By: David Bressoud @dbressoud

The first four installments of My Mathematical Journey explored the origins of my early textbooks and favorite projects. This may seem strange since I initially built my reputation as a research mathematician at Penn State where I was on the faculty from 1977 until 1994. I chose to begin with those because they drew on my earliest experiences with mathematics, but the time has now come to explain the circuitous route by which I became a mathematician. But first a math problem:

Problem: The Rogers-Ramanujan identities, first discovered by L.J. Rogers in 1894 and independently rediscovered by S. Ramanujan, are a pair of power series identities given in Figure 1. Major P.A. MacMahon recognized that, when viewed as generating functions, they had a combinatorial interpretation: The number of ways of writing a non-negative integer as a sum of positive integers congruent to plus or minus r modulo 5, where r = 1 or 2, is equal to the number of ways equals the number of ways of writing that integer as a sum of integers that differ by at least two, with 1 allowed as one of these integers if and only if r = 1. Explain how to get this result from the functional identities.

As an example, with r = 1, 10 can be written as 9+1, 6+4, 6+1+1+1+1, 4+4+1+1, 4+1+1+1+1+1+1, or 1+1+1+1+1+1+1+1+1+1, six ways. It can also be written as 10, 9+1, 8+2, 7+3, 6+4, 6+3+1, six ways. With r = 2, 10 can be written as 8+2, 7+3, 3+3+2+2, 2+2+2+2, four ways. It can also be written as 10, 8+2, 7+3, 6+4, four ways.

When I entered college, I had no intention of becoming a mathematician. I did not even have any idea of what that meant except the need to pursue a PhD and get a job at a college or university. I had done very well in high school mathematics, but in my first year at Swarthmore I was much more attracted toward a major in philosophy or religious studies. By the end of that first year, however, I was already anxious to get into the real world. I recognized that a math major would be the easiest way to graduate in three years since I had entered with credit for a full year of calculus. Taking one extra course per semester plus two summer courses at Penn in 1970 would enable me to graduate in 1971. I began setting my sights on entering the Peace Corps upon graduation.

My undergraduate mathematics preparation was idiosyncratic. In addition to linear algebra and several variable calculus, I studied modern algebra using Fraleigh’s book and modern applied algebra taught by Gene Klotz using Birkhoff and Bartee. Gene would later write a review of that text for the Monthly (Vol. 79, No. 5 (May, 1972), pp. 529-530) in which he complained about how poorly it had worked for our class. As mentioned in an earlier column, I studied elementary topology out of Gemignani’s wonderful little book. The high point of my mathematical education was a seminar led by Dave Rosen using the first volume of William Feller’s An Introduction to Probability Theory and Its Applications. We were paired up, with one pair each week responsible for leading the class on a section of the book. Deciphering and then explaining Feller was fun.

While my immediate goal was the Peace Corps, I realized that I would need to plan for some sort of career. If I wasn’t going to pursue a PhD, the only option I saw was to prepare for high school teaching. Here also my academic preparation was idiosyncratic. I took the Introduction to Teaching with Alice Broadhead. This gave me an opportunity to observe and try my hand in front of a class at the small high school that then existed in Swarthmore. We would meet at her house over tea to discuss our experiences. I took Introductory Psychology and a seminar in Educational Psychology. In my last semester I did student teaching with Tom Werner at what was then the Nether Providence High School. By then I had been accepted by the Peace Corps and knew that my assignment would be somewhere in the Eastern Caribbean, and so I earned my last required education credits with an independent study exploring the development of the educational system in what had been the British West Indies. That was the extent of my preparation for teaching. I received temporary certification to teach in Pennsylvania for three years. I still have two years remaining on that certificate (more on this in a later column).

Tom had warned me that one could not afford to raise a family on a high school teacher’s salary, and that I would need to think of a second job. He was self-employed as a coin dealer, something he could do mostly during the summer. This did dampen my ardor for high school teaching.

My lack of exposure to analysis showed up just before graduation. Back then, the math department had an exit exam. One of the questions was to prove the intermediate value theorem. I had no idea how to approach this. I later learned that my performance on this exam was so abysmal that the department questioned whether I should graduate. But, after all, I was not really going to use this mathematics, so they let me slip through. I was so confident that graduate school was not in my future that I did not take any GRE exams.

My Peace Corps assignment was to the Clare Hall School in Antigua, West Indies. It was a brand new school in a village outside the capital. I was the first—and only–maths teacher during my two years, teaching 1st and 2nd form (7th and 8th grade). There is so much to say about this experience. I came to appreciate the beauty and complexities of the peoples of the Eastern Caribbean, and I came to see the United States from a perspective that has never left me. But this is the story of my mathematical journey.

Dave Rosen at Swarthmore became my connection to the mathematical community, regularly sending me problems from The Monthly. I had also taken two textbooks with me to work on. One was Gemignani’s Elementary Topology. I was determined to work through every exercise, which I accomplished. I then turned to Nevanlinna and Paatero’s Introduction to Complex Analysis.

This was a book that one of the Swarthmore faculty discarded at the end of the year. I picked it up thinking that this was a subject I should learn. To say I struggled is an understatement. I would later identify with what I read of Newton’s attempt to understand Euclid’s Elements. He found himself constantly going back to the beginning and trying to work his way a bit further into it.

As much as I enjoyed teaching, I found that I missed the challenge of higher mathematics. I decided that graduate school would be the right next step. I joined with two other volunteers on Antigua to request a special administration of the GRE on that island, which was granted. My original intention was to earn a PhD so that I could get a university appointment where I could work on the preparation and support of mathematics teachers. If I had known more about it, I might have headed toward mathematics education for a graduate degree. But back then I am not sure whether it was even possible to combine graduate level mathematics with graduate work in mathematics education.

For personal reasons, I needed to stay in the Philadelphia area. Penn was the natural choice, but my undergraduate performance had been less than stellar. They were willing to accept me as a student, but without any financial support. It was Dave Rosen who recommended Temple University, also in Philadelphia. Temple offered me a full scholarship for my first and last years with teaching assistantships in between. In 1965 Temple had become a “state-related” university, which meant that it was supported by the state without formally becoming a state institution. It is a status Temple shares with Penn State, the University of Pittsburgh, and Lincoln University. This marked the start of Temple’s rise as a research university.

At the time I applied, Temple had two world-class mathematicians: Jim Stasheff in algebraic topology and Emil Grosswald in analytic number theory. Rosen had recommended Temple with the specific suggestion that I work with Grosswald. In those years Stasheff and Grosswald alternated teaching first-year graduate students in their subject. In even years, Stasheff would teach the introductory algebraic topology. In odd years, Grosswald would teach complex analysis. I entered in 1973 and managed to connect with Grosswald. John McCleary, who would go on to teach at Vassar, entered Temple the following year, worked with Stasheff, and became an algebraic topologist. My year of struggling with Nevanlinna and Paatero paid off. Under Grosswald’s inspired guidance the pieces fell into place, and I excelled in this course.

As I described earlier, I knew no number theory when Grosswald agreed to take me on. He started me on Ireland and Rosen (no relation to Dave Rosen), then set me loose with the 4th edition of Hardy and Wright. That is where I fell in love with number theory: the breadth, the challenges, the beauty of the arguments. And that is where I first encountered the Rogers-Ramanujan identities, whose proof seemed unnecessarily complicated. One of my proudest achievements some years later was the discovery of an “easy” proof of this identity (Journal of Number Theory, 1983, 16:2, 235–241).

In 1975 Bryan Birch published a list of 40 identities involving various combinations of the functions that appear in the Rogers-Ramanujan identities. These were stated without proof by Ramanujan and discovered by Birch in an Oxford library. (Yes, Oxford, not Cambridge. Not clear how they came to be there.) All of the identities involve functions that are modular forms, and so each lives in a finite dimensional space. As Birch observed, at least in theory they each could be proven by determining the dimension and verifying that their power series expansions agreed at the requisite number of terms. I had just finished studying modular forms with Grosswald. He decided that this would be an appropriate topic for my thesis.

Determining these spaces turned out to be far more difficult than either of us had expected. As I searched for a way in, I learned that Ramanujan had sent several of these identities to Hardy during his last year of life back in India and that the full list had circulated among Hardy’s acquaintances. L.J. Rogers and G.N. Watson were among those who had access. Rogers published proofs of several of the identities in 1921, Watson several more in 1933, establishing a total of 16 of the identities. I realized that their methods, formal manipulation of q-series, could be applied to the others. I succeeded in proving an additional 15. More than that, I recognized that these methods could be extended, placing most of these identities within families of results.

In 1989 Tony Biagioli employed some very impressive use of modular forms to prove eight of the nine remaining identities for his doctoral thesis. The last also eventually fell. An excellent account of this search with elegant proofs and generalizations of all but five was published by B.C. Berndt, G. Choi, Y.-S. Choi, H. Hahn, B.P. Yeap, A.J. Yee, H. Yesilyurt, and J. Yi as a Memoir of the AMS in 2007, with Yesilyurt eventually completing their approach for the remaining five.

Once I had discovered the papers of Rogers and Watson, most of this work went very fast. By the summer of 1976 my results were ready. I had become a research mathematician. Now all I had to do was write up the thesis, get my results published, and find a job.

Looking back on the events that led to my development as a research mathematician, I realize how much I benefited from being in the bulls-eye of privilege. This is not to discount how hard I worked while in graduate school. Never before or since have I worked so hard. None of this came easily. But I did have so many advantages that only recently have I come to fully appreciate.

Beyond the fact that I am white and male, I had two college-educated parents. My father had earned a BFA at Syracuse and worked as a graphic artist at Bethlehem Steel. My mother had a degree in music education from Eastman and a Master’s in Education from Lehigh. She taught elementary school. We certainly were not wealthy, but we were comfortable. Travel was important. We have strong family ties to both Peru and France. Visiting this family was considered a priority, outranking the need for a new car. College was assumed, with the understanding that my parents would find the means to see that finances would not limit my choice.

Growing up in Bethlehem, Pennsylvania in the 1950s and ‘60s I was in a city that had a strong economy, a school system that offered many advantages including solid Advanced Placement courses, and two institutions of higher education, Lehigh University and Moravian College. I was able to benefit from the proximity of Lehigh, both its summer high school program and its library. I also was born at an opportune time, in 1950. My parents’ incomes were buoyed on the strong economy, and I approached my teen years as the country was ramping up to promote education in mathematics and the natural sciences. Furthermore, I was exactly the favored demographic for this effort.

I cannot know what might have been different had I not been white and male. Throughout my trajectory, I received constant validation that I belonged on the route I was taking. That almost certainly would not have been true.

As I reported in my column from May, 2021, Taking Responsibility for Justice, Equity, Diversity, and Inclusion, AMS and other societies are now acknowledging their historical role in erecting barriers against women and underrepresented minorities, especially African-Americans. It is too easy simply to regard those as mistakes made in the past, “Thank God we are no longer like that.” The barriers today are far more subtle and easy to ignore. Over the past dozen years of digging into what happens in college calculus, I have come to realize the importance of that validation that I belonged.

The team that has worked with me on the MAA studies of college calculus has revealed the fragility of that sense of belonging among women, African-Americans, and others. Most of those from these underrepresented groups who come into college with aspirations for a math-intensive major believe that they belong despite gender or racial stereotypes because they have excelled in mathematics in high school. Any setbacks early in their college career can be particularly damaging. The woman who has been top of her high school mathematics classes and who receives a C in her first college calculus exam is easily led to believe that she was mistaken in confidence that she was capable of succeeding in mathematics.

We saw this in the substantial numbers of women who dropped out of a math-intensive track after receiving a B for the course, or occasionally even with an A but not a high A, women who gave as their reason for dropping out that their grades were not good enough. Men never gave that reason when their course grade was an A or a B.

The kinds of fears and insecurities that can play out are beautifully illustrated in Chapter 8, “Getting an A,” of Paul Tough’s insightful book The Inequality Machine: How College Divides Us (originally published as The Years that Matter Most: How College Makes or Breaks Us). It follows the story of a young Hispanic women who had excelled in mathematics in her under-resourced high school where calculus was not offered. She enrolled in calculus at the University of Texas, Austin, finding herself competing against students who were repeating a course they had passed in high school. Her struggles led her to doubt not just whether she belonged in calculus but whether she even belonged at this university. This chapter shows what it took to build her self-confidence and ability to tackle unfamiliar and challenging mathematical ideas.

The answer has nothing to do with remediation or lowering expectations. In fact, those are exactly the wrong approaches because they reinforce a sense of not belonging. What is needed is far harder and more personal. It involves communicating the inevitability of such setbacks and equipping students to deal with and overcome them.

The CBMS Statement on Equity, Diversity, and Inclusion in the Mathematical Sciences, adopted by the CBMS Council at its meeting on December 3, 2021, envisions “a community of mathematical scientists where all of our colleagues and students are valued and in which we all work and learn together with respect and dignity.” Valuing all of our students, treating all of them with respect and dignity, is not simply a matter of avoiding the discriminatory practices of the past. It requires positive actions that acknowledge the fears and insecurities of our students and help them learn how to overcome the obstacles that they face.

One of the beauties of working with q-series is that so often they have combinatorial interpretations, inviting alternate proofs that involve direct combinatorial manipulation. If two sets are known to be equinumerous, it would be nice to have a direct bijection that implies this fact. The Rogers-Ramanujan identities lack this. There is a known bijection, discovered by Garsia and Milne, but it is neither direct nor particularly insightful into why the two sets are always in one-to-one correspondence. But that is not the task here.

The starting point to solving the problem posed at the start of this column is an observation made by Euler. Consider the product of infinite geometric series given in Figure 2. When we multiply it out, we get a power series for which the coefficient of q^n is the number of ways of writing n as a sum of 1’s, 2’s, and 3’s: The power of q from the first series tells us the number of 1’s, the power of q from the second indicates the number of 2’s, and the power of q from the third contributes 3’s. As an example the coefficient of q^6 is 7 because 6 can be written as 3+3, 3+2+1, 3+1+1+1, 2+2+2, 2+2+1+1, 2+1+1+1+1, or 1+1+1+1+1+1. We call this product the generating function for the number of partitions into parts less than or equal to 3.

This product can also be written as 1/(1-q)(1-q^2)(1-q^3). If we include the binomials for all positive integer powers of q, the coefficient of q^n in the power series expansion is the number of partitions of n, the number of ways of writing n as a sum of positive integers with no restrictions on what those integers might be. We now see that the right-hand side of the Rogers-Ramanujan identities is the generating function for partitions into parts that are congruent to plus or minus 1 modulo 5 in the first case, congruent to plus or minus 2 modulo 5 in the second.

The left side is slightly more complicated. Consider the kth term of the summation. The denominator is the generating function for partitions into parts less than or equal to k. But it is also the generating function into at most k parts with any restriction on the size of the parts. To see this, consider the Ferrers graph in Figure 3. Each row represents one of the parts in a partition into parts less than or equal to 5. If we read down the columns, we get a partition into at most 5 parts. The direct bijection is easily seen. Note that we have at most five parts. We can make this exactly five parts if we add parts of size 0 as needed.

The power of q in the numerator, k^2, is the sum of the first k odd integers, k^2 = 1 + 3 + 5 + … + (2k-1). Taking our partition into exactly five parts, some of which might be 0, we add 1 to the smallest, 3 to the next, and so on. This produces a unique and well-defined partition into parts that differ by at least 2. For the second Rogers-Ramanujan identity, the power of q in the numerator is k^2+k = 2 + 4 + 6 + … + 2k. Adding these to the k parts generated by the denominator again yields parts that differ by at least two, but now 1 is no longer possible as one of the parts.

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