### From his Obituary

On Thursday, February 15, 2024, William Wells ”Bill” Adams of Silver Spring, Maryland, died peacefully. Beloved husband of Elizabeth Shaw Adams and devoted father of Ruth and Sarah (Dwight Shank) Adams, loving grandfather of Aaron, Hannah and Dylan, and dear brother of the late Carol Steele. Bill was a first generation college student who received his undergraduate degree from UCLA and his Ph.D. from Columbia University. He devoted most of his career to teaching and research in Mathematics at the University of Maryland, writing many articles and a few books on Number Theory. Bill was an avid birder who spent much of his free time looking to the trees in search of ever more elusive birds to add to his extensive life list (2,746 species). Bill shared his birding passion with his wife, Liz, and daughter, Ruth, many friends, and his grandson, Dylan. When not birding, Bill could be found ushering at many theaters in the area, cheering on his grandchildren in all their activities, solving Kenken, reading, playing cards, boating in Southold and cheering on the Washington Nationals and Maryland Terps.

#### Report on the Occasion of his Retirement Mathematical Research

William Adams received his Ph.D. from Columbia University in 1964 under Serge Lang. He then had positions at Berkeley, UCLA, and the Institute for Advanced Study. In 1966, he was hired as Associate Professor by Maryland and he was promoted to Professor in 1971.

The most significant results of the first twenty years of Bill’s career were in the areas of Diophantine approximation and transcendence theory. The main purpose of Diophantine approximation is to study the approximation of real numbers by rational numbers. An easy application of the box principle, or of continued fractions, shows that, for a given real number *α*, there are infinitely many pairs of integers *p, q *such that *|qα − p| < *1*/q*. For all *α *outside a set of measure zero, the number of solutions of this inequality with 1 *≤ q ≤ x *is asymptotic to 2 log *x *as *x → ∞*. One of Bill’s early, striking results is that the number of solutions to *|qe − p| < *1*/q *with *|q| ≤ x *is asymptotic to a constant times (log *x/ *log log *x*)^{3/2} as *x → ∞*. This shows that *e *does not behave like most real numbers in this regard.

It is well-known that if *α *is a quadratic irrational, then *|qα − p| < *1*/*(*q**√*5) has infinitely many solutions, and that 1*/**√*5 is the best constant that works for all *α*. Now suppose that *K ⊂ ***R** is a cubic extension of **Q** and let *{*1*, β*_{1}*, β*_{2}*} *be a basis of *K *as a **Q**-vector space. Let *c*_{0} *> *0 be the infimum of constants *c *such that

*|qβ*_{1} *− p*_{1}*| < *(*c/q*)1*/*2*, |qβ*_{2} *− p*_{2}*| < *(*c/q*)1*/*2

has infinitely many solutions in integers *p*_{1}*, p*_{2}*, q*, and let *C*_{0} be the supremum of the *c*_{0} as *K *and *β*_{1}*, β*_{2} vary. Cassels and Davenport showed that 2*/*7 *≤ C*_{0} *≤ *46*−*1*/*4(more generally, for any *β*_{1}*, β*_{2} *∈ ***R** such that *{*1*, β*_{1}*, β _{2}*

*}*is linearly independent over

**Q**). In a series of papers, Bill completely settled the question in the cubic case by showing that in fact

*C*

_{0}= 2

*/*7.

In the area of transcendence theory, Bill developed a *p*-adic analogue of Gel'fond and Schneider’s theory of transcendence and algebraic differential equations. This gave new proofs of the transcendency of the *p*-adic numbers *e**α* and *α**β*, where *α *and *β *are *p*-adic numbers algebraic over **Q** (satisfying some standard conditions), and also gave the first *p*-adic transcendence measures for such numbers. He also proved that if *β *has degree *r ≥ *4 over **Q**, then the transcendence degree over **Q** of **Q**(*α**β**, α**β*2*, . . . , α**β**r−*1) is at least 2.

In the late 1980s, Bill started working in computational ring theory, in partic ular the theory of Gröbner bases. Recall that a polynomial ring in one variable over of field is a principal ideal domain and the generator of an ideal can be found from a set of generators, essentially by the Euclidean algorithm. This is no longer possible for polynomials in several variables. The Gröbner basis algorithm, which is in some ways a cross between the Euclidean algorithm and the Gaussian reduction algorithm from linear algebra, is a replacement that finds generators of ideals in rings of polynomials in several variables. The method, which found early success in Hironaka’s work on resolution of singularities, is now indispensable in many computer algebra calculations. One of Bill’s most important contributions to the subject is his book *An Introduction to Gröbner Bases*, written jointly with Philippe Loustaunau. This is a well-written introduction to the subject and is described by *Math. Reviews *as “excellent” (I have read the book and heartily agree). Bill has written many papers on Gröbner bases and their applications. As an example of how wide reaching the techniques are, consider the following result from a paper written by Adams jointly with Berenstein, Loustaunau, Sabadini, and Struppa. Let *K *be a compact subset of *n*-dimensional quaternionic space **H**^{n}, with *n > *1, such that **H**^{n }*\K *is connected. If *f *is a regular function on **H**^{n }*\K*, then *f *extends to an entire function. This is of course the analogue of Hartogs’ theorem in several complex variables. It was proved by Pertici, but the proof in the present paper is almost purely algebraic.

They consider the Cauchy–Fueter complex of differential operators whose solution sheaf is the sheaf of regular functions of several quaternionic variables, and they study a free resolution of this complex. Gröbner basis techniques allow them to prove the vanishing of some of its Ext-modules and explicitly calculate the degrees of syzygies. It is quite surprising that these techniques work even in this non-commutative setting. This and related papers are featured in the book *Analysis of Dirac Systems and Computational Algebra *by Colombo, Sabadini, Sommen, and Struppa (Birkhäuser, 2004).

#### Teaching Accomplishments

Bill had a distinguished career in teaching. On the graduate level, he directed 13 Ph.D. theses and 4 M.A. theses and has taught numerous graduate courses. On the undergraduate level, he excelled in both large lectures and small sections, and he was the advisor for many undergraduates. He gave several talks to high school students, including one to the US Mathematical Olympiad Team. In the summer of 1990, he and L. Washington co-mentored three students from the Research Science Institute (including future Fields Medalist Terry Tao).

Three times, for a total of 8 years, Bill served as Associate Chair for Un dergraduate Studies. He also served on numerous curriculum development and review committees.

#### Service

Bill had an impressive service record at both the national and the local levels. He served for two and one half years as the Program Director for Algebra/Number Theory for the National Science Foundation. He was the Editor for Number Theory for the *Proceedings of the American Math. Society *for 9 years, and was Associate Editor of the *Journal of Symbolic Computation *for 7 years.

In the early 1990’s, Bill was Project Director for the JPBM Committee on Professional Recognition and Rewards. This was a major committee sponsored by the AMS, MAA, and SIAM, and Bill personally conducted 19 site visits at various universities during his directorship.

At the university level, Bill served on the Faculty Senate and many com mittees for the Faculty Senate, including chair of the Senate Faculty Affairs Committee, several program review committees, and several search commit tees. At the department level, in addition to being Undergraduate Chair, he served on numerous committees.

Bill helped design and was the main organizer of the new Developmental Mathematics Program in the mathematics department. This very successful program was featured as the cover story in the December 2003 issue of *Focus*, a publication of the MAA.Finally, Bill deserves much credit for building the number theory group at the University of Maryland. He arrived at Maryland at the same time as

L. Goldstein and H. Jacquet. In the next few years, G. Cooke, T. Kubota, D. Garbanati, M. Razar, and S. Kudla were hired. Bill was one of the main organizers of the Special Year in Number Theory in 1977-1978, which featured talks by many of the top number theorists in the world and which resulted in the hiring of L. Washington and D. Zagier. Bill’s efforts over the years greatly enhanced the university’s international reputation.

Bill played an unintended role in the history of the University. He openly opposed an action of the administration in response to some campus protests in the late 1960’s. In retaliation, the administration turned down his promotion to full professor. The chair of the math department resigned in protest, and Kirwan (for whom our building is named) became acting chair, starting him on the path that led to becoming chancellor of the Maryland system. During the controversy, one of the regents told Bill Adams, ”You’re too idealistic to work in a university.” No comment, except that Bill was very idealistic and principled, and a wonderful person.