Welcome to the long anticipated end of 2020, or at least as close to it as we're willing to get this year! This is the seventh annual edition of our year-ending tradition of celebrating the biggest stories in math from the twelve preceding months, at the end of which, we'll identify the biggest math story of the year.

The criteria we use in selecting the stories that make our year-end list is that the maths they involve must either resolve long-standing questions, have real practical applications, or otherwise have real world impact. If you want more, deeper information about the topics covered in our summary of math stories for 2020, be sure to follow the links to additional background material.

We've found three main themes around which to organize this year's edition. Let's get straight to it!

Cracking Old Problems

2020 saw a lot of progress toward solving problems that have been challenging mathematicians for 50, 90, 100, and 270 years. The youngest problem in this category was the deciphering of the "340 Cipher", which California's Zodiac Killer taunted police with a half century ago.

The cipher was finally cracked in 2020 by a team of amateur codebreakers, David Oranchak, Sam Blake, and Jarl Van Eycke, who combined their skills in encryption, applied maths, and computer programming along with some basic intuitions to break the killer's code. The featured video in this section is the fifth installment in their series describing how they tested over 650,000 variations of the cipher to work out how to successfully read it.

The 90-year old problem that was finally solved in 2020 dealt with Keller's Conjecture, which predicts that if you cover a space with identical tiles, at least two of the tiles would have to share an edge. That proposition seems straightforward, but Keller really took it to the next level, and then some, by conjecturing that it would also apply in higher dimensions. Previous generations of mathematicians proved Keller's conjecture would work up through at least six dimensions and also proved it wouldn't hold in eight-or-higher dimensional spaces, leaving an open question of whether it would or would not hold in seven dimensions.

In 2020, professional mathematicians Joshua Brakensiek, Marijn Heule, John Mackey, and David Narváez used a cluster of 40 computers to finally prove, once and for all, that the Keller Conjecture is true in seven dimensions. The practical accomplishment here was the application of discrete graph theory to search for nonlinear codes called "cliques", which in addition to being useful for determining whether the Keller Conjecture might hold for a given dimensional space, can also be applied to speed data transmission.

The 100 year old problem that was solved during 2020 relates to how fluids diffuse within a closed space, which has been an highly relevant issue during 2020 given the airborne transmission of the SARS-CoV-2 coronavirus. Previously, solving problems involving diffusion processes could only be done approximately, requiring extensive computing resources. Mathematician Luca Giuggioli worked out how to more easily frame these kinds of problems for direct solution, which offers great promise for solving problems across a wide number of practical applications much more efficiently. In an ordinary year, this story would be a leading contender for the biggest math story of the year, but this is 2020.

At 270 years, the oldest problem is also a geometry problem, but involves a hungry goat, which makes it perhaps the most fun math story of the year. Quanta Magazine describes the scenario:

Here’s a simple-sounding problem: Imagine a circular fence that encloses one acre of grass. If you tie a goat to the inside of the fence, how long a rope do you need to allow the animal access to exactly half an acre?

Believe it or not, it wasn't until Ingo Ullisch applied techiques from the mathematical field of complex analysis that a mathematical formula was developed to precisely determine the length of the goat's rope to solve the problem!

Finding the Limits

2020 was a year in which more mathematicians gained direct experience in confronting Gödel's incompleteness theorems, which sets limits on what problems can be solved.

In 2020, some mathematical laws that have previously been advanced to describe real world observations have been found wanting. For example, Zipf's Law, a relatively simple relationship that has long been used to model population growth in cities, has turned out to not be complex enough to capture dynamic changes in populations over time.

Meanwhile, Benford's Law, which describes how often the digits 1 through 9 appear as the first digit of values within a dataset that has been successfully used to detect accounting fraud, turned out to be the wrong tool to use to detect potential election fraud in the U.S.

Those previous examples represent the narrowing of the boundaries where the mathematical principles involved may be successfully used. But 2020 also saw a major expansion of the boundaries of problems that can be solved with the introduction of a new proof that establishes that quantum computers may be used to solve problems conventional computers cannot. Quanta Magazine explains how the proof developed by Henry Yuen, Zhengfeng Ji, Anand Natarajan, Thomas Vidick, and John Wright expands the boundaries for what problems can be solved with the new computational technology:

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.

“It was a complete surprise,” said Miguel Navascués, who studies quantum physics at the Institute for Quantum Optics and Quantum Information in Vienna.

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.

This accomplishment is a very big deal, which will drive a lot of development for quantum computing technology. Again, in an ordinary year, this story would be a leading contender for the title of the biggest math story of the year, but it came in 2020.

A Year Defined by Mistakes

The featured video in this section is about the world's first documented math mistake, which dates back several thousand years to the days of ancient Sumeria. The video is relevant because two significant math mistakes that took place in 2020 have affected the lives of billions of people, which makes 2020's math mistakes the biggest math story of the year.

Both mistakes, perhaps unsurprisingly at this point of the year, are related to the coronavirus pandemic. Both mistakes came early in the pandemic and drove public policy responses to the spread of SARS-CoV-2 coronavirus infections, and in turn, are directly responsible for the year's economic recessions in many regions of the world.

The first math mistake that helped define 2020 came from an epidemiological model developed by Neil Ferguson at Imperial College London (ICL). In March 2020, Ferguson and public health officials heavily promoted the projected outcome of the ICL model's "do nothing" scenario, which projected 2.2 million deaths in just the U.S. and another 500,000 deaths in the U.K. to justify the early policy response of imposing lockdowns, the government mandated closure of businesses and orders for residents to stay at home.

But the assumptions encoded within the ICL model were providing results that proved to be off by several orders of magnitude from reality. Worse, the mistakes evaded detection by the ICL model's promoters because the code itself was virtually impenetrable, filled with excessively complex patches, kludges and other band-aid type programming and no connection to "ground truth" data for its 450 parameters. These issues did not become known until Ferguson finally made the code available for public review, with the outcome that trillions of dollars of economic losses around the world occurred because of the unstable projections that emerged from the ICL model's "black box".

Ultimately, not even Neil Ferguson believed his model's results. After it was discovered he had repeatedly violated the stay-at-home orders he promoted, he was forced to resign in disgrace from his role as a government adviser in the U.K.

The second math mistake that helped define 2020 appears to have resulted from the misplacement of a decimal point by the U.S. National Institutes of Health.

The error was such that it amplified the NIH's estimates of projected deaths from novel coronavirus infections in a way that seemed to provide confirmation of the ICL model's flawed projections. A post-mortem of the math error describes its impact:

Sampling bias in coronavirus mortality calculations led to a 10-fold increased mortality overestimation in March 11, 2020, US Congressional testimony. This bias most likely followed from information bias due to misclassifying a seasonal influenza IFR as a CFR, evident in a NEJM.org editorial. Evidence from the WHO confirmed that the approximate CFR of the coronavirus is generally no higher than that of seasonal influenza. By early May 2020, mortality levels from COVID-19 were considerably below predicted overestimations, a result that the public attributed to successful mitigating measures to contain the spread of the novel coronavirus.

Because of these mistakes, 2020 was all too often defined by the wrong policies being put into place at great harm and cost to society in fear of a projected massive death toll from the coronavirus. No wonder then that one of the more important math-related headlines for the year was "Modeling COVID-19 data must be done with extreme care, scientists say".

To be fair, maths done properly have greatly aided the development of vaccines and more appropriate public policy responses to the coronavirus pandemic. Never-the-less, these two mistakes together represent the biggest math story of 2020, helping to make the year what it was: a truly "Kushim"-ed up affair (watch the video to catch the reference!)

Update 29 December 2020: Just published peer-reviewed research highlights the ICL model's deficiencies. For a nontechnical summary of findings, see the press release via MedicalXpress: Model used to evaluate lockdowns was flawed.

Previously on Political Calculations

The Biggest Math Story of the Year is how we've traditionally marked the end of our posting year since 2014. Here are links to our previous editions, along with our coverage of other math stories during 2020:

• The Biggest Math Story of the Year (2014)
• The Biggest Math Story of 2015
• The Biggest Math Story of 2016
• The Biggest Math Story of 2017
• The Biggest Math Story of 2018
• The Biggest Math Story of 2019
• The Biggest Math Story of 2020
• The Survival Function of Roman Emperors
• The Math Helping AI See the World Better
• A Proof for Batchelor's Law
• How Fast Could China's Coronavirus Spread?
• Maths on the Back of an Envelope
• Estimating Economic Impacts from a Disaster Shock
• A Pie Chart for Pi Day
• Simulating an Epidemic
• What Is m?
• How Do False Test Outcomes Affect Estimates of the True Incidence of COVID-19
• Why the Media Can't Do Math
• Very Similar, But Different, Triangles
• Governor Cuomo and the Coronavirus Models
• John Conway's Game of Life
• The Hand That Draws The Circles That Draw The Hand
• Slicing Up Conway's Knot and Learning To Think Four Dimensionally
• Diffusion and Random Walks
• The Mathematical Packing Efficiency of Penguins
• 602,214,076,000,000,000,000,000
• Benford's Law and the Trustworthiness of COVID-19 Case Counts
• Of Storks, Statistics, and Cigarettes
• How Pooling COVID-19 Test Samples Is Super Beneficial
• The Limits of Microsoft Excel
• Three and a Half Thousand Year Old Fractions
• Four Minutes of Infinite Patterns
• When Benford's Law Breaks Down
• Replacing Zipf's Law in Modeling Growth of Cities

We've now officially reached the end of our year, since this is Political Calculations final post for 2020. Thank you for passing time with us this year and have a Merry Christmas and a wonderful holiday season. We'll see you again in the New Year, which we'll kick off with another annual tradition by presenting a tool to help you find out what your paycheck will look like in 2020 after the U.S. government takes its cut from it....

But before we go, we have one last video, from the invaluable Quanta Magazine, featuring their take on the biggest breakthroughs in math and computer science in 2020:

See you in the new year!