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# Category Archives: Algebra and Number Theory

## The Simplex, the Cyclic polytope, the Positroidron, the Amplituhedron, and Beyond

A quick schematic road-map to these new geometric objects. The positroidron can be seen as a cellular structure on the nonnegative Grassmanian – the part of the real Grassmanian G(m,n) which corresponds to m by n matrices with all m by … Continue reading

## TYI 25: The Automorphism Group of the Symmetric Group

True or False: The group of automorphisms of the symmetric group , n ≥ 3 is itself.

Posted in Algebra and Number Theory, Test your intuition
Tagged Test your intuition, the symmetric group
7 Comments

## Test your intuition 24: Which of the following three groups is trivial

Martin Bridson We have three finitely presented groups A is generated by two generators a and b and one relation B is generated by three generators a, b, c and three relations , . C is generated by four generators a, b, c, d … Continue reading

Posted in Algebra and Number Theory, Test your intuition
Tagged Martin Bridson, Test your intuition
10 Comments

## 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 Ramanujan graphs
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

## 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

## 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 Euclid, Greek mathematics
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

## 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

## 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 Alex Levitzki. Yaacov Levitzki, Shimshon Amitsur
7 Comments