He has parametrized actions of the modular group on projective lines over Galois fields. This method has proven to be so effective and rewarding, that its wide uses can be seen in Combinatorial Group Theory, Algebraic Number Theory, and Theory of Group Graphs. His graphical technique helped to solve George Abram Miller’s problem (1901) on alternating groups as homomorphic images of the modular group^{[2]}

He has also invented a new algebraic structure known as Locally Associative LA-semigroup^{[3]}, and has done some fundamental research on LA-semigroups producing some significant results in this theory. Consequently, a number of useful mathematical results have emerged which otherwise were applicable under restricted conditions only.

Qaiser Mushtaq is a Professor at Quaid-i-Azam University^{[4]}, and a former Syndicate Member, Quaid-i-Azam University, Islamabad. He is an Honorary Full Professor at the Mathematics Division, Institute for Basic Research, Florida, USA. He did his B.Sc. from University of the Punjab; M.Sc. and M.Phil. from Quaid-i-Azam University; and D.Phil. from the University of Oxford.

He was a doctoral student of Professor Graham Higman at Oxford. He has collaborated in research with legends like the late Professor Graham Higman (Oxford) and the late Professor Gian-Carlo Rota (MIT). He has been an invited speaker at Oxford University, MSRI Berkeley, Harvard University, Massachusetts Institute of Technology and Southampton University; and an invited speaker at several international conferences.

He has supervised, as a sole supervisor, the highest number of M.Phil. and Ph.D. students in Pure Mathematics in Pakistan (see the Mathematics Genealogy Project^{[5]}). As a result, he has established a research group in Pakistan, the largest of its kind, which is producing high level original research in mathematics.

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He is an editor of the Asian-European Journal of Mathematics^{[6]} (World Scientific). He is an associate editor of the Bulletin of the Southeast Asian Mathematical Society^{[7]} (Springer-Verlag). Additionally, he is an editor of the Quasigroups and Related Systems^{[8]} (Maldova Academy of Sciences) and the Bulletin of the Malaysian Mathematical Society^{[9]}, an advisory editor of the Journal of Interdisciplinary Mathematics^{[10]}, an associate editor of the journal Advances in Algebra and Analysis, a reviewer for the Mathematical Reviews of the American Mathematical Society (USA) and the Zentrablatt fur Mathematik (Springer-Verlag, Germany).

Professor Mushtaq has over a hundred research papers to his credit. He has written and edited several books, some of which are, Mathematics: The Islamic Legacy^{[11]} (which received a prize from the National Book Council of Pakistan), published by UNESCO and other international publishers; A Course in Group Theory, and Discrete Lectures in Mathematics. He has also written books on topics other than mathematics. They are, Focus on Pakistan, and Pakistan: An Introduction. He was also an invited writer for the monumental book, comprising six volumes, entitled the History of Civilizations of Central Asia^{[12]}, published by UNESCO (translated into several foreign languages).

He is also famous for his analytical writings and articles on history, mathematics, science, education, and philosophy. He has been an active opposer of the use of impact factors^{[13]} and citation counts of the Higher Education Commission. He led the movement against its use which he believed has damaged the growth of mathematics in Pakistan. One of his essays has been published by the American Mathematical Society which won appreciation and support by many mathematicians from all over the world. The International Mathematical Union has included it in its report^{[14]} on the use of impact factors. Professor Mushtaq also founded the well known mathematical quarterly, PakMS Newsletter^{[15]}, in Pakistan.

Professor Mushtaq has founded the internationally acclaimed mathematical society, the Pakistan Mathematical Society^{[16]}. He has also founded the series of international conferences, namely IPMC^{[17]}, in Pakistan, which now has become an icon of mathematics in Pakistan and is a well known international mathematical activity. The conference takes place every August in Islamabad. He is a member of the American Mathematical Society, the Oxford Society, the London Mathematical Society, and the Punjab Mathematical Society. He has been the President of the Brunei Darussalam Mathematics Society, and he is the current President of the Pakistan Mathematical Society.

Professor Mushtaq was one of the pioneers who started the ‘Islamic Society’ at Oxford. He was its vice president and secretary in 1980 to 1983. He was one of the Pakistanis at Oxford University who rejuvenated the ‘Oxford Union Pakistan Society’ of which he was the Vice President from 1981 to 1982.

In Pakistan, he started the ‘Mathematical Seminar Series’ at Quaid-i-Azam University in 1983 and developed it into an institution recognized nationally. At Quaid-i-Azam University he founded the ‘Algebra Forum’ which has held advanced level seminars on algebra in particular and on various academic topics of general interest.

He reformed and restructured the Mathematical Society of Brunei Darussalam as its President.

He was an overseas scholar of the Royal Commission for the Exhibition of 1851^{[18][19]} in 1980 and a senior Fulbright Scholar in 1990. Professor Mushtaq was elected an Associate Member of the International Centre for Theoretical Physics, Trieste, Italy in 1991.

Professor Mushtaq’s international recognition is evident by the numerous citations of his research work and several international and national awards for his contribution in mathematics. He has been awarded the Chowla Medal (1977); Salam Prize (1987); Mathematician of the Year Award (1987) by the National Book Council of Pakistan; Gold Medal of Honour (1987) from USA; Mathematician of the Year Award (1990) by the National Book Council of Pakistan; M. Raziuddin Siddiqi Gold Medal (1991) from the Pakistan Academy of Sciences; 5th Khwarizmi Award (1992) from the President of Iran; Young Scientist of the South Award (1993) from Third World Academy of Sciences, Italy; 5th National Education Award (1999) by the National Education Forum, and the Gold Medal in Mathematics (2000) from the Pakistan Academy of Sciences.

“We do fundamental research, not only to acquire results solely, but because the process is an ennobling one,” Graham Higman told Qaiser Mushtaq in an interview on Pakistan Television in 1987. “It is one that makes you more worthwhile than before; it is something that if you cut yourself off from, you are making yourself less human than you ought to be.” This has been quoted in the obituary of Graham Higman in the Telegraph on 26th May 2008^{[20]}. Like Graham Higman, Professor Mushtaq is one of those mathematicians who care less for what mathematics can do for scientific advance than for its life affirming properties.

**^**http://designtheory.org/library/encyc/topics/cayley.pdf**^**http://www.patrick-reichert.de/publikationen/BelyiVortrag/Vortrag.pdf**^**http://www.springerlink.com/content/t845k34483852383/fulltext.pdf?page=1**^**http://math.qau.edu.pk/profile.php?id=bL8q6B6**^**http://genealogy.math.ndsu.nodak.edu/id.php?id=56529**^**http://ejournals.wspc.com.sg/aejm/mkt/editorial.shtml**^**http://seams-bull-math.scnu.edu.cn/en/bwh.asp**^**http://www.quasigroups.eu/**^**http://math.usm.my/bulletin/html/editorial.htm**^**http://www.iospress.nl/loadtop/load.php?isbn=09720502**^**http://www.dkagencies.com/doc/from/1065/to/1123/bkId/DK6465233211690941927930151371/details.html**^**http://portal.unesco.org/en/ev.php-URL_ID=18313&URL_DO=DO_TOPIC&URL_SECTION=201.html**^**http://www.ams.org/notices/200707/index.html**^**http://www.mathunion.org/Publications/Report/CitationStatistics**^**http://www.pakms.org.pk/newsletter.htm**^**http://www.pakms.org.pk/**^**http://www.pmc.org.pk/**^**http://www.royalcommission1851.org.uk**^**http://www.1851alumni.org.uk/search**^**http://www.telegraph.co.uk/news/obituaries/2032991/Professor-Graham-Higman.html

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