*Dr Imran Anwar, an ardent **mathematician**, is currently an Associate Professor at the Abdus Salam School of Mathematical Sciences, Lahore. He has research interests in Commutative Algebra, Algebraic Geometry, and Enumerative Combinatorics. Recently, Spectra had the opportunity to speak to Dr Imran about the transformation of his initial fascination with mathematics to a career choice, some of the greatest heros in mathematics, and the need to reform the scientific and mathematical education culture in Pakistan.*

*The interview has been condensed and edited for the purposes of clarity and brevity.*

**Spectra: Could you talk about your journey of mathematical education? What experiences and people were especially influential?**

**Dr Imran Anwar:** I have liked mathematics since my early school days, as it gives me the pleasure to solve difficult questions that others are afraid of handling. I had always wanted to choose a career where I could apply my strength in understanding and explain various mathematical concepts. As a student, I happened to be a keen listener; the thought of competing with my teacher was always in the back of my mind. Many times, I used to prepare topics on my own beforehand and calibrated my concepts with that of my teachers, while I sat in their lectures. This was a strange attitude that I carried throughout my life as a student. I am lucky that I got outstanding teachers who expected more from me than my capacity. Teachers’ expectations from a student are far more valuable than the encouragement. My attitude was severely tested when I applied for an admission in the School of Mathematical Sciences, Lahore (now known as Abdus Salam School of Mathematical Sciences, ASSMS). The admission was purely based on the admission test, followed by an intensive interview. Many teachers, who had taught me mathematics at a different stage of my academic career were also competing for the admission. It was a mixed feeling, but I succeeded in securing admission at the school, beating my respected teachers.

Initially, I went through a phase of unlearning and relearning of various mathematical concepts in an entirely new perspective. I got the opportunity to experience a deeper understanding of simple looking concepts. I started enjoying the abstract for more sophisticated theories and concepts. We learnt a great deal from seminars and class tutorials in addition to conventional lectures and it helped me grow more confidence. The journey comprised many ups and downs, however, I continued as a mathematical learner in my professional career. My PhD advisor was Prof. Dorin Popescu from Romania. As a mentor, I found him a challenging supervisor as he was very demanding who would get perturbed very quickly on any mistake, but simultaneously he was also very encouraging.

Consequently, for my PhD thesis research work, I got rewarded with a publication in the well-reputed* Journal of Algebra*. I am lucky that I got many great mentors, including Dr Amer Iqbal who is an exceptional individual. Fortunately, I also got the chance to work with him as a colleague at the Abdus Salam School of Mathematical Sciences. His career and his character as a mathematician and as a physicist has inspired me in many ways. Sometimes, you learn by viewing things from others’ perspective: I learnt many things through Dr Amer Iqbal’s narrative on various abstract mathematical concepts. It is very important to be surrounded by someone who inspires you to do big things in your academic career. I am still in the phase of building my academic career with many accomplishments in a very challenging environment of ASSMS.

**Who are some of the people in mathematics that you look towards for inspiration? **

I always regard David Hilbert as my inspiration in mathematics. His character is exceptional, and his contributions to mathematics and physics are commendable. As a draftsman of mathematics of the 20^{th} century, he was well-articulated in his address to the International Congress of Mathematicians in the year 1900, where he announced a list of 23 problems. These problems played an empirical role in the development of modern mathematics one way or another. Hilbert’s interaction with Albert Einstein and providing his expertise to give a mathematical proof of the General Theory of Relativity was extremely impressive. He is among the pioneers of quantum physics and his distinctive character as an axiomatic physicist is well portrayed in the article *Planting in his Neighbor’s Garden: David Hilbert and Early Göttingen Quantum Physics* by Arne Schirrmacher. The more you learn about him, the more fascinated you are.

“

David HilbertA mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street.”

**There is often a distinction made between ‘pure’ and ‘applied’ mathematics. Can you elucidate on this distinction? Do you think this distinction is a useful way of characterizing mathematics?**

It is deplorable that many people took this division as their sectarian divisions. I mainly agree with the narrative of German Mathematician Felix Klein.

*Mathematics has grown like a tree, which does not start at its tiniest rootlets and grow merely upward, but rather sends its roots deeper and deeper and at the same time and rate that its branches and leaves are spreading upward. *

In our country, applied mathematicians exist in larger numbers as compared to pure mathematicians. The approach of pure mathematics is altogether different from that of applied mathematics. Many people criticize pure mathematicians on the utility of their abstract theories. A famous English mathematician G. H. Hardy wrote a book “A mathematician’s apology” addressing the debate between pure and applied mathematics. His stance on the matter was mainly based on the following narrative:

*It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done. Exposition, criticism, appreciation, is work for second-rate minds.*

We need an intellectual culture in our institutions that cultivates brilliant ideas and research. We need to give space by accepting and valuing the quality. It is entirely unfair to compare one with the other without taking care of the nature of the carried-out work.

**Since you formally entered the field of mathematics, what changes have you seen in your field? What is the current scope of your research?**

The classical study of algebra is the abstract axiomatic framework. Due to its abstract nature, it requires more conceptual vitality to get an understanding about the structure. The inception of combinatorial commutative algebra made it fascinating and attractive as it includes a discrete geometrical object one can visualize. The crux of research mainly lies in translating the problems from geometry to algebra and solving it with the algebraic tools and vice versa. There is a long list of exciting problems with a bunch of renowned conjectures (means open problems) and it is a dream for any mathematician to answer one of them. I am lucky enough to give a partial proof of Stanley’s conjecture on Stanley’s decomposition in my PhD thesis research. Later on, I worked on various other problems but kept on changing my track within my domain of commutative algebra and discrete geometry. It is very challenging to build and maintain an excellent research profile, high standards of teaching, along with carrying out lots of academic administrative workload. I am not quite sure how I successfully navigated the stormy situations that defined my career in mathematics.

**In Pakistan, the odds are stacked against math and science education here due to various reasons. What could possibly be done to rectify the situation?**

A simple answer to stop jobbery or tyranny in the education sector is promoting quality teachers. The role of scientific societies has been entirely neglected in Pakistan. If one looks at the developed countries, there are powerful scientific societies like the American Mathematical Society and the European Mathematical Society. These societies are responsible for steering higher standards in making policies and providing a forum for talented young minds for recruitments. The Higher Education Commission (HEC) did not play any role in establishing scientific societies. All policies regarding hiring teaching and promotions are being addressed quite vaguely that are totally in contrast with the international practices. Due to these weak policies, politics and incompetency are stronger than competency and merit in the country. The promotion of each subject should be led by someone who knows the subject and its nature.

In contrast to international standards, in Pakistan due to HEC policies, pseudo-mathematicians have taken charge of policymaking, killing the essence of mathematics in the country. I strongly recommend HEC to play its role in formulating and empowering scientific societies in Pakistan, keeping AMS as a benchmark.

**Many people do not understand the role of mathematics in science and other disciplines. What do you think is the role of mathematicians in shaping public policy?**

One has to be blind to state any statement like that. Slowly, we have become accustomed to electronic transactions and electronic coding. Nowadays, there seems to be a number for everything. Even our physical attributes are being mathematized using the science of biometrics by such things as eye scans and the geometry of hands. Gradually, we are becoming conditioned to multi-faceted types of e-data and e-money. Banks already transfer huge sums of money electronically daily. Electronic cash has already begun to redefine the banking industries job force, and more changes are on the way once digital cash comes into widespread use. The understanding and utility of Artificial Intelligence and policies regarding its inception will heavily be based on mathematics. One thing is certain; mathematics will play an essential role in our pocketbooks. As a teacher to engineering students, I came across these questions many times. Over the years, I built a good collection of short video documentaries that I used to show in my classes to address these questions. It is somewhat important as a mathematics teacher to keep oneself updated to respond to such questions properly. In fact, this is the real charm and fun of teaching mathematics to engineering majors.

**What advice would you give to the students in high school who would like to pursue a career in mathematics? What is some good undergraduate program they can look to pursue?**

I believe that awareness about mathematics is strongly linked with knowing about mathematicians and exploring various mathematical theories through problem-solving. Problems and their solutions are at the crux of mathematics. Mathematical skills are based on conceptual understanding of subtle structures of logic. I strongly recommend young students to spend a fair amount of time reading books, blogs, AMS Feature columns, and mathematical sciences articles published in NATURE and Quanta Magazine. With the utility of social media, I feel it is more convenient to read and disseminate knowledge. A student should always stay curious. It is essential to spend time in critical reading and doing exercises. A right blend of standard video recorded lectures and documentaries is a great source of knowledge and motivation. Those who are interested in knowing about mathematics and career as a mathematician must visit AMS link.

It is harder to recommend an excellent undergraduate program of mathematics in the country; there are competent individuals in every department. Faculty members at large are somehow either de-motivated towards teaching due to extraneous research productivity load or are ineffective. All institutions are at par one way or the other in competence and capacity.

I would say a lot more depends on students now, how they put their efforts to get more out of it— it’s about finding a solution out of the box.

**How do you think we can popularize math education, specifically logic, in our schools?**

We don’t have any academic interactions between schools and universities. MATH CIRCLES is the best solution to motivate school kids. During my visit to Dalhousie University, Canada, I used to spot school kids after 4 pm in the mathematics departments, where MATH CIRCLE volunteers engaged these kids with intense mathematical theories in a fun way. Moreover, there is an outreach program as well, where MATH CIRCLE volunteers book their visits to some schools and spend time with the school kids (usually one period) in each class. I like this well-established model. Through these MATH CIRCLES, the volunteers can help students to nurture their logical thinking and motivate them toward mathematics. It involves serious skills ranging from logic, algebra, geometry, number theory, advanced combinatorics and topology.

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