Introduction

K. T. Arasu is a world class scholar, and excellent scholarship is the common thread through his research, teaching, and professional service. While Arasu’s training and research is essentially in combinatorics and discrete mathematics, the scope of his research has expanded over the years. Arasu now works with communication engineers all over the world and publishes in excellent engineering journals, including IEEE which is top class, and attracts funds from the signal processing community and communication engineering. His total research funding is about 1.5 million dollars, a phenomenal amount for a mathematician doing basic as well as applied research. Arasu has engaged dozens of students in his research over the years, including many from the WSU College of Engineering and Computer Science in addition to students in the College of Science and Mathematics. Also, having garnered supplemental funding for an NSF grant, Arasu supported and provided training to 12 high school teachers, spending nearly 100 hours per summer over four summers with these teachers, to give them some exposure to recent activities in discrete mathematics, cryptography and applicable algebra, to facilitate the development of discrete mathematics into the high school curriculum in this modern era of computing and communication.

Here is a more detailed perspective on Arasu’s career accomplishments. In the fall of 1983, after completing his Ph.D. at Ohio State University, he joined Wright State University as Assistant Professor. In 1988 Arasu was recognized in the Dayton Daily News for being awarded a prestigious Alexander von Humboldt Fellowship supporting a year’s research at the University of Giessen, Germany during 1988-89 with Professor Jungnickel, an international figure in design theory. Arasu gave an invited address at the Institute of Mathematics at the University of Minnesota in June 1988 at the special year in coding theory and design theory, and he presented a keynote address at the International Conference on Information and System Sciences in Baden-Baden, Germany in August 1988. With his research program progressing extremely well, Arasu rose rapidly through the ranks, earning promotion to Professor in 1994. Arasu’s early research success was merely a launching pad to greater heights. Over the years Arasu has established an international reputation based on a spectacular record of scholarship. His research activities have always revolved around problems in discrete mathematics and combinatorics that require applying algebraic and number theoretic techniques and, because the combinatorial objects he studies have immediate applications in a variety of disciplines, including cryptography, communication engineering, and statistics, his research area is known as “applicable algebra; and a major thrust of his research is in Design Theory, a branch of Combinatorics.

Arasu is extremely knowledgeable about what’s going on in his field; he maintains constant communication via email with researchers all around the world. His boundless energy lets him work on many different problems simultaneously, and he is never afraid to share his ideas with others—this is the reason he has so many joint papers, including collaborations with students and world-class scholars alike! The most important of Arasu’s many contributions is to the study of difference sets—important combinatorial objects. His survey of difference sets served as a major stimulus for research in the subject by listing unsolved problems and conjectures and updating the state of knowledge as to which finite Abelian groups could support a difference set (most of the new results being due to Arasu himself!). On another front, for years mathematicians and statisticians puzzled over why all difference sets known to give rise to certain weighing designs had the so-called Singer parameters. Arasu’s brilliant analysis unraveled the mystery. Much of Arasu’s work impacts modern communications. Virtually all signals transmitting information—from wireless to digital TV—exploit special binary sequences for synchronization and randomization purposes. Arasu’s contributions to the understanding of these special binary sequences are truly brilliant. In his work with former student Kevin Player and John Dillon, NSA, he obtained

“…ternary sequences with ideal autocorrelation which thereby prove a number of conjectures including the notorious ‘Lin conjecture’ which had resisted all attempts at resolution ever since its enunciation in Alfred Lin’s 1998 USC Ph.D. dissertation.”

In a communication several years ago, Dillon summarized this contribution as follows.

“Absolutely incredible! Arasu’s latest work will revolutionize the pseudo-random sequence business! It initiates the most profound development in the already rich history of this vital subject. At the very least, it will change the face of Combinatorics forever!”

Dillon closed his communication with the following summary of Arasu’s research contributions at the time.

“Be assured that the research of K. T. Arasu is of the highest quality and is directed at a problem area in the mainstream of mathematics. His results so far have been ASTOUNDING and have been a major impetus to the remarkable recent progress seen in this interesting and important area of research. He is a leader in this area and I can only expect many more exciting innovations from him in the future.”

Dr. Warwick de Launey, a highly regarded Australian mathematician working on similar problems, provided the following more succinct but likewise compelling endorsement.

“Since (Arasu’s) first paper was published in 1985, he has published over 70 papers. Over the same time period, the field has been transformed, and Dr. Arasu has been personally involved in nearly all of the major advances.”

Furthermore, de Launey offered the following assessment.

“To measure the importance of a researcher, one can look at a number of things. Several things come to mind: the importance of the field, the importance of the applications, the number of publications, the number of collaborations, the number of good students fostered, the quality and importance of the breakthroughs made by the researcher, the number of times the researcher has been invited to speak at conferences, the number of grants the researcher has received, and the number of awards the researcher has received. Dr. Arasu does well in all the above measures, and this is quite unusual.”

Several years ago, Professor D. K. Ray-Chaudhuri provided a long list of Arasu’s scholarly accomplishments. Two are worthy of special note. First, Dr. Ray-Chaudhuri mentioned with distinction Arasu’s series of invited talks on perfect sequences at the NATO meetings in 1988. Also, quoting Ray-Chaudhuri,

“The high point of Arasu’s career came when he (jointly with Dillon, Jungnickel and Pott) gave the first solution of the so-called Waterloo problem that has been open for over 20 years. This paper appeared in JCT(A), in which they characterized all the Singer difference sets whose complements lift to a relative difference set with the forbidden subgroup of order 2.”

The quality and applicability of Arasu’s work have enhanced his fundability. In mathematics funding is very difficult to obtain, and mathematics research grants are typically tiny in comparison to laboratory science disciplines, yet Arasu has enjoyed extraordinary success obtaining sustained funding from top federal agencies. Following an initial grant from the National Security Agency (NSA) for 1987-89, Arasu has enjoyed continuous funding since 1990 from one or more of NSA, the National Science Foundation (NSF), and the Air Force Office of Scientific Research (AFOSR), with major grants totally about 1.5 million dollars.

Other aspects of Arasu’s research record are commensurate with his exceptional funding success and speak to his international stature in discrete mathematics and areas of application. He has over 90 papers published or to appear to date, and the number could perhaps be doubled, except he made a commitment to himself not to publish mediocre results, nor to publish in mediocre journals. Several times he’s been invited to talk at the Oberwolfach meeting, Germany—small meetings of 40 to 50 experts from all over the world, by invitation only, to meet for a week to lecture and interact.

Major invited addresses he’s given include: a plenary talk plus an invited talk at the International Conference on Applications of Modern Algebra in Computer Science at Manipal University, India (January, 2013), a plenary talk at the International Conference on Mathematical Sciences 2012 in Nagpur, India (December, 2012), an invited talk at the International Conference on Applied Mathematics, Modeling and Computational Science Laurier Centennial Conference at Wilfred-Laurier University in Waterloo, Canada (July, 2011), an invited lecture at the International Workshop on Combinatorial Designs at the National University of Singapore (June, 2011), an invited talk at the NATO workshop on Security in Opatija, Croatia (June, 2010), three lectures at the China International Workshop on Combinatorics and Computing, China (June, 1997), distinguished speaker at the Design Theory session at the Denison OSU meeting (May, 1996), the keynote address at the International Conference on Information and System Sciences in Baden-Baden, Germany (August 1988), four lectures delivered at the NATO workshop in Germany (August 1998), and an invited address at the Hadamard Centenary Conference in Australia (December, 1993). Arasu has given some 115 invited colloquia and seminars at various universities in USA, Canada, Germany, Netherlands, Italy, U.K., Singapore, Malaysia, Hong Kong, India, China, and Japan. The number of recent major invited talks speaks well to his current stature in the international academic community.

In recognition of his accomplishments, Arasu was the winner of the WSU Presidential Award for Research Excellence in 2001, as well as the Trustee’s Award for Faculty Excellence in 2005.

Arasu is an exceptional teacher, and his hallmark in teaching is his involvement of students in his research—undergraduate and graduate students alike! Arasu began involving students in his research in 1990. Over the past 23 years he has supported over 70 undergraduate and 30 graduate students with his NSA and NSF grants, including many students from engineering and computer science, as well as students in mathematics and statistics. One of Arasu’s students, Nancy Voss, went on from a modest start in MTH 102 (now DEV 0950) to earn a BS in Mathematics from WSU. While in the undergraduate program, she worked with Arasu and is a co-author on two papers in international journals, the Journal of Algebra and Designs, Codes and Cryptography. Paraphrasing Arasu, they settled some open questions raised in the research of Alex Pott, Arasu’s German collaborator for the von Humboldt Fellowship, and the Germans were surprised by the short proof obtained, as they had tried and failed.

In a letter of support for a past award, Kevin Player, a former student of Arasu, who was at that time a leading scientist at NSA, said the following.

“Professor K. T. Arasu has been the most influential man in my academic and professional life. I am where I am today because of this man’s mentoring, teaching abilities and leadership.”

Kevin went on to attest to Arasu’s exceptional teaching skills, stretching students to achieve not what they can do but what they could do. We discussed extensively Arasu’s involvement of students in his research, because that is so extraordinary in a department lacking a Ph.D. program with a paucity of master’s thesis students. Arasu’s teaching in the classroom is also excellent. Student evaluations consistently rate Arasu as a highly effective teacher. It is noteworthy that he is a recipient of a Teaching Excellence Award in 1987-88.

It is fitting to speak further of Arasu’s contributions in the realm of professional service. Arasu is an Associate Editor of the prestigious international research journal Designs, Codes and Cryptography, and also on the editorial board of the journal Bulletin of Kerala Mathematical Association. The former in particular is evidence of the very high esteem in which he is held by knowledgeable professionals. Obviously, such editorial work is also vital, substantial professional service. Furthermore, Arasu annually reviews proposals for NSF and NSA. He has served as a panelist at NSF six times. He referees about a dozen papers per year for leading journals. He was editor of a special volume on difference sets and related topics for the Journal of Statistical Planning and Inference and has served as editor for three other volumes of conference proceedings. Arasu has been an organizer of four international conferences. He spearheaded a unique research initiative—a 3-day workshop on difference sets at Ohio State University. This was a milestone event, to which he attracted the leading researchers from all over the world. Its Proceedings, edited by Arasu, remains a beautiful and very important reference for this discipline.

Research Accomplishments

Arasu’s research activities have always revolved around problems in discrete mathematics and combinatorics that require applying algebraic and number theoretic techniques. His research publications always have the flavor of being algebraic.

The main tools he employs are: representation theory, algebraic number theory, Gauss sums, Jacobi sums, group theory, group rings, field theory, Galois theory, finite geometry, and linear algebra. So he can be classified as a person working in applicable algebra and combinatorial design theory.

He primarily studies combinatorial objects that possess a nice automorphism group, mostly a “regular” group. Hence the object under study takes the form of a group ring element satisfying certain nice equations, which are dictated by the “regularity” assumptions of the underlying combinatorial objects. Alternatively, these can be looked at as a class of matrices admitting a regular group action, so-called “group invariant matrices”. Good classes of Hadamard matrices, weighing matrices and generalized weighing matrices are examples of such matrices. Because of their immediate applications to variety of disciplines (cryptography, communication engineering, statistics – to name a few), research in this area is known as “applicable algebra”. Arasu’s research activities have been supported by National Security Agency, Air Force Office of Scientific Research and National Science Foundation.

Arasu has proved several characterization theorems of these objects; constructed many infinite classes of them as well. His major discoveries include obtaining several new classes of perfect binary and quaternary arrays. He also has interests on other problems: self-dual codes, for instance. Recently he has succeeded in constructing some new such codes whose minimum distance beats the previously known ones.

He keeps his research students busy by engaging them to work on many open cases of classes of Hadamard/weighing matrices. He consistently has a group students at various levels working with him (Undergraduate, master’s and Ph.D. students), whose research supported by NSF or NSA. He also engaged some high school Math teachers for 4 summers giving them research experience to work on problems in finite fields and cryptography. NSF funded this project. . His research is highly regarded that he is very successful in securing grant supports from NSF, AFOSR and NSA. His research support so far has totaled over a million dollars.

His thesis was on a topic that generalizes the notion of group difference sets to what he called "difference lists", which has by now a standard term in this area. As remarked in the recent encyclopedean book on Design theory by Beth, Jungnickel and Lenz, Arasu's approach in his Ph.D. thesis mirrors the approach that Eric Lander of Harvard University took in his Ph.D. thesis at Oxford University. But Arasu's approach is somewhat simpler because it only uses group rings with integer coefficients, as opposed to Lander's who uses the so-called K-matrices. Since his Ph.D. in 1983, Arasu has published over 100 papers (plus 4 submitted papers). His research primarily focuses on combinatorial designs that admit some nice automorphism groups. Difference sets are the simplest such objects, arising from a symmetric design with a regular automorphism group. In his early work, he generalized a theorem of Wilbrink on planar difference sets to arbitrary abelian difference sets and obtained some nice characterizations. Lander's monograph lists several open cases of abelian difference sets; Arasu was the first one to work on these open cases, he filled over a dozen of these cases, some with existence proofs and the rest were shown to be nonexistent. It was his work that made others get interested in this table and now it is complete (last bit was done by Joel Iiams). Arasu also worked on projective and affine planes with cyclic automorphism groups, equivalently planar and affine difference sets. His paper with Jungnickel in JCT(A) (1989) is worth mentioning: they prove that if a cyclic affine difference set with n = 4 mod 8 exists, then n = 4. A similar result for n = 2 mod 4 implying n = 2 was proved by Arasu in his Discrete Math paper (1989) . This was a progress toward the prime power conjecture in the even order case. His work with Pott (1992) in Discrete Math deals with the n = 8 mod 16 case and provides further restrictions, but the conclusion n must be 8, is still unresolved.

Arasu also has several papers on various generalizations of difference sets - so-called divisible difference sets, relative difference sets. He has also published a paper on a class of strongly regular graphs, in JCT(A), jointly with Jungnickel, Ma and Pott).

In 1992, he disproved a conjecture on dicyclic designs made by Bhat-Nayak. His discovery (1993) of the new class of Menon difference sets, equivalently perfect binary arrays, (joint with Davis, Jedwab and Sehgal) has drawn the attention of communication engineers and combinatorial design theorists. In this article they construct two new families of perfect binary arrays, thereby answering an open question of Chan, Siu and Ma raised in 1992.

His new difference sets (with Sehgal - JCT(A) 1995) were the first examples of the so-called building sets, in the terminology of the seminal work of Davis and Jedwab on the unification paper. As Davis and Jedwab mention, it was this example of Arasu and Sehgal that led them to investigate building blocks in general.

The high point of Arasu's career came when he (jointly with Dillon, Jungnickel and Pott) gave the first solution of the so-called Waterloo problem that has been open for over 20 years. This paper appeared in JCT(A), in which they characterize all the Singer difference sets whose complements lift to a relative a difference set with the forbidden subgroup of order 2.

In 1995, Arasu and Xiang proved a multiplier theorem (JCD) which is perhaps the best theorem known so far, regarding multipliers. (not counting the adhoc improvements made in some special cases by Qiu Weishung and his Chinese colleagues).

Arasu also has also published in statistical journals that deal with designs - he has constructed new families of nested row-column designs, new classes of weighing matrices, group divisible designs to name a few.

Recently he constructed a new difference set (with Chen) in Z43 X Z5, a case that was open in the unification work of Davis and Jedwab. (this paper appeared in Designs, Codes and Cryptography).

The affine analog of the Waterloo problem - namely, which relative difference sets with classical parameters (i.e. trace = 1 objects in a finite extension of GF(q)) admit a lifting to other relative difference sets has been completely beaten to death by Arasu's recent work with Dillon, Leung and Ma. (this paper appeared in JCT(A)).

Because of his expertise in difference sets, Arasu was invited to write a chapter in the Encyclopedia for Electrical and electronic engineers. This was published by Wiley. Among his many invited presentations in international conferences, I must mention the NATO meeting in August 1998, to which Arasu was invited to deliver a series of talks on perfect arrays.

In addition, Arasu has also contributed to selfdual codes over finite fields, weighing matrices, complex Hadamard matrices, Hadamard and Conference matrices, almost perfect sequences and several variations thereof. I have drawn my comments only to some selected works of Arasu.

Testimonials

Dr. Bulutoglu of Airforce Institute of Technolgy says:

``In the summer of 2018, I had the honor of hosting Dr. K. T. Arasu and his PhD student Dr. Jefferey R. Hollon for 12 weeks within the U.S. Air Force Research Lab Summer Faculty Fellowship Program sponsored by the Air Force Office of Scientific Research. I gave them some open problems and papers to work on. At the end of the short period of 12 weeks they came up with a theorem that provided an elegant solution to one of the open problems in Ding [J. Combin. Des. 16 (2008), 164-171]. Moreover, they established the connection between two well-known binary sequences within the combinatorics literature.

Dr. Arasu also provided me with a partial solution to an open problem in E(s^2)-optimal supersaturated designs that came out of my dissertation from 17 years ago. All these three contributions will appear in papers that will be submitted to the most respected journals in combinatorics.''

Dr. Manil Mohan of Indian Institute of technology (IIT), Roorkie, says:

``I met Prof Arasu in January 2017 at Wright State University. Even though my research area is different from his, we started working together on the topic "Optimization problems involving orthogonal matrix constraints". This research area was totally new for me and his magical words made me comfortable. We found an interesting connection of Entropy of orthogonal matrices with Hadamard, conference and weighing matrices. His earlier works in weighing matrices was a surprise for me and it helped me to steer the subject easily. We were able to write three promising papers out of this research project and one is already published. I enjoyed a lot while working with him and look forward to work more in his areas of expertise.''

Conclusion

In summary, Professor K.T. Arasu is a world-class scholar and the impact of his research extends beyond his own discipline. His record of research and scholarship is phenomenal, his substantial involvement of undergraduate and graduate students in his research is the highest level of teaching and service, his classroom teaching is outstanding and inspiring, his hundreds of summer hours devoted to area high school teachers is exceptional, and his contributions to not only his own discipline but to areas of engineering, especially communication theory, and to students from engineering disciplines, are without equal!

We are pleased to organize this conference to honor K. T. Arasu on the happy occasion of his 65th birthday.