# Edray Herber Goins

### Born: 29 June 1972 in Los Angeles, USA

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**Edray Herber Goins**was born in South Central Los Angeles and, given the Acknowledgement in his thesis, seems to have been brought up by his mother, Eddi Beatrice Goins, and her godfather, William Herber Dailey. He writes:-

Edray Goins was brought up along with his brother. From the Abstract of the talkThe author would like to give ultimate credit to his mother, Eddi Beatrice Goins, and her godfather, William Herber Dailey, for constant guidance, support, and protection over the years.

*From the Diary of a Black Mathematician: My Journey from South Central to Studying Dessins d'Enfants*he gave in 2015 to the University of Michigan's "Dr Marjorie Lee Browne Colloquium" we have the following details. In 1939, Annie Beatrice Brown gave birth in Marshall, Texas, to Eddi Beatrice Brown. She spent a majority of the years 1936 - 1942 taking classes in education at Bishop College - alternating between taking classes in education and being a stay-at-home mother. Eddi Beatrice Brown became a teacher, married Goins, and in 1972 she gave birth to Edray Herber Goins, the subject of this biography, in South Central Los Angeles.

Edray said that his mother and other teachers in the Los Angeles Unified public school system that he attended encouraged and motivated him to study hard and to take on additional learning challenges [9]:-

He was highly successful at high school and in 1989 won the Bronze Medal in Mathematics from the Los Angeles Academic Decathlon.Even at young age, Edray had a thirst for knowledge. He asked his teachers to let him study ahead or subjects not part of the curriculum.

The Preliminary Scholastic Assessment Test at the time that Goins was at high school consisted of a mathematics section and a verbal section. The National Merit Scholarship Corporation established the National Achievement Scholarship Program in 1964 which aims to recognize outstanding black American high school students. Black students may enter both this programme and the National Merit programme by taking the Preliminary Scholastic Assessment Test. In 1989 Goins received a National Merit Scholarship Honourable Mention, and in the following year he graduated from High School and was awarded a National Achievement Scholarship. This was not the only scholarship he received before entering the California Institute of Technology, Pasadena, California, for in 1990 he also received a Roy A Wilkins Scholarship from the National Association for the Advancement of Colored People, a Sigma Pi Phi Scholarship, and a Robert A Millikan Physics Scholarship from the California Institute of Technology.

At the California Institute of Technology he quickly realised that he wanted be become a mathematics researcher. He said [4]:-

He showed his abilities from his first year, being awarded the Morgan Ward Mathematics Prize in both 1991 and 1992. The California Institute of Technology gives the following information about this prize:-I knew at a pretty early stage in my life - my freshman year of college, to be exact - that I wanted to become a research mathematician.

He also won a 1991-92 American Physical Society Scholarship. InThe Morgan Ward Prize was established by the Department of Mathematics in1963to honour the memory of Professor Morgan Ward in recognition of his long service to mathematics and to the Institute. The competition is open to all freshmen and sophomores, regardless of major. An entry consists of a mathematical problem together with a solution or a significant contribution toward a solution. ... The problem may have any source, but this source should be stated in the entry. The entries may be judged on the basis of the nature of the problem, originality and elegance of the solution.

*Physics Today*(November 1991) details of these scholarships are given:-

Indeed Goins' American Physical Society Scholarship was renewed for the year 1992-93. Also in 1993 he was awarded a Los Angeles Philanthropic Foundation scholarship which is for students who have a financial aid need and are involved in extra-curricular school activities and community service. At the California Institute of Technology, Goins was advised by the mathematics professor Dinakar Ramakrishnan and the theoretical physics professor Steven C Frautschi. In June 1994, Goins graduated with a B.S. majoring in Mathematics and Physics. He had won several awards during his undergraduate years at the California Institute of Technology, namely the Dean's Cup for Service (1993), the Doris S Perpall Speaking Award for best presentation in the Humanities (1993), and the Rodman W Paul History Prize (1994).The American Physical Society has awarded corporate-sponsored scholarships for the1991-92academic year to22minority students who are majoring in physics. ... Scholarships are awarded to African-American, Hispanic American or Native American students who are high school seniors or college freshmen or sophomores. Selections are based on academic grades and achievement, personal statements ... and letters of reference. Each scholarship consists of $2000, which can be used for tuition, room and board, and can be renewed once.

In 1994 Goins was awarded a National Science Foundation Graduate Research Fellowship (Honourable Mention) and a National Physical Science Consortium Graduate Fellowship which enabled him to begin research for a Ph.D. at Stanford University. During the summers of 1995 and 1996 he held a Summer Internship from June to August at the National Security Agency, Fort George Meade, Maryland. In 1996 he was given the Outstanding Graduate Student award from the Black Community Services Center, Stanford University. His thesis advisors at Stanford were Daniel Willis Bump and Karl Cooper Rubin. In 1999 he received a number of awards and honours from Stanford University for his outstanding contributions: the James W Lyons Award for Service, Stanford University; the Graduate Service Award, Graduate Student Council, Stanford University; and the Outstanding Graduate Student, Chicano/Latino Graduate Student Association, Stanford University. He was awarded a Ph.D. in September 1999 for his thesis

*Elliptic Curves and Icosahedral Galois Representations*. Goins gives the following acknowledgements in his thesis:-

Although the date on which he submitted his Ph.D. thesis is shown clearly on a copy on the web as "August 1999," and "Copyright by Edray Herber Goins 1999," Stanford University has a copy of the thesis on their website dated "October 2002" and has "Copyright by Edray Herber Goins 2003." This is puzzling but the 1999 date is surely correct since that copy has the signatures of the examiners on it while the 2002 copy does not. In his thesis Goins looked at cases of Artin's Conjecture and he gives the background at the start of the thesis:-The author would like to thank his advisor, Dan Bump, for suggesting the original problem; his unofficial co-advisor Karl Rubin for his invaluable suggestions and the realization that the elliptic curve is isogenous to its conjugate; and graduate student William Stein for his help in calculating coefficients via Cremona's modular symbol algorithm. This thesis could not have been completed without their help. The author would also like to thank the following persons for helpful conversations: David Carlton, Jordan Ellenberg, Ralph Greenburg, Kenneth Ribet, and Richard Taylor. Papers from Ki-ichiro Hashimoto and Yuji Hasegawa were also helpful in gaining insight into the properties of Q-curves. All of the research was sponsored by a generous fellowship from the National Physical Science Consortium(NSPC)and the National Security Agency(NSA).

In August 1999 spent a month as a Mathematical Sciences Research Institute, Berkeley, California, then, in September 1999, he became a postdoctoral fellow at the Institute for Advanced Study, Princeton, New Jersey, holding this position until August 2000. He did however spend April 2000 as a Visiting Scholar at Harvard University. After his time at Princeton, he returned to Berkeley in August 2000 where he was again a postdoctoral fellow until December 2000, spending October 2000 as a Visiting Scholar at Purdue University. He spent January 2001 - June 2001 at the Max Planck Institut für Mathematik in Bonn, Germany, returning to the United States to take up the position of Irvine Foundation Instructor of Mathematics at the California Institute of Technology at Pasadena in August 2001. He continued to hold this Instructorship until August 2003, although he spent September 2001 - June 2002 as a Visiting Scholar at Harvard University. Remaining at the California Institute of Technology at Pasadena, he was appointed as Taussky-Todd Instructor of Mathematics in September 2003. In January 2004 he was featured inIn1917, Erich Hecke proved a series of results about certain characters which are now commonly referred to as Hecke characters; one corollary states that one-dimensional complex Galois representations give rise to entire L-series. He revealed, through a series of lectures at Princeton's Institute for Advanced Study in the years that followed, the relationship between such representations as generalizations of Dirichlet characters and modular forms as the eigenfunctions of a set of commuting self-adjoint operators. Many mathematicians were inspired by his ground-breaking insight and novel proof of both the analytic continuation of the L-series and its functional equation. In the1930's, Emil Artin conjectured that a generalization of such a result should be true; that is, irreducible complex projective representations of finite Galois groups should also give rise to entire L-series. He came to this conclusion after proving himself that both3-dimensional and4-dimensional representations of the simple group of order60, the alternating groupA_{5}, give rise to L-series with singularities. In the spirit of Hecke, he phrased his conjecture in terms of both the analytic continuation of the L-series and its functional equation. It is known, due to the insight of Robert Langlands in the1970's relating Hecke characters with Representation Theory, that in order to prove the conjecture it suffices to prove that such representations are automorphic. This conjecture and has been the motivation for much study in both Algebraic and Analytic Number Theory ever since.

*Black Issues in Higher Education*as one of the "2004 Emerging Scholars of the Year." He showed that he fully deserved this honour when in August 2004 he was appointed as Assistant Professor of Mathematics at Purdue University in West Lafayette, Indiana.

Goins' first two papers appear in the book

*Council for African American Researchers in the Mathematical Sciences*. Vol. III which was published by the American Mathematical Society in 2001. The book contains papers from the 3rd Conference for African American Researchers in the Mathematical Sciences held at Morgan State University, Baltimore, MD, June 17-20, 1997 and the 5th Conference held at the University of Michigan, Ann Arbor, MI, June 22-25, 1999. One of the two papers is

*The fractional parts of n/k*written jointly with M R Currie and the other is the single-authored paper

*Artin's conjecture and elliptic curves*. As the first of these gives Goins' address as Stanford University while the second gives his address as Institute for Advanced Study Princeton, we conjecture that the first paper was presented at the 1997 conference while the second was presented at the 1999 conference. The paper

*Artin's conjecture and elliptic curves*contains results from his thesis.

Several of Goins' papers are particularly attractive and at least the problems they studied is understandable without a deep knowledge of mathematics. For example his 2006 paper

*Heron triangles via elliptic curves*(written with Davin Maddox) has a review which begins:-

The Abstract of the paper states:-A Heron triangle is a triangle whose sides a, b, c and area n are all rational numbers. In this very pleasant paper, the authors look at the following question: Given n, how can we decide whether there exists a Heron triangle of area equal to n? This question is similar to the congruent number problem, which asks to determine, given n, whether there is a Pythagorean triangle with rational sides and area n.

Another lovely paper is Goins' paperGiven a positive integer n, one may ask if there is a right triangle with rational sides having area n. Such integers are called congruent numbers, and are closely related to elliptic curves of the formy^{2}=x^{3}-n^{2}x. In this paper, we generalize this idea and show that there is a correspondence between positive integers n associated with arbitrary triangles with rational sides having area n and the family of elliptic curvesy^{2}=x(x-nτ)(x+nτ^{-1})for nonzero rationalτ.

*Palindromes in different bases: A conjecture of J Ernest Wilkins*(2009). We quote a review by Clemens Fuchs:-

Goins was awarded many research grants, most having the theme of promoting mathematical participation by minority groups. For example, the conferences 'Purdue Research in Mathematics Experience (PRiME)' brought in:-In this amusing paper the author proves that there exist exactly203positive integers N, which are all explicitly listed, such that N is a palindrome in base10with d ≥2digits(d =1would be trivial)and N is at the same time a palindrome with d digits in a base b ≠10. The smallest such N is22, the largest is9986831781362631871386899, and the possible d's are d =2,3,4,5,6,7,9,11,13,15,17,19,21,23,25. This verifies a conjecture by J Ernest Wilkins that there is no such N with8digits and that the only even d's for which there is such an N are d =2,4,6. The result is obtained from two simple lemmas(both suggested by Wilkins)and exhaustive brute-force computer search implemented in 'Mathematica'. The computer code and the complete output list are given. The parallel search on twenty processors took about fifteen months. The paper ends with an open problem for future research.

A goal of the conference... outside speakers, women of colour in the mathematical sciences, to discuss their professional journey from being an undergraduate student to being a member of the professoriate.

*Underrepresented Students in Topology and Algebra Research Symposium*in 2013 was to bring together young researchers in algebra and topology from diverse backgrounds and to expose undergraduate students to research opportunities. Grants won by Goins in 2016, 2017 and 2018 set up conferences with the aim to "produce a more diverse professional mathematical workforce."

In 2018 Goins published the fascinating article

*Three Questions: The Journey of One Black Mathematician*in which he writes [3]:-

You can read a version of that article, in which he explains why on his personal journey to become a full professor these question were so significant, at THIS LINK.I am the president of the 'National Association of Mathematicians', a professional organization that seeks to increase the public awareness of issues of importance to underrepresented minorities in the mathematical sciences, and I would like to modify these questions and ask you the following:

'How many African American mathematicians have you graduated?'

'How many African American mathematicians have you hired?'

'Why?'

Shortly after writing this article, Goins resigned from Purdue University and accepted a position at Pomona College from July 2018. He explained his decision in a number of articles and interviews, for example in [4]. The article [5]

*For a Black Mathematician, What It's Like to Be the 'Only One'*, published in

*The New York Times*in February 2019, looks at Goins' decision and comments on the reaction, both positive and negative. You can read a version of the article at THIS LINK.

**Article by:** *J J O'Connor* and *E F Robertson*

**List of References** (9 books/articles)

**Mathematicians born in the same country**

**Additional Material in MacTutor**

**Honours awarded to Edray Herber Goins**

(Click the link below for those honoured in this way)

1. | Bharucha-Reid lecturer | 2006 | |

2. | David Blackwell lecturer | 2009 | |

3. | Claytor-Woodard lecturer | 2011 |

**Other Web sites**