In order to fully understand the quantum world, we may have to develop a new realm of mathematics.
Mathematics might be more of an environmental science than we realize. Even though it is a search for eternal truths, many mathematical concepts trace their origins to everyday experience. Astrology and architecture inspired Egyptians and Babylonians to develop geometry. The study of mechanics during the scientific revolution of the 17th century brought us calculus.
.. The bizarre world of quantum theory — where things can seem to be in two places at the same time and are subject to the laws of probability — not only represents a more fundamental description of nature than what preceded it, it also provides a rich context for modern mathematics. Could the logical structure of quantum theory, once fully understood and absorbed, inspire a new realm of mathematics that might be called “quantum mathematics”?
.. Ideas that originate in particle physics have an uncanny tendency to appear in the most diverse mathematical fields. This is especially true for string theory.
.. in the quantum world everything that can happen does happen.
.. Mirror symmetry illustrates a powerful property of quantum theory called duality: Two classical models can become equivalent when considered as quantum systems, as if a magic wand is waved and all the differences suddenly disappear. Dualities point to deep but often mysterious symmetries of the underlying quantum theory. In general, they are poorly understood and an indication that our understanding of quantum theory is incomplete at best.
.. A succinct way to summarize that theory is that mass tells space how to curve, and space tells mass how to move.
.. It is comforting to see how mathematics has been able to absorb so much of the intuitive, often imprecise reasoning of quantum physics and string theory, and to transform many of these ideas into rigorous statements and proofs.
A Long-Sought Proof, Found and Almost Lost
When a German retiree proved a famous long-standing mathematical conjecture, the response was underwhelming.
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.”
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.”
.. 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.”
.. 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
.. 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.
.. 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.
.. 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,”
.. 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.
.. Richards said a variation on the inequality could help statisticians better predict the ranges in which variables like stock prices fluctuate over time.
.. 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.”
Proof That Computers Can’t Do Everything (The Halting Problem)
This video gives an informal presentation of Alan Turing’s Halting Theorem, a serious, highly influential result in computer science.
Meet the Math Professor Who’s Fighting Gerrymandering With Geometry
Moon Duchin is an associate professor of math and director of the Science, Technology and Society program at Tufts. She realized last year that some of her research about metric geometry could be applied to gerrymandering — the practice of manipulating the shape of electoral districts to benefit a specific party, which is widely seen as a major contributor to government dysfunction.
.. In redistricting, one of the principles that’s taken seriously by courts is that districts should be compact. The U.S. Constitution does not say that, but many state constitutions do, and it’s taken as a kind of general principle of how districts ought to look.
.. So our first aim was to think like mathematicians about compactness and look at all the definitions that already exist, and compare them and try to prove theorems about the relationships between the definitions.
.. this landscape of compactness had a hole in it. It’s not that mathematicians hadn’t been working on it; it’s that [geometry experts] hadn’t entered the legal conversation.
.. the Polsby-Popper score; that’s the area of a district compared to the area of a circle of the same perimeter, given as a percentage.
.. It turns out that if you take a district and you look at the average distances between all of its points, then the bigger that is, the less compact, once you normalize by the diameter. That meant that I already had published theorems that, I think, cast some light on the districting problem.
.. The point of the summer school isn’t just to bring people together so we can convince them of our ideas. We also want to pool ideas and see, putting all those brilliant people in one place, can we make some progress on what’s been a pretty intractable problem?
.. Bouncing things off of diverse audiences has taught me things I didn’t already know about how rhetorically accessible different ideas are. This is well-known to educators: Once you achieve a certain level of expertise, it can be hard to find the difficult spots and the reasoning anymore because they’re so familiar to you.
.. the “efficiency gap.” It was a new metric of partisan gerrymandering that, for the first time, a court said they liked. The way it was devised was that the people who created it, they went back and they read all of Justice Anthony Kennedy’s written decisions about measuring gerrymandering. By reading his words and by reading what he said he found convincing and less convincing, they designed a statistic to appeal to him. He’s that vaunted median justice. If you can come up with something that will be convincing to Anthony Kennedy, then you’ve probably just changed the outcome.