Showing papers in "Notices of the American Mathematical Society in 2015"

Seven Characteristics of Successful Calculus Programs

Abstract: In these days of tight budgets and pressure to improve retention rates for science and engineering majors, many mathematics departments want to know what works, what are the most productive means of improving the effectiveness of calculus instruction. This was the impetus behind the study of Characteristics of Successful Programs in College Calculus undertaken by the Mathematical Association of America. The study consisted of a national survey in fall 2010, followed by case study visits to seventeen institutions that were identified as “successful” because of their success in retention and the maintenance of “productive disposition,” defined in [NRC 2001] as “habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.” Our survey revealed that Calculus I, as taught in our colleges and universities, is extremely efficient at lowering student confidence, enjoyment of mathematics, and desire to continue in a field that requires further mathematics. The institutions we selected bucked this trend. This report draws on our experiences at all seventeen colleges and universities but focuses on the insights drawn from those universities that offer a PhD in mathematics, the universities that both produce the largest numbers of science and engineering majors and that often struggle with how to balance the maintenance of high-quality research with attention to undergraduate education. Case studies were conducted in the fall of 2012 at five of these universities: two large public research universities, one large private research university, one public technical university, and one private technical institute. We shall refer to these as: PrTI: Private Technical Intitute. Private university. Data from nine sections of calculus with an average enrollment of 33.

Investigating and improving undergraduate proof comprehension

Abstract: Undergraduate mathematics students see a lot of written proofs But how much do they learn from them? Perhaps not as much as we would like – every professor knows that students struggle to make sense of the proofs presented in lectures and textbooks Of course, written proofs are only one resource for learning; students also attend lectures and work, independently or with support, on problems But, because mathematics majors are expected to learn much of their mathematics by studying proofs, it is important that we understand how to support them in reading and understanding mathematical arguments This observation was the starting point for the research reported in this article Our work uses psychological research methods to generate and analyse empirical evidence on mathematical thinking, in this case via experimental studies of teaching interventions and quantitative analyses of eye-­‐movement data What follows is a chronological account of three stages in our attempts to better understand students’ mathematical reading processes and to support students in learning to read effectively

Professional Development in Teaching for Mathematics Graduate Students

Abstract: The purpose of this article is twofold: to give a sense of the current lay of the land in the preparation of mathematics graduate student teaching assistants (TAs) and to describe the collegiate mathematics education research base informing the next generation of college mathematics instructor preparation. We anchor discussion in three common types of TA preparation programs, each represented in one of the quotes above. Notably, the first quote represents a sink or swim experience that is becoming rare in US PhD-granting mathematics departments. Preparation of TAs for their instructional roles has blossomed in the last twenty years. The second quote is representative of current practices in many departments. The third quote illustrates the activities in innovative departments that are already implementing best practices suggested by research in collegiate mathematics education: sustained professional growth about teaching and learning. To highlight the challenges and benefits of spending time paying attention to teaching, we provide information from postsecondary and related secondary-level educational research of several types. This includes basic and applied educational research that identifies good instructional practices, examines experiences TAs bring with them to teaching, provides frameworks for the structuring of TA preparation, and gives insight into the kinds of mathematics-specific and teaching-specific knowledge that needs to be developed among TAs. And, once a program for supporting TAs to learn and grow as instructors is put in place, evaluation research explores the implementation of efforts to improve TAs’ teaching and the related impacts on undergraduate student learning. We close with promising practices and sketch anticipations for the future of the field of related research. Though not common thirty years ago, today most doctorateand master’s-granting institutions provide some kind of TA preparation for teaching [4a]. The content for this professional development often comes from mathematicians offering their collective wisdom from practical Jessica M. Deshler is assistant professor of mathematics at West Virginia University. Her email address is deshler@ math.wvu.edu.

Gauss' Hidden Menagerie: From Cyclotomy to Supercharacters

Abstract: Gaussian periods, when viewed appropriately, exhibit a dazzling and eclectic host of visual qualities. This brief survey reviews the historical context and summarizes our current knowledge of graphical properties of Gaussian periods.

Fruit Flies and Moduli: Interactions between Biology and Mathematics

Abstract: Possibilities for using geometry and topology to analyze statistical problems in biology raise a host of novel questions in geometry, probability, algebra, and combinatorics that demonstrate the power of biology to influence the future of pure mathematics. This expository article is a tour through some biological explorations and their mathematical ramifications. The article starts with evolution of novel topological features in wing veins of fruit flies, which are quantified using the algebraic structure of multiparameter persistent homology. The statistical issues involved highlight mathematical implications of sampling from moduli spaces. These lead to geometric probability on stratified spaces, including the sticky phenomenon for Frechet means and the origin of this mathematical area in the reconstruction of phylogenetic trees.

The myth and the medal

Abstract: The weekend before the start of the 2014 International Congress of Mathemati- cians, I contributed an op-ed to the New York Times about the history of the Fields Medal. The article treated two topics that are familiar to many mathematicians, if not necessarily to the general public. I suggested that a previously unrecognized connection between these two stories, discovered in the course of my research, helped illuminate the relationship between politics and modern mathematics. The first relatively wellknown story involved the lack of a Nobel Prize in mathematics and the creation of the Fields Medal some three decades later. While the criteria, compensation, and other aspects of these awards have always been quite different, many have seen the Fields Medal’s origin as a response to the

WHAT IS...an Automatic Sequence

Abstract: A sequence s(n)n≥0 is called k-automatic if s(n) is a finite-memory function of the base-k digits of n. This means that some computer with only finitely many possible states can compute s(n) for any n by reading the base-k digits of n one at a time (beginning with the least significant digit) and following a transition rule that specifies the next state of the computer as a function of both the current state and the current digit being read. Each possible state of the computer has an associated output value, and the result of the computation is the output value corresponding to the state of the computer after it has read the final digit. A computer of this kind is called an automaton, hence the name “automatic sequence.” For example, consider an automaton with only two states, q1 and q2, that reads binary representations of integers. Suppose the automaton starts in state q1 and performs transitions according to the function δ : {q1, q2} × {0,1} → {q1, q2} given by the following table.

Mathematical Biology is Good for Mathematics

Abstract: A bout ten years ago I wrote an article, “Why is Mathematical Biology so Hard?” for these Notices intending to explain why the applications of mathematics to biology would be very different than the traditional applications to physics and engineering [42]. A lot has happened since then. Mathematical biology has grown from a small field, containing relatively few mathematicians, to a major branch of applied mathematics. The reasons for this growth, which are implicit in the discussion below, are not the point of this article, nor do I want to encourage mathematicians to switch to mathematical biology. Rather, I want to make the case that mathematical biology benefits all mathematicians; it is good for the health of mathematics as a whole.

Malaysia airlines flight MH370: Water entry of an airliner

Abstract: O n March 8, 2014 Malaysia Airlines Flight MH370 disappeared less than an hour after take-off on a flight from Kuala Lumpur to Beijing. The Boeing 777-200ER carried twelve crew members and 227 passengers. On March 24 the Malaysian Prime Minister announced that “It is therefore with deep sadness and regret that I must inform you that ...Flight MH370 ended in the Southern Indian Ocean.” Though the exact fate of Flight MH370 remains undetermined, the available evidence indicates a crash into the ocean. However, disturbing as this is, not all emergency water landings, referred to as “ditching” when they are controlled, end in tragedy. In the “Miracle on the Hudson,” on January 15, 2009, Capt. Chelsey B. “Sully” Sullenberger and his crew successfully ditched US Airways Flight 1549, an Airbus A320200, in the Hudson River after a loss of power due to a bird strike on take-off from La Guardia Airport. There was no loss of life. Figure 1 and the video animation referenced on the second page of this article show our “representation” of a commercial airliner, a Boeing

Mathematics and teaching

Abstract: Induction I was not born to be a mathematician. Like many, I was drawn to mathematics by great teaching. Not that I was encouraged or mentored by supportive and caring teachers—such was not the case. It was instead that I had as teachers some remarkable mathematicians who made the highest expression of mathematical thinking visible and available to be appreciated. This was like listening to fine music with all of its beauty, charm, and sometimes magical surprise. Though not a musician, I felt that this practice of mathematical thinking was something I could pursue with great pleasure and capably so, even if not as a virtuoso. And I had the good fortune to be in a time and place where such pursuits were comfortably encouraged. The watershed event for me was my freshman (honors) calculus course at Princeton. The course was directed by Emil Artin, with his graduate students John Tate and Serge Lang among its teaching assistants. It was essentially a Landau-style course in real analysis (i.e., one taught rigorously from first principles). Several notable mathematics research careers were launched by that course. Amid this cohort of brilliant students, I hardly entertained ideas of an illustrious mathematical future, but I reveled in this ambience of beautiful thinking, and I could think of nothing more satisfying than to remain a part of that world. It was only some fifteen years later that I came to realize that this had not been a more-or-less standard freshman calculus course. Certain mathematical dispositions that were sown in that course remain with me to this day, and influence both my research and my teaching. First is the paramount importance of proofs as the defining source of mathematical truth. A theorem is a distilled product of a proof, but the proof is a mine from which much more may often be profitably extracted. Proof analysis may show that the argument in fact proves much more than the theorem statement captures. Certain hypotheses may not have been, or only weakly, used, and so a stronger conclusion might be drawn from the same argument. Two proofs may be observed to be structurally similar, and so the two theorems can be seen to be special cases of a more unifying claim. The most agreeable proofs explain rather than just establish truth. And the logical narrative clearly distinguishes the illuminating turn from technical routine. Artin himself once reflected on teaching in a review published in 1953.

The Mathematics Community and the NSA: Encryption and the NSA Role in International Standards

Abstract: “AMS Should Sever Ties with the NSA” (Letter to the Editor), by Alexander Beilinson (December 2013); “Dear NSA: Long-Term Security Depends on Freedom”, by Stefan Forcey (January 2014); “The NSA Backdoor to NIST”, by Thomas C. Hales (February 2014); “The NSA: A Betrayal of Trust”, by Keith Devlin (June/July 2014); “The Mathematical Community and the National Security Agency”, by Andrew Odlyzko (June/July 2014); “NSA and the Snowden Issues”, by Richard George (August 2014); “The Danger of Success”, by William Binney (September 2014); “Opposing an NSA Boycott” (Letter to the Editor), by Roger Schlafly (November 2014).

What Is...a Graphon

Abstract: Graphons were introduced in 2006 by Lovasz and Szegedy as limits of graph sequences. Graphon theory not only draws on graph theory (graphs are special types of graphons), it also employs measure theory, probability, and functional analysis. At only a few years old, the theory is developing quickly and finding new applications. The material for this talk, and most of the notation, was taken exclusively from Lovasz’s new book [1].

A Medieval Mystery: Nicole Oresme's Concept of Curvitas

Abstract: I n a paper published in 1952, J. L. Coolidge (1873–1954) appreciates that the story of curvature is “unsatisfactory” [2], and he points out that “the first writer to give a hint of the definition of curvature was the fourteenth century writer Nicole Oresme, whose work was called to my attention by Carl Boyer.” Then Coolidge comments: “Oresme conceived the curvature of a circle as inversely proportional to the radius; how did he find this out?” The scholarly conditions of the fourteenth century make this discovery phenomenal and the question as to how it was achieved worth researching. In the present article we describe how a fourteenthcentury scholar (i) gave a correct definition for curvature of circles and attempted to extend it to general curves, (ii) tried to apply curvature to understand the behavior of real-life phenomena, and (iii) produced in his research a statement that anticipates the fundamental theorem of curves in the plane. In various cases Oresme’s work is not cited when the history of curvature is discussed (e.g., [5], [8]), while some authors (e.g., [1], p. 191) make note of his contribution to this concept. Several scholars have even concluded that the medieval sciences contributed very little to the modern scientific revolution. In addressing this perception, Edward

William P. Thurston, 1946--2012

Abstract: W illiam Paul Thurston, known universally as Bill, was an extraordinary mathematician whose work and ideas revolutionized many fields of mathematics, including foliations, Teichmüller theory, automorphisms of surfaces, 3-manifold topology, contact structures, hyperbolic geometry, rational maps, circle packings, incompressible surfaces, and geometrization of 3-manifolds. Bill’s influence extended far beyond his incredible insights, theorems, and conjectures; he transformed the way people think about and view things. He shared openly his playful, ever curious, near magical and sometimes messy approach to mathematics. Indeed, in his MathOverflow profile he states, “Mathematics is a process of staring hard enough with enough perseverance at the fog of muddle and confusion to eventually break through to improved clarity. I’m happy when I can admit, at least to myself, that my thinking is muddled, and I

The Method of Archimedes: Propositions 13 and 14

Abstract: N o area of mathematics has attracted more international attention in the past decade than the Palimpsest of Archimedes. The 1998 auction at Christie’s, followed by collaborative work centered at the Walters Art Museum led to traveling museum exhibits, newspaper articles, television specials, and dozens of presentations. Mathematicians and other scholars attracted a new and significant audience. The singed, battered, faded, mildewed, damaged 10th century manuscript—the world’s oldest copy of The Method of Archimedes—sold for $2 million “under the hammer.” Mathematicians and classical scholars have long wondered just how close Archimedes (287–212 BC), a mechanical genius, had come to formulating modern calculus. The clues would surely lie in Propositions 13 and 14, if only they could be read. Though now transcribed, the content may contain copyists’ errors. In the

Mathematical Research in High School: The PRIMES Experience

Abstract: (...) Seriously? Is it really possible for tenthand eleventh-graders to do original mathematical research? Yes! Christina and Joseph, as well as over a hundred other students, have done their research at PRIMES (Program for Research In Mathematics, Engineering, and Science: web.mit.edu/primes), which we’ve been running in the MIT mathematics department since January 2011. Every year we receive numerous questions about our program from prospective students and their parents and also from academics who want to organize a similar program. Here we’d like to answer some of these questions, to share our experience, and to tell a wider mathematical community how such a seemingly impossible thing as mathematical research in high school can actually be done. (...) How do you select projects? Can my student be told to prove the Twin Primes Conjecture in PRIMES? P.E.: Famous open problems don’t usually make good projects, but we don’t assign “toy projects” with known solutions either. Students delve into real research, with all its uncertainties, disappointments, and surprises. Finding cutting-edge projects requiring a minimal background is one of the trickiest tasks in running PRIMES. Here are some features we want to see in a PRIMES project:

Reflections on Paul Erdős on His Birth Centenary, Part II

Abstract: This is Part II of the two-part feature on Paul Erdős following his centennial. Part I had contributions from Krishnaswami Alladi and Steven G. Krantz, Vera T. Sos and Laszlo Lovasz, Ronald Graham and Joel Spencer, Jean-Pierre Kahane, and Mel Nathanson. Here in Part II we have six articles from contributors Noga Alon, Dan Goldston, Andras Sarkozy, Jozsef Szabados, Gerald Tenenbaum, and Stephan Ramon Garcia and Amy L. Shoemaker.

In Memory of Jacob Schwartz

Abstract: A few months after Jack Schwartz’s death on March 2, 2009, a memorial gathering was held for him at the Courant Institute of New York University, his home base for almost fifty years. The well-planned program included a few piano pieces and almost a dozen speakers, each of whom had collaborated with Jack in one or more of his many activities. Those present would have heard the speakers talk about:

The Nesin Mathematics Village in Turkey

Abstract: In August 2014, my wife, Mathura, and I were on a three-week visit to Turkey. I participated in a conference on algebra and number theory in Samsun on the Black Sea coast in northern Turkey, and after that we went on a one-week sightseeing tour of Turkey. Professor Ali Bülent Ekin of the University of Ankara, who was one of the organizers of the conference and my host in Turkey, graciously offered to take us on a 1,500-mile journey to see several ancient historical sites of the Greek, Roman, and Ottoman periods. One of the places we visited was Selçuk, a town which is close to the city of İzmir and known for the well-preserved ruins of Ephesus, as well as the not-so-well-preserved ruins of the Temple of Artemis, one of the Seven Wonders of the Ancient World. After a long, enjoyable, but tiring day of

Puzzling the 120–cell

Abstract: A burr puzzle is a collection of notched wooden sticks [2, page xi] that fit together to form a highly symmetric design, often based on one of the Platonic solids. The assembled puzzle may have zero, one, or more internal voids ; it may also have multiple solutions. Ideally, no force is required. Of course, a puzzle may violate these rules in various ways and still be called a burr.

The exterior algebra and central notions in mathematics

Abstract: Definition Here we define the exterior algebra using standard machinery from algebra. Let V be a vector space overk, and denote byV⊗p thep-fold tensor product V ⊗k V ⊗k · · ·⊗k V . The free associative algebra on V is the tensor algebra T(V) = ⊕ p≥0 V⊗p which comes with the natural concatenation product (v1⊗· · ·⊗vr )·(w1⊗· · ·⊗ws)=v1⊗· · ·vr⊗w1⊗· · ·⊗ws . LetR be the vector subspace of V⊗kV generated by all elements v⊗v where v ∈ V . The exterior algebra is the quotient algebra of T(V) by the relations R. More formally, let 〈R〉 be the two-sided ideal in T(V) generated by R. The exterior algebra E(V) is the quotient algebra T(V)/〈R〉. The product in this quotient algebra is commonly denoted by ∧. Let e1, . . . , en be a basis for V . We then have ei ∧ ei = 0, since ei ⊗ ei is a relation in R. Similarly, (ei + ej)∧ (ei + ej) is zero. Expanding this 0 = ei ∧ ei + ei ∧ ej + ej ∧ ei + ej ∧ ej , we see that ei ∧ ej = −ej ∧ ei . In fact, we obtain v ∧w +w ∧ v = 0 for any v,w in V . Hence when the characteristic of k is not 2, the exterior algebra may be defined as T(V)/〈S2V〉 where S2V = {v ⊗w +w ⊗ v |v,w ∈ V} are the symmetric two-tensors in V ⊗ V . The pth graded piece of E(V), which is the image of V⊗p, is denoted as ∧pV . We shall in the following indicate: • How central notions in various areas in mathematics arise from natural structures on the exterior algebra. • How the exterior algebra or variations thereof are a natural tool in these areas. Combinatorics I: Simplicial Complexes and Face Rings For simplicity denote the set {1,2, . . . , n} as [n]. Each subset {i1, . . . , ir} of [n] corresponds to a monomial ei1 ∧ ei2 ∧ · · · ∧ eir in the exterior algebra E(n). For instance, {2,5} ⊆ [6] gives the monomial e2 ∧ e5. It also gives the indicator vector (0,1,0,0,1,0) ∈ Z2 (where Z2 = {0,1}), with 1’s at positions 2 and 5. We may then consider e2 ∧ e5 to have this multidegree. This one-toone correspondence between subsets of [n] and monomials in E(n) suggests that it can be used to encode systems of subsets of a finite set. The set systems naturally captured by virtue of E(n) being an algebra are the combinatorial simplicial complexes. These are families of subsets ∆ of [n] such that if X is in ∆, then any subset Y of X is also in ∆. Example 2. Let n = 6. The sets {1,2}, {3,4}, {3,5}, {4,5,6}, together with all the subsets of each of these four sets, form a combinatorial simplicial complex. The point of relating these to the algebra E(n) is that combinatorial simplicial complexes on [n] are in one-to-one correspondence with Z2 -graded ideals I in E(n) or equivalently with Z2 -graded quotient rings E(n)/I of E(n): To a simplicial complex ∆ corresponds the monomial ideal I∆ generated by {ei1 ∧ · · · ∧ eir | {i1, . . . , ir} 6∈ ∆}. Note that the monomials ei1 ∧ · · · ∧ eip with {i1, . . . , ip} in∆ then constitute a vector space basis for the quotient algebra E(∆) = E(V)/I∆. We call this algebra the exterior face ring of ∆. For the simplicial complex in the example above, E(∆) has a basis: • degree 0: 1, • degree 1: e1, e2, e3, e4, e5, e6, April 2015 Notices of the AMS 365 • degree 2: e1 ∧ e2, e3 ∧ e4, e3 ∧ e5, e4 ∧ e5, e4 ∧ e6, e5 ∧ e6, • degree 3: e4 ∧ e5 ∧ e6. Although subsets {i1, . . . , ir} of [n] most naturally correspond to monomials in E(n), one can also consider the monomial xi1 · · ·xir in the polynomial ring k[x1, . . . , xn]. (Note, however, that monomials in this ring naturally correspond to multisets rather than to sets.) If one associates to ∆ the analog monomial ideal in this polynomial ring, the quotient ring k[∆] is the Stanley-Reisner ring or simply the face ring of ∆. This opens up the arsenal of algebra to study ∆. The study of E(∆) and k[∆] has particularly centered around their minimal free resolutions and all the invariants that arise from such. The study of k[∆] was launched around 1975 with a seminal paper by Hochster [29] and Stanley’s proof of the Upper Bound Conjecture for simplicial spheres; see [44]. Although one might say that E(∆) is a more natural object associated to ∆, k[∆] has been preferred for two reasons: (i) minimal free resolutions over k[x1, . . . , xn] are finite in contrast to over the exterior algebra E(n), (ii) k[∆] is commutative and the machinery for commutative rings is very well developed. Since 1975 this has been a very active area of research, with various textbooks published: [44], [6], [34], and [22]. For the exterior face ring, see

Albert Baernstein II 1941--2014

Abstract: Albert (Al) Baernstein made significant and original contributions to classical analysis. An outstanding and scrupulous scholar and teacher, he had a special talent for developing his own methods to settle long-standing problems. During his long service at Washington University, he trained a large cadre of PhD students, both from the US and abroad, and their contributions to complex and harmonic analysis continue. Al’s own research impact lies in his approach to symmetrization, now encapsulated by the term Baernstein star function (Baernstein ?-function). Symmetrization is an ancient concept in mathematical analysis and geometry. Its most famous result is the isoperimetric principle: among all plane regions of fixed area, the one of least perimeter is the disk. The general situation is concerned with a class of functions F or domains D, and the issue is

Prime Numbers: A Much Needed Gap Is Finally Found

Abstract: In May of 2013 the Annals of Mathematics accepted a paper [Z], written by Yitang Zhang and showing “bounded gaps for primes,” that is, the existence of a positive constant (specifically mentioned was 70 million) with the property that infinitely many pairs of primes differ by less than that constant. Zhang’s result created a sensation in the number theory community, but much more broadly as well. I don’t know what it says about the current state of the world or of mathematics or maybe just of me, but I began writing these words by going to Google and typing in “zhang, primes, magazine.” Among the first 10 out of more than 74,000 hits, I found references to articles on this topic by magazines with the names Nautilus, Quanta, Nature, Discover, Business Insider , and CNET . (Within a week of my beginning this, there has appeared a long article [W] in the The New Yorker .) I can’t begin to guess how many more there have been. I understand that there is also a movie and, I guess, probably television interviews as well. Zhang has since won a number of prizes, including a MacArthur Fellowship, the Ostrowski Prize, the Rolf Schock Prize of the Royal Academy of Sciences (Sweden), and a share of the Cole Prize of the American Mathematical Society. There have also been quite a number of professional papers written about the mathematics and its ensuing developments. Thus, when I was invited by Steven Krantz to write this article and I requested a few days to think it over, my overriding concern naturally was: “What can I possibly write that is not simply covering well-trodden ground?” This is my excuse for what

Response to the Elizabeth Green Article

Abstract: On July 23, 2014, the New York Times Magazine carried an article by journalist Elizabeth Green entitled “Why Do Americans Stink at Math?” (Green’s article can be found at: www.nytimes.com/2014/07/27/magazine/why-do-americans-stink-at-math. html?_r=0). The (somewhat provocative) question posed by Green has since been resonating with the mathematical community. We present here reactions to the article from six diverse points of view.

Geometry of Data and Biology

Abstract: Introduction The analysis of large high-dimensional data sets is a necessity in a wide variety of fields, including statistics, engineering and signal processing, physics, biology and medicine. While in the field of statistics data has always been at the center of attention, in the past several years the nature of many data sets has changed in a way that requires novel approaches, both from a theoretical and a practical perspective. Modern data sets may be very large but are very often high-dimensional, meaning each data point has a long list of attributes or coordinates, and this is often the case: This happens frequently in biological data (e.g., a genetic profile has easily more than 104 entries). While a large number n of data points is beneficial for statistical analysis, the high-dimension D of the data is a “curse” in the sense that, without further assumptions or model on the data, for many classical statistical inference and function approximation techniques to work n is required to scale exponentially in D, a truly gargantuan requirement (think about what 210 4 looks like) [11]. Various types of assumptions on the data are usually made to avoid this curse, including parametric and nonparametric statistical models, as well as geometric models. These are not disjoint approaches but rather different languages to express modeling assumptions and provide a priori information about the structure of data. These hypotheses may often be interpreted geometrically in terms of imposing that the data is intrinsically low dimensional, and this property is

Herbert S. Wilf (1931--2012)

Abstract: DOI: http://dx.doi.org/10.1090/noti1247 received both the Steele Prize for Seminal Contributions to Research (from the AMS, 1998) and the Deborah and Franklin Tepper Haimo Award for Distinguished Teaching (from the MAA, 1996). During his long tenure at Penn he advised twenty-six PhD students and won additional awards, including the Christian and Mary Lindback Award for excellence in undergraduate teaching. Other professional honors and awards included a Guggenheim Fellowship in 1973–74 and the Euler Medal, awarded in 2002 by the Institute for Combinatorics and its Applications. Herbert Wilf’s mathematical career can be divided into three main phases. First was numerical analysis, in which he did his PhD dissertation (under Herbert Robbins at Columbia University in 1958) and wrote his first papers. Next was complex analysis and the theory of inequalities, in particular, Hilbert’s inequalities restricted to n variables. He wrote a cluster of papers on this topic, some with de Bruijn [1] and some with Harold Widom [2]. Wilf’s principal research focus during the latter part of his career was combinatorics. In 1965 Gian-Carlo Rota came to the University of Pennsylvania to give a colloquium talk on his then-recent work on Möbius functions and their role in combinatorics. Wilf recalled, “That talk was so brilliant and so beautiful that it lifted me right out of my chair and made me a combinatorialist