Category Archives: Algebra and Number Theory

New Ramanujan Graphs!

Margulis’ paper Ramanujan graphs were constructed independently by Margulis and by Lubotzky, Philips and Sarnak (who also coined the name). The picture above shows Margulis’ paper where the graphs are defined and their girth is studied. (I will come back to the question … Continue reading

Posted in Algebra and Number Theory, Combinatorics, Open problems | Tagged | 10 Comments

Andrei

Andrei Zelevinsky passed away a week ago on April 10, 2013, shortly after turning sixty. Andrei was a great mathematician and a great person. I first met him in a combinatorics conference in Stockholm 1989. This was the first major … Continue reading

Posted in Algebra and Number Theory, Combinatorics, Obituary | Tagged | 5 Comments

Primality and Factoring in Number Fields

Both PRIMALITY – deciding if an integer n is a prime and FACTORING – representing an integer as a product of primes, are algorithmic questions of great interest. I am curious to know what is known about these questions over … Continue reading

Posted in Algebra and Number Theory | Tagged , | 9 Comments

Test Your Intuition (16): Euclid’s Number Theory Theorems

Euclid’s Euclid’s book IX on number theory contains 36 propositions. The 36th proposition is: Proposition 36.If as many numbers as we please beginning from a unit are set out continuously in double proportion until the sum of all becomes prime, … Continue reading

Posted in Algebra and Number Theory, Test your intuition | Tagged , | 16 Comments

The AC0 Prime Number Conjecture

Möbius randomness and computational complexity Last spring Peter Sarnak gave a thought-provoking lecture in Jerusalem. (Here are the very interesting slides of a similar lecture at I.A.S.) Here is a variation of the type of questions Peter has raised. The Prime … Continue reading

Posted in Algebra and Number Theory, Computer Science and Optimization | Tagged , , , , | 13 Comments

Octonions to the Rescue

Xavier Dahan and Jean-Pierre Tillich’s Octonion-based Ramanujan Graphs with High Girth. Update (February 2012): Non associative computations can be trickier than we expect. Unfortunately, the paper by Dahan and Tillich turned out to be incorrect. Update: There is more to … Continue reading

Posted in Algebra and Number Theory, Combinatorics, Computer Science and Optimization, Open problems, Physics | Tagged , , , | 11 Comments

The Amitsur-Levitzki Theorem for a Non Mathematician.

Yaacov Levitzki The purpose of this post is to describe the Amitsur-Levitzki theorem: It is meant for people who are not necessarily mathematicians. Yet they need to know two things. The first is what matrices are. Very briefly, matrices are rectangular arrays … Continue reading

Posted in Algebra and Number Theory | Tagged , | 7 Comments

The Thompson Group

Update (july 2009): A detailed posting on the Thompson group appeared on “Geometry and the Imagination,” Danny Calegary’s blog. In spite of two recent preprints one claiming that the Thompson group is amenable and the other claiming the opposite, the problem appears … Continue reading

Posted in Algebra and Number Theory, Conferences | 3 Comments