#### Head of Mathematics & Computer Science Department

Professor Mohammad N.  Abdulrahim has a Bachelor degree in mathematics in 1990  and an MS degree in mathematics in 1992 from American University of Beirut. He then completed his Ph.D. in Mathematics (Algebra) from The Pennsylvania State University- USA  in 1995 where he had a position  at the Department of Mathematics at the Altoona campus as well as the  university park campus from 1995-1998. Beside teaching at Lebanese American University in Beirut, Professor Abdulrahim has been an assistant professor at the Department of Mathematics & Computer Science at Beirut Arab University since 1998. He got promoted to an associate professor in 2005 and a professor in 2011. During his position at Beirut Arab University,  he paid many visits to The Pennsylvania State University which was essential in  triggering his research in the field of braid group representations, which was  his main scope of  research. He is also a reviewer at the American Mathematical Society (AMS)  since 2015. Currently, he is the chairman of the Department of Mathematics & Computer Science at the Faculty of Science at Beirut Arab University since 2008. He has several publications in group representations and his main interest is to study irreducible representations of the braid group and its normal subgroup, pure braid group.  Open questions as of whether or not such representations are faithful are always of great significance to his research. Research. He  has more than 40 publications in this field.

## Teaching

Fall 2015/2016:

• Abstract Algebra
• Functional Analysis
• Research Topics
• Number Theory
• Galois Theory
• Universal Algebra

Spring 2015/2016:

• Topology Topics in Abstract Algebra
• Abstract Algebra I
• Modules and Rings
• Representation Theory

Fall 2016/2017:

• Topology
• Functional Analysis
• Number Theory
• Galois Theory
• Universal Algebra

Spring 2016/2017:

• Real Analysis I
• Abstract Algebra I
• Abstract Algebra II
• Topics in Abstract Algebra
• Senior Research in Mathematics
• Senior Project
• Modules & Rings
• Algebraic Topology

Fall 2017/2018:

• Topology
• Linear Algebra
• Abstract Algebra II
• Functional Analysis
• Galois Theory
• Universal Algebra

Spring 2017/2018:

• Real Analysis II
• Abstract Algebra I
• Multivariable Calculus
• Senior Project
• Modules & Rings
• Algebraic Topology

## Research

### Research Interests

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.

### Publications

1.    M.N. Abdulrahim, A  Faithfulness Criterion of the Gassner representation of the Pure Braid Group, Proceedings of the American Mathematical Society, V. 125, no. 5, May 1997, 1249-1257.

2.    M.N.Abdulrahim, Complex Specializations of the Reduced Gassner Representation of the Pure Braid Group, Proceedings of the American Mathematical Society, V. 125, no. 6, June 1997, 1617-1624.

3.    M.N.Abdulrahim, The reduced Gassner Representation restricted to a normal free subgroup of the Pure Braid Group. Archive Der Mathematik, V. 69, 1997, 8-12.

4.    M.N. Abdulrahim, On the Composition of the Burau Representation and the Natural Map B_n \to B_{nk}. Journal of Algebra and its Applications, V. 2, no. 2, June 2003, 169-175.

5.    M.N. Abdulrahim & Woo Lee, The Hecke algebra representations of the braid groups B_2 and B_3. Int. J. Appl. Math, V. 14, no. 4, 2003, 325-331.

6.    M.N. Abdulrahim, A braid element with eigenvalues as distinct powers of a complex number. Int. J. Appl. Math, V. 15, no. 1, 2004, 17-28.

7.    M.N. Abdulrahim, The  Number of Descending  paths from a given Young Diagram to a smaller one. Mathematical  Sciences Research  Journal, V.8, no. 10, 2004, 312-317

8.    M.N. Abdulrahim, On Embeddings of Pure Braid Groups P_n \to P_{n+k}, Int. J. Appl. Math, V. 16, no. 1, 2004, 1-12

9.    M.N. Abdulrahim, Pure Braids as Automorphisms of free Groups. Journal of Algebra and its Applications, V. 4, no. 4, 2005, 435-440.

10.    M.N. Abdulrahim & E. Formanek, Tensor Products of Specializations of the Burau Representation, Journal of Pure and applied Algebra, V. 203, no. 1-3, 2005, 104-112.

11.    M.N. Abdulrahim, Wada’s Representation and the natural map B_n \to B_{2n}, Int. J. Pure and Appl. Math, V.18, no.2, 2005, 137-142.

12.     M.N. Abdulrahim, Generalizations of  the  standard Artin representation are unitary, Int. J. Math. Math. Sci., no. 8, 2005, 1321-1326.

13.    M.N. Abdulrahim, Wada’s Representations of the Pure Braid Group, International Journal of Mathematics, V.16, no. 7, 2005, 757-763.

14.    M.N. Abdulrahim, Wada's representation is of Burau type. International  Journal of  Algebra &  Computation, V. 17 , no.3,  2007, 557-565.

15.    M.N. Abdulrahim & N. Kassem, The interplay between linear representations of the braid group. Int. J. Math. Math. Sci. 2007, Art. ID 16186, 9 pp.

16.    M.N. Abdulrahim, The Kernel of Wada's Representation restricted to a normal  free subgroup of the Pure braid group, International journal of Applied Mathematics, V. 21, no. 2, 2008, 239-245.

17.    M.N. Abdulrahim & D. Santina, The Hecke Algebra Representation of the braid group $\;B_3$ is conjugate to a unitary representation,  International journal of Applied Mathematics, V. 21, no. 3, 2008, 369-374

18.    M.N. Abdulrahim, On the Composition of Wada's representation and the natural map $\; P_n \to P_{n+k}$}, Advances in Algebra, V. 1, no. 1, 2008, 47-55.

19 . M.N. Abdulrahim & M.Al-Tahan, Complex Specializations of Krammer’s Representation  of the Braid Group, B_3,  J. Math. Stat. , V.4, no. 4, 2008, 213-216.

20 .M. N. Abdulrahim, Tensor Products of the Gassner representation of the pure braid group, Open Math J., V. 2, 2009, 12-15

21. M.N. Abdulrahim, Tensoring Three irreducible linear representations of the  Braid Group, Linear  and Multi Linear Algebra ,V.57, no. 8, 2009, 749-758.

22. M. N. Abdulrahim, Gassner Representation of the Pure Braid Group, P_4,Int.  J. Algebra, V. 3, no. 16, 2009, 793-798.

23.   M.N. Abdulrahim, On the irreducibility of linear representations of the pure braid group, Tamkang Journal of Mathematics, V. 41, no.3, 2010, 283-292.

24. M.N. Abdulrahim & M.Al-Tahan, On a class of irreducible representations of the braid group, B_n, International journal of Applied Mathematics, V. 23, no. 4, 2010, 681-691.

25. M.N. Abdulrahim & M.Al-Tahan, Krammer's Representation of the Pure Braid  Group, P_3, Int. J. Math. Math. Sci. 2010, Art. ID 806502, 10 pp.

26. M.N. Abdulrahim & G. H. Abboud, Pure Braid group representations of low degree,  Int.  J. Algebra, V. 5, no. 21-24, 2011, 1011-1020.

27. W.M. Zeid & M.N. Abdulrahim, Irreducibility of the Tensor product of specializations of the Burau representation of the braid groups, ISRN Algebra, V. 2011, Article ID 104263, 9 pages.

28. H.A.Tarraf & & M.N. Abdulrahim,    Higher Dimensional Irreducible Representations of the Pure Braid group, Journal of Mathematics Research, V.3, no.4, 2011, 141-151

29.  M.N. Abdulrahim & Hariri, Mariam, Another proof of the faithfulness of the  Lawrence-Krammer representation of the braid group $B\sb 3$. ISRN Algebra  2012, Art. ID 858959, 13 pp .

30. M.Chreif & M.N. Abdulrahim, Irreducible representations of the pure braid group P_n, International journal of Applied Mathematics, V. 25, no. 5, 2012, 723-728

31 G. H. Abboud & M. N.Abdulrahim, On the irreducibility of Wada’s representation of the pure braid group P_4, , Open Math J., V. 5, 2012, 15-18

32. W.M. Zeid & M.N. Abdulrahim, Tensor product of Wada’s representations of the braid group, B_3, Int. Math. Forum, V. 7, no. 29-32, 2012, 1413-1418

33. H.A.Tarraf  & & M.N. Abdulrahim, Tensor Product of Krammer’s representations of the Pure braid group, P_3, Advances in Pure Math, 2012,2, 190- 194

34. M.N. Abdulrahim & I.Daakur,  Wada’s representations of the pure braid group of high degree, Theoretical Mathematics & Applications, V.2, no. 1, 2012, 117-125.

35.  M.Al-Tahan, M.N.Abdulrahim & S.S. Habre, Multi-parameter Burau representations, Tamkang Journal of Mathematics, V. 44, no.1, 2013, 91-98.

36. M.Al-Tahan, M.N. Abdulrahim, A new six Dimensional representation of the braid group on three strands and its irreducibility and unitarizability, British Journal of Mathematics & Computer Science, V.3, no.3, 2013, 275-280

37. M.Al-Tahan, M.N. Abdulrahim, An Overview of Wada’s representations, Journal of Mathematical & Computational Science, V.3,no.6, 2013, 1430-1443

38. M.Al-Tahan, M.N. Abdulrahim & S.S. Habre, Spin Representations with negative indices, Tamkang Journal of Mathematics, V.45, no. 4, 2014, 367-374.

39. N.Maanna, M.N. Abdulrahim, Tuba’s Representation of the pure braid group on three strands, British Journal of Mathematics & Computer Science V.4, no. 16, 2014,2381-2402.

40.M.Dally, M.N. Abdulrahim, On the irreducibility of Artin’s group of graphs, Journal of Mathematics Research, V.7,  no.2, 2015, 117-124.

41 M.Chreif & M.N. Abdulrahim, Attacks on the Faithfulness of the Burau Representation of the Braid Group $B_4$ Journal of Mathematics Research, V.8,  no.1, 2016, 5-9.

42    M. . Chreif & M. N. Abdulrahim,  On the Irreducibility of the complex specialization of the representation of the Hecke algebra of the complex reflection group G , Journal of Linear & Topological Algebra, V.5, No.4, 2016, 263-270

43    M. Chreif & M. N. Abdulrahim , The Hecke Algebra  representation of  the complex reflection group G is unitary, JP   Journal of Algebra, Number Theory and Applications, V.39, No.2, 2017, 151-164

## Activities

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.