## The 3SUM Assumption Is Wrong?

* A new result on our three body problem *

Allan Grønlund and Seth Pettie are leaders in algorithm design and related problems.

Today I want to give a quick follow up on our discussion of 3SUM based on a recent paper of theirs. Read more…

## Our Three Body Problem

* The three body problem, computer theory style *

Ellis Horowitz is one of the founders of the theory of algorithms. His thesis with George Collins in 1969 had the word “algorithm” in the title: “Algorithms for Symbolic Integration of Rational Functions.” He is known for many things, including an algorithm that after forty years is still the best known.

Today I want to talk about this algorithm, and one of the most annoying open problems in complexity theory. Read more…

## Laplace’s Demon

* Demons and other curiosities *

Pierre-Simon Laplace was a French scientist, perhaps one of the greatest ever, French or otherwise. His work affected the way we look at both mathematics and physics, among other areas of science. He may be least known for his discussion of what we now call Laplace’s demon.

Today I want to talk about his demon, and whether predicting the future is possible.

Can we predict the past? Can we predict the present? Can we predict the future? Predicting the past and predicting the present sound a bit silly. The usual question is: Can we predict the future? Although I think predicting the past—if taken to mean “what happened in the past?”—is not so easy.

## Diagonalization Without Sets

* Avoiding actual infinities *

Carl Gauss is of course beyond famous, but he had a view of infinity that was based on old ideas. He once wrote in a letter to Heinrich Schumacher in 1831:

so protestiere ich zuvörderst gegen den Gebrauch einer unendlichen Größe als einer vollendeten, welcher in der Mathematik niemals erlaubt ist. Das Unendliche ist nur eine

façon de parler, indem man eigentlich von Grenzen spricht, denen gewisse Verhältnisse so nahe kommen als man will, während anderen ohne Einschränkung zu wachsen gestattet ist.

Today we want to show that the famous diagonal argument can be used without using infinite sets. Read more…

## The Cantor-Bernstein-Schröder Theorem

*And whose theorem is it anyway? *

Georg Cantor, Felix Bernstein, and Ernst Schröder are each famous for many things. But together they are famous for stating, trying to prove, and proving, a basic theorem about the cardinality of sets. Actually the first person to prove it was none of them. Cantor had stated it in 1887 in a sentence beginning, “One has the theorem that…,” without fanfare or proof. Richard Dedekind later that year wrote a proof—importantly one avoiding appeal to the axiom of choice (AC)—but neither published it nor told Cantor, and it wasn’t revealed until 1908. Then in 1895 Cantor deduced it from a statement he couldn’t prove that turned out to be equivalent to AC. The next year Schröder wrote a proof but it was wrong. Schröder found a correct proof in 1897, but Bernstein, then a 19 year old student in Cantor’s seminar, independently found a proof, and perhaps most important, Cantor himself communicated it to the International Congress of Mathematicians that year.

Today I want to go over proofs of this theorem that were written in the 1990’s not the 1890’s. Read more…

## An Old Galactic Result

* A cautionary tale *

Karl Sundman was a Finnish mathematician who solved a major open problem in 1906. His solution would have been regarded as paradigm-“shifty” had it been a century or so earlier. It employed power series to represent the solutions of equations, namely the equations of the 3-body problem famously left unsolved by Isaac Newton. The tools of analysis needed to regard a convergent power series as defining a valid real number had been well accepted for a century, and the explicit giving of series and convergence proof even met the demands of mathematical constructivism of his day.

Today Ken and I want to explain why that problem is nevertheless still considered open, even though Sundman solved it over a hundred years ago.

This is a cautionary tale: For some problems there are solutions and there are solutions. There are solutions that make you famous and there are solutions that no one cares about. Unfortunately for Sundman his solution, which is correct mathematically, is the latter type; hence, his lack of fame—did you know about him? He did win honors from the top French and Swedish academies of science for this and other work in mechanics, and he has a Moon crater and an asteroid named for him. Read more…

## Shifts In Algorithm Design

* How to find approximate page rank fast, among other things *

Jennifer Chayes is the current director of a research lab in Cambridge—that is Cambridge Massachusetts—for a company called Microsoft. She is famous for her own work in many areas of theory, including phase transitions of complex systems. She is also famous for her ability to create and manage research groups, which is a rare and wonderful skill.

Today Ken and I wish to talk about how to be “shifty” in algorithm design. There is nothing underhanded, but it’s a different playing field from what we grew up with. Read more…