers to an incredibly vast set of problems. The correspondence between entanglement and computing came as a jolt to many researchers.

The proof’s co-authors set out to determine the limits of an approach to verifying answers to computational problems. That approach involves entanglement. By finding that limit the researchers ended up settling two other questions almost as a byproduct: Tsirelson’s problem in physics, about how to mathematically model entanglement, and a related problem in pure mathematics called the Connes embedding conjecture.

Undecidable Problems

Turing defined a basic framework for thinking about computation before computers really existed. In nearly the same breath, he showed that there was a certain problem computers were provably incapable of addressing. It has to do with whether a program ever stops.

Typically, computer programs receive inputs and produce outputs. But sometimes they get stuck in infinite loops and spin their wheels forever. When that happens at home, there’s only one thing left to do.

“You have to manually kill the program. Just cut it off,” Yuen said.

Turing proved that there’s no all-purpose algorithm that can determine whether a computer program will halt or run forever. You have to run the program to find out.

In technical terms, Turing proved that this halting problem is undecidable — even the most powerful computer imaginable couldn’t solve it.

After Turing, computer scientists began to classify other problems by their difficulty. Harder problems require more computational resources to solve — more running time, more memory. This is the study of computational complexity.

Ultimately, every problem presents two big questions: “How hard is it to solve?” and “How hard is it to verify that an answer is correct?”

Interrogate to Verify

When problems are relatively simple, you can check the answer yourself. But when they get more complicated, even checking an answer can be an overwhelming task. However, in 1985 computer scientists realized it’s possible to develop confidence that an answer is correct even when you can’t confirm it yourself.

The method follows the logic of a police interrogation.

In computer science terms, the two parties in an interrogation are a powerful computer that proposes a solution to a problem — known as the prover — and a less powerful computer that wants to ask the prover questions to determine whether the answer is correct. This second computer is called the verifier.

To take a simple example, imagine you’re colorblind and someone else — the prover — claims two marbles are different colors. You can’t check this claim by yourself, but through clever interrogation you can still determine whether it’s true.

Put the two marbles behind your back and mix them up. Then ask the prover to tell you which is which. If they really are different colors, the prover should answer the question correctly every time. If the marbles are actually the same color — meaning they look identical — the prover will guess wrong half the time.

“If I see you succeed a lot more than half the time, I’m pretty sure they’re not” the same color, Vidick said.

By asking a prover questions, you can verify solutions to a wider class of problems than you can on your own.

researchers showed that by interrogating two provers separately about their answers, you can quickly verify solutions to an even larger class of problems than you can when you only have one prover to interrogate.

Prior to the new work, mathematicians had wondered whether they could get away with approximating infinite-dimensional matrices by using large finite-dimensional ones instead. 

 Now, because the Connes embedding conjecture is false, they know they can’t.

“Their result implies that’s impossible,” said Slofstra.

SS Chern – A Mathematician

Shiing-Shen Chern (/ɜːrn/Chinese陳省身pinyinChén XǐngshēnMandarin: [tʂʰən.ɕiŋ.ʂən]; October 28, 1911 – December 3, 2004) was a Chinese-American mathematician and poet.

He made fundamental contributions to differential geometry and topology. He has been called the “father of modern differential geometry” and is widely regarded as a leader in geometry and one of the greatest mathematicians of the twentieth century, winning numerous awards and recognition including the Wolf Prize and the inaugural Shaw Prize.[1][2][3][4][5][6][7]

In memory of Shiing-Shen Chern, the International Mathematical Union established the Chern Medal in 2010 to recognize “an individual whose accomplishments warrant the highest level of recognition for outstanding achievements in the field of mathematics”.[8]

Quanta Highlights Math Achievements

Sources: Quanta, 2019

https://www.quantamagazine.org/with-category-theory-mathematics-escapes-from-equality-20191010/

Lurie’s books are the single, authoritative text on infinity categories. They are completely rigorous, but hard to completely grasp. They’re especially poorly suited to serving as reference manuals — it’s difficult to look up specific theorems, or to check that a specific application of infinity categories that one might encounter in someone else’s paper really works out.

https://www.quantamagazine.org/foundations-built-for-a-general-theory-of-neural-networks-20190131/

Neural networks aim to mimic the human brain — and one way to think about the brain is that it works by accreting smaller abstractions into larger ones. Complexity of thought, in this view, is then measured by the range of smaller abstractions you can draw on, and the number of times you can combine lower-level abstractions into higher-level abstractions — like the way we learn to distinguish dogs from birds.

power in taking small pieces and combining them at greater levels of abstraction instead of attempting to capture all levels of abstraction at once.

https://www.quantamagazine.org/random-surfaces-hide-an-intricate-order-20190702/

Mathematicians have proved that a random process applied to a random surface will yield consistent patterns.

https://www.quantamagazine.org/how-artificial-intelligence-is-changing-science-20190311/

generative modeling asks how likely it is, given condition X, that you’ll observe outcome Y.

Cognitive Technologies … Sentences –> Symbols

Source: Dominic Cummings blog, Jun 2019

Language and writing were cognitive technologies created thousands of years ago which enabled us to think previously unthinkable thoughts. Mathematical notation did the same over the past 1,000 years. For example, take a mathematics problem described by the 9th Century mathematician al-Khwarizmi (who gave us the word algorithm):

Once modern notation was invented, this could be written instead as:

x2 + 10x = 39

Michael Nielsen uses a similar analogy. Descartes and Fermat demonstrated that equations can be represented on a diagram and a diagram can be represented as an equation. This was a new cognitive technology, a new way of seeing and thinking: algebraic geometry. Changes to the ‘user interface’ of mathematics were critical to its evolution and allowed us to think unthinkable thoughts (Using Artificial Intelligence to Augment Human Intelligence, see below).

Similarly in the 18th Century, there was the creation of data graphics to demonstrate trade figures. Before this, people could only read huge tables. This is the first data graphic:

screenshot 2019-01-29 00.28.21

The Jedi of data visualisation, Edward Tufte, describes this extraordinary graphic of Napoleon’s invasion of Russia as ‘probably the best statistical graphic ever drawn’. It shows the losses of Napoleon’s army: from the Polish-Russian border, the thick band shows the size of the army at each position, the path of Napoleon’s winter retreat from Moscow is shown by the dark lower band, which is tied to temperature and time scales (you can see some of the disastrous icy river crossings famously described by Tolstoy). NB. The Cabinet makes life-and-death decisions now with far inferior technology to this from the 19th Century (see below).

screenshot 2019-01-29 10.37.05

If we look at contemporary scientific papers they represent extremely compressed information conveyed through a very old fashioned medium, the scientific journal. Printed journals are centuries old but the ‘modern’ internet versions are usually similarly static. They do not show the behaviour of systems in a visual interactive way so we can see the connections between changing values in the models and changes in behaviour of the system. There is no immediate connection. Everything is pretty much the same as a paper and pencil version of a paper. In Media for Thinking the Unthinkable, Victor shows how dynamic tools can transform normal static representations so systems can be explored with immediate feedback. This dramatically shows how much more richly and deeply ideas can be explored. With Victor’s tools we can interact with the systems described and immediately grasp important ideas that are hidden in normal media.

Circles Disturbed

Source: Princeton University Press, 2012

Circles Disturbed delves into topics such as the way in which historical and biographical narratives shape our understanding of mathematics and mathematicians, the development of “myths of origins” in mathematics, the structure and importance of mathematical dreams, the role of storytelling in the formation of mathematical intuitions, the ways mathematics helps us organize the way we think about narrative structure, and much more.

Related Resource: JStor, date indeterminate

CHAPTER 1 From Voyagers to Martyrs: Toward a Storied History of Mathematics

 (pp. 1-51)

AMIR ALEXANDER
CHAPTER 6 Visions, Dreams, and Mathematics

(pp. 183-210)

BARRY MAZUR

Mathematicians can hardly avoid making use of stories of various kinds, to say nothing of images, sketches, and diagrams, to help convey the meaning of their accomplishments and their aims. As Peter Galison points out in chapter 2, we mathematicians often are nevertheless silent—or perhaps even uneasy—about the role that stories and images play in our work.

If someone asks usWhat is X?, whereXis some mathematical concept, we boldly answer, for we have been well trained in the art of definition. All the fine articulations of logical structure are at our fingertips. If, however, someone…

CHAPTER 8 Mathematics and Narrative: Why Are Stories and Proofs Interesting?

(pp. 232-243)

BERNARD TEISSIER

There are many types of narrations, from origin myths to the ship logs of maritime explorers, from children’s bedtime stories to works of literature—including poetry—and theater. We might also recall here Kipling’s joke in one of his letters from Japan about the person who, having borrowed a dictionary, gives it back with the comment that the stories are generally interesting, but too diverse.

The concept of narration varies with location and is not easy to define. Is a haiku a narration? Is Heraclitus’spanta rhei—“all things are flowing”—the concise narration of a part of his experience…