The braid group, Bv, has a well- known representation due to Artin in the group of automorphisms of the free group with n generators. Applying the Fox derivatives and Magnus representation, we get the classical Burau representation which is of degree n. D.D. Long gave a new derivative of the Burau representation by using an action of the automorphism group of the group of a free group with n generators on the representation variety of the free group and the group G, where G is a compact semi simple Lie group . We say that a group is linear if it admits a faithful representation into the general linear group for some natural number n and some field K. The question of faithfulness of the braid group has been the subject of research for a long time. It has been shown that the Burau representation is faithful for values of n smaller than 4 and non faithful for values on larger than 4. However, the faithfulness of the Burau representation in the case n=4 is still unknown. Our target then is to construct possible linear irreducible representations of the braid group and hope we could find one that is proved to be faithful. We need also to mention that the concept of faithfulness plays an important role in the field of cryptography, where there is an undergoing significant research in that field.
1. Colloquium Lecture at The American University of Beirut, Lebanon, 2009 . Title of the talk: Tensor Product of Specializations of the Burau Representation.
2. Colloquium Lecture at The American University of Beirut at the conference of Lie Groups and Related Topics, April 23rd, 2014.
3. Organized a workshop in mathematics in March 2009 (Workshop on Representation Theory and Number Theory ) as a partner with American University of Beirut in the LSMS conference in April 26th, 2012.
4. Organized a workshop on Partial Differential Equations in April 22nd, 2015
5. Organized A workshop on Mathematical Analysis on March 15th, 2016.
- A reviewer of American Mathematical Society (Member since 2015)
- A reviewer of Journal of Mathematics & Statistics (Since 23/12/2021)|