Jon was born and raised in Minnesota, learned mathematics at Gustavus Adolphus College, in Budapest, and finally earned his PhD at the University of Oregon. After positions at the Universities of Toronto, Georgia, and Oklahoma, he's now on the faculty at Oregon State University. He thinks it a shame how badly math fares in the modern educational-industrial complex and occasionally tries to do something about it when not indulging his other interests. Email: [email protected]
by Jonathan Kujawa So, some stuff has happened in the world of math since I wrote my last 3QD essay at the beginning of September. In May, there was a breakthrough result for the Erdos Unit Distance Conjecture. In July, an explicit counterexample to the Jacobian Conjecture was announced. In September, one version of the…
by Jonathan Kujawa Lots of people enjoy a good puzzle. Take Sudoku puzzles. Like many good puzzles, the rules of Sudoku are simple. The challenge is to figure out the implications of each possible move and to use those implications to reason our way to the solution. You probably know how this goes: you try…
by Jonathan Kujawa John Conway invented the Game of Life in 1970, and it remains an active area of research and play fifty-six years later. John Conway was a deeply original mathematician who made important contributions to multiple areas of mathematics. Indeed, when Dr. Conway passed away in 2020, I wrote an entire 3QD essay…
by Jonathan Kujawa Human intuition is a marvelous thing. With scant evidence, we can make assessments, judgments, and predictions that are often surprisingly close to correct. But our intuition can also lead us astray. Worse, an intuitive idea can be virtually impossible to give up, even when we know it is wrong. It is a…
by Jonathan Kujawa In the past few months, there has been a flurry of activity at the intersection of AI and mathematics. There is a keen interest in applying AI to mathematics. One reason is that mathematics provides a useful benchmark on the progress of AI’s ability to reason and solve problems. After all, to…
by Jonathan Kujawa One common theory of intelligence is that it is an emergent phenomenon. One-on-one, termites aren’t exactly geniuses. But get enough of them together, and they build incredibly elaborate communities that can be 30 meters across with sophisticated ventilation systems to control heating and cooling. And these communities can last for thousands of…
by Jonathan Kujawa A few months ago, here at 3QD, we learned that even the humble number two has the ability to lead us to mathematical riches. It is the base of the world’s largest known prime number and the root of irrationality. And I, for one, was surprised to learn that every counting number…
by Jonathan Kujawa Mathematics is an inexhaustible subject. Every time you think you’ve mined out a vein, you hit a new gem. There is no clear reason why this should be true, but all evidence shows that it is mathematics all the way down [1]. I once saw a talk by a mathematician near retirement…
by Jonathan Kujawa A few hours’ drive north of my home is the Laser Interferometer Gravitational-Wave Observatory (LIGO) in Hanford, Washington. LIGO was designed and built to detect gravitational waves. When the LIGO project was started in the 1980s, gravitational waves were a purely theoretical phenomenon predicted by Einstein’s theory of general relativity. General relativity…
by Jonathan Kujawa Recently, Chris Drupieski and I released a new research paper. If we were a tech company, we would announce the paper at a lavish event. There, every lemma would be amazing, every proposition would be magic, and every theorem would be world-changing. Instead, we put it on the arXiv. The arXiv (pronounced…
by Jonathan Kujawa The humble 2. It’s not big, like the Brobdingnagian numbers. It’s not nothing, like zero. It’s not the first something, like one. It’s hard to imagine much can be said about the unremarkable two. Of course, Covid gave us a newfound appreciation for the power of exponential doubling. If you know of…
by Jonathan Kujawa Among our many flaws, humanity prefers to be shortsighted. We tend to set our priorities by what we see right in front of us. This makes some sense. After all, there is no point in worrying about storing food for the winter when facing down a saber toothed tiger. But eventually winter…
by Jonathan Kujawa Four years ago was peak COVID-19. I was thinking about moving furniture, which led to this essay here at 3QD. The topic was the problem of moving a sofa down a hallway with a 90° corner: The problem is to find the “couch” of largest possible area that could be smoothly pushed…
by Jonathan Kujawa Prime numbers are the atoms of arithmetic. Just as a water molecule can be broken into two hydrogen and one oxygen atoms, 12 can be broken into two 2s and a 3. Indeed, the defining feature of a prime number is that it cannot be factored into a nontrivial product of two…
by Jonathan Kujawa Nearly two years ago we talked at 3QD about the Busy Beaver problem [1]. Since then, the beavers have been busy. As discussed in that essay, the Busy Beaver problem measures how complicated a computation might be. It does so by measuring how long a Turing might run before stopping. A Turing machine…
by Jonathan Kujawa One of the great pleasures in life is learning about something today that you couldn’t have imagined yesterday. The infinite richness of mathematics means I get to have this experience regularly. However much I think I know, it is a drop in the ocean of things yet to be learned. And even…
by Jonathan Kujawa My nieces, Hannah and Sydney, came to visit for the weekend. Since they’re 8 and 9, the delightful Guardian Games in downtown Corvallis was a must-stop. Along with games galore, they have an amazing assortment of puzzles. They have puzzles with micro-sized pieces, puzzles with jumbo-sized pieces, puzzles with only a few…
by Jonathan Kujawa The State of Georgia has a lottery guaranteed to turn an enterprising 3QD reader into a millionaire [1]. Playing the lottery can be fun. It is a tradition for Anne and me to buy lottery tickets when stopping for gas on long road trips. Imagining what you’d do with our new wealth…
by Jonathan Kujawa In 2016, here and here at 3QD, we talked about some of the inherent paradoxes in democratic voting [1]. We discussed Arrow’s Impossibility Theorem, along with related results like the Gibbard-Satterthwaite Theorem. They tell us that there is no way to convert the individual preferences of the voters into a single group…
by Jonathan Kujawa Functions are machines that take in a number as an input, apply some rule to that input, and generate an output. The input might be cost, temperature, wind speed, a politician’s favorability rating, or whatever you like. The output could tell you the resulting profit, windchill, chance of a tornado forming, the…