Do our computers live in a simulation?

René Descartes is famous for countless things in mathematics—Cartesian products, Cartesian coordinates, Descartes’ rule of signs, the folium of Descartes. He is also famous for his work in philosophy and the notion of an evil genius. The evil genius presents a full illusion of a reality, and “fools” Descartes into believing there is a reality, while actually there is none.

Today Ken and I want to talk about the evil genius, and its relationship to the simulation hypothesis.

Comments on three papers from the Conference on Computational Complexity

Michael Saks is Chair of the Program Committee of this year’s Conference on Computational Complexity (CCC). He was helped by Paul Beame, Lance Fortnow, Elena Grigorescu, Yuval Ishai, Shachar Lovett, Alexander Sherstov, Srikanth Srinivasan, Madhur Tulsiani, Ryan Williams, and Ronald de Wolf. I have no doubt that they were faced with many difficult decisions—no doubt some worthy papers could not be included. The program committee’s work does not completely end after the list of accepted papers is posted, but it is not too early to thank them all for their hard work in putting together a terrific program.

Today I wish to highlight three papers from the list of accepted ones. Read more…

How can we possibly see atoms?

John Sidles is a medical researcher and a quantum systems engineer. His major focus is on quantum spin microscopy for regenerative medicine. He is both Professor of Orthopedics and Sports Medicine in the University of Washington School of Medicine, and co-director of the UW Quantum Systems Engineering Lab. Watching various injury troubles at the Sochi Winter Olympics makes us wonder whether quantum sports medicine is an idea whose time has come. Well beyond some media’s overheated references to our athletes as “warriors” is a nice reality: John’s main project is for healing those injured in the armed services.

Today Ken and I wish to talk about John and his work in general. We especially like his title of quantum systems engineer. Read more…

What is better than how (?)

 History of filters source.

Wilhelm Cauer was a German mathematician and engineer who worked in Göttingen and the US between the two world wars. He is associated with the term “black box,” although he apparently did not use it in his published papers, and others are said to have used it before. What Cauer did do was conceive a computing device based on electrical principles. According to this essay by Hartmut Petzold, Cauer’s device was markedly more advanced and mathematically general than other ‘analog devices’ of the same decades. He returned to Germany in the early 1930′s, stayed despite attention being drawn to some Jewish ancestry, and was killed in the last days of Berlin despite being on the Red Army’s list of scientists whose safety they’d wished to assure.

Today Ken and I wish to talk about black boxes and white boxes, no matter who invented them, and their relation to computing.

How far can trivial ideas go?

Klaus Roth is famous for many results, but two stand out above all others.
One sets limits on Diophantine approximations to algebraic numbers, and the other sets limits on how dense a set can be and have no length-three arithmetic progression. He has won many awards for his work, including a Fields Medal and the Sylvester Medal.

Today I want to try and amuse you with a simple proof that is related to his work on progressions.

It snowed in Atlanta, and we are closed, for the foreseeable future—where is global warming?

Atlanta is frozen. Here is what we looked like the other night.

Today I thought I might talk about the weather in Atlanta, and its connection to mathematical paradoxes. Read more…

More on the crypto approach to the Jacobian Conjecture

Arno van Essen is one of the world experts on the Jacobian conjecture (JC)—we have discussed his work before here. He has made many contributions to it, with my favorite being: To Believe Or Not To Believe: The Jacobian Conjecture. I like his attitude about conjectures: I think we should be more skeptical about our own. Oh well, few of my colleagues feel this way about ${\mathsf{P \neq NP}}$, for example.

Today I want to update a previous discussion on the JC, and prove a new theorem. Read more…