A simple but interesting issue with analyzing high-dimensional data

Peter Landweber, Emanuel Lazar, and Neel Patel are mathematicians. I have never worked with Peter Landweber, but have written papers with Larry and Laura Landweber. Perhaps I can add Peter one day.

Today I want to report on a recent result on the fiber structure of continuous maps.

Teaching automata theory

Noam Chomsky is famous for many many things. He has had a lot to say over his long career, and he wrote over 100 books on topics from linguistics to war and politics.

Today I focus on work that he pioneered sixty years ago.

Yes sixty years ago. The work is usually called the Chomsky hierarchy(CH) and is a hierarchy of classes of formal grammars. It was described by Noam Chomsky in 1956 driven by his interest in linguistics, not war and politics. Some add Marcel-Paul Schützenberger’s name to the hierachry. He played a crucial role in the early development of the theory of formal languages—see his joint
paper with Chomsky from 1962. Read more…

A possible error with mathematical ramifications

 Non-technical fact-check source

Dan Brown is the bestselling author of the novel The Da Vinci Code. His most recent bestseller, published in 2013, is Inferno. Like two of his earlier blockbusters it has been made into a movie. It stars Tom Hanks and Felicity Jones and is slated for release on October 28.

Today I want to talk about a curious aspect of the book Inferno, since it raises an interesting mathematical question. Read more…

Some CS reflections for our 700th post

 MacArthur Fellowship source

Lin-Manuel Miranda is both the composer and lyricist of the phenomenal Broadway musical Hamilton. A segment of Act I covers the friendship between Alexander Hamilton and Gilbert du Motier, the Marquis de Lafayette. This presages the French co-operation in the 1781 Battle of Yorktown, after which the British forces played the ballad “The World Turned Upside Down” as they surrendered. The musical’s track by the same name has different words and melodies.

Today we discuss some aspects of computing that seem turned upside down from when we first learned and taught them.

We revisit a paper from 1994

Richard Lipton is, among so many other things, a newlywed. He and Kathryn Farley were married on June 4th in Atlanta. The wedding was attended by family and friends including many faculty from Georgia Tech, some from around the country, and even one of Dick’s former students coming from Greece. Their engagement was noted here last St. Patrick’s Day, and Kathryn was previously mentioned in a relevantly-titled post on cryptography.

Today we congratulate him and Kathryn, and as part of our tribute, revisit a paper of his on factoring from 1994.

What is the role of theory today?

Anna Gilbert and Atri Rudra are top theorists who are well known for their work in unraveling secrets of computation. They are experts on anything to do with coding theory—see this for a book draft by Atri with Venkatesan Guruswami and Madhu Sudan called Essential Coding Theory. They also do great theory research involving not only linear algebra but also much non-linear algebra of continuous functions and approximative numerical methods.

Today we want to focus on a recent piece of research they have done that is different from their usual work: It contains no proofs, no conjectures, nor even any mathematical symbols.
Ernie Croot, Vsevolod Lev, and Péter Pach (CLP) found a new application of polynomials last month. They proved that every set ${A \subseteq U = \mathbb{Z}_4^n}$ of size at least ${|U|^{1-\epsilon}}$ has three distinct elements ${a,b,c}$ such that ${2b = a + c}$. Jordan Ellenberg and Dion Gijswijt extended this to ${\mathbb{F}_q^n}$ for prime powers ${q}$. Previous bounds had the form ${|U| n^{-c}}$ at best. Our friend Gil Kalai and others observed impacts on other mathematical problems including conjectures about sizes of sunflowers.