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Serge Lang

Serge Lang was an influential French-American mathematician and activist, noted for his number theory work, influential textbooks, and controversial views.

1. Early Life and Education

Serge Lang's early life in France and his subsequent education in the United States laid the foundation for his distinguished career in mathematics and his later public activism.

1.1. Birth and Family

Lang was born in Saint-Germain-en-Laye, a town near Paris, France, on May 19, 1927. He grew up with a twin brother, who later became a basketball coach, and a sister, who pursued a career as an actress. As a teenager, Lang and his family emigrated to the United States, settling in California.

1.2. Education

After moving to California, Lang completed his secondary education, graduating from Beverly Hills High School in 1943. He then pursued his undergraduate studies at the California Institute of Technology, earning his degree in 1946. Lang continued his academic journey at Princeton University, where he obtained his PhD in mathematics in 1951. His doctoral dissertation was titled "On quasi algebraic closure," and he completed his research under the esteemed supervision of Emil Artin. Following his doctoral work, Lang held faculty positions at several prominent institutions, including the University of Chicago, Columbia University (starting in 1955, from which he resigned in 1971 due to a dispute), and Yale University, where he spent most of his career and held an honorary professorship.

2. Mathematical Career

Serge Lang's mathematical career was marked by profound contributions across diverse fields, the formulation of significant conjectures, and his involvement with the influential Bourbaki group.

2.1. Research Areas

Lang's research spanned numerous principal fields within mathematics. Early in his career, he focused on the geometric analogues of class field theory and Diophantine geometry. He later expanded his work into Diophantine approximation and transcendental number theory. His interests also encompassed modular forms and modular units, the concept of a "distribution" on a profinite group, and value distribution theory. Additionally, he contributed to the study of algebraic groups.

2.2. Key Theorems and Conjectures

Throughout his career, Lang formulated and proved several major theorems and conjectures that significantly advanced various mathematical disciplines. Among his notable contributions are the Schneider-Lang theorem, which he proved, and several conjectures in Diophantine geometry, including the Mordell-Lang conjecture, the Bombieri-Lang conjecture, the Lang-Trotter conjecture, and the Lang conjecture on analytically hyperbolic varieties. He also introduced the Lang map, contributed to the Katz-Lang finiteness theorem, and was associated with the Lang-Steinberg theorem (related to Lang's theorem) in algebraic groups.

2.3. Bourbaki Group

Serge Lang was an active member of the Bourbaki group (Nicolas Bourbaki), an influential collective of primarily French mathematicians. This group is known for its efforts to reformulate mathematics on a rigorous and abstract basis, producing a series of textbooks that have significantly shaped modern mathematical education and research. Lang's involvement with Bourbaki underscores his commitment to the foundational aspects of mathematics and his collaborative spirit within the global mathematical community.

3. Mathematical Writing and Teaching

Serge Lang was renowned for his prolific output of influential mathematics textbooks and his distinctive, passionate approach to teaching.

3.1. Influential Textbooks

Lang was an exceptionally prolific writer of mathematical texts, often completing entire books during his summer vacations. The majority of his works were aimed at the graduate level, though he also authored undergraduate texts. His most celebrated work, Algebra, a graduate-level introduction to the subject, became a highly influential text that underwent numerous updated editions. This book is credited with transforming the way graduate algebra was taught, significantly impacting all subsequent graduate-level algebra textbooks. It incorporated ideas from his teacher, Emil Artin, and some of the most insightful passages in his Algebraic Number Theory also reflect Artin's influence, preserving ideas that might otherwise have remained unpublished. He also prepared a book on group cohomology for the Bourbaki group and authored several calculus texts, including A First Course in Calculus and Calculus of Several Variables.

3.2. Teaching Style and Pedagogy

Lang was known for his energetic and passionate teaching style, which was sometimes confrontational. He was described as a teacher who would occasionally throw chalk at students if he felt they were not paying sufficient attention. A colleague recalled his dedication to clarity and truth in the classroom, quoting him as saying, "Our two aims are truth and clarity, and to achieve these I will shout in class." This approach reflected his deep commitment to ensuring students grasped complex mathematical concepts with precision.

4. Activism and Public Engagement

Serge Lang was a prominent public intellectual and activist, known for his strong opinions and willingness to challenge established norms and figures. His activism, while driven by a desire for truth and accountability, also included controversial stances that drew significant criticism.

4.1. Opposition to the Vietnam War

Lang was a staunch socialist and a fervent opponent of the Vietnam War. He actively participated in the anti-war movement, volunteering for the 1966 anti-war campaign of Robert Scheer, which he documented in his book The Scheer Campaign. His commitment to the cause led to his resignation from his position at Columbia University in 1971, in protest over the university's handling of anti-war demonstrators.

4.2. Critiques of Academia and Science

Lang dedicated considerable effort to challenging individuals and institutions he believed were disseminating misinformation or misusing science and mathematics. He sharply criticized the 1977 Survey of the American Professoriate, an opinion questionnaire sent to thousands of college professors by Seymour Martin Lipset and E. C. Ladd. Lang argued that the survey contained numerous biased and leading questions, leading to a public and acrimonious conflict detailed in his book The File: Case Study in Correction (1977-1979).

In 1986, Lang launched a notable "one-man challenge" against the nomination of political scientist Samuel P. Huntington to the National Academy of Sciences. Lang labeled Huntington's research, particularly his use of mathematical equations to assert that South Africa was a "satisfied society," as "pseudoscience." He contended that Huntington's work presented "the illusion of science without any of its substance." Despite support for Huntington from the Academy's social and behavioral scientists, Lang's challenge proved successful, and Huntington was rejected for Academy membership twice. Huntington's supporters, however, maintained that Lang's opposition was politically motivated rather than scientifically based. Lang extensively documented these events in "Academia, Journalism, and Politics: A Case Study: The Huntington Case," which comprises the initial 222 pages of his 1998 book Challenges.

Lang meticulously maintained extensive "files" of his political correspondence and related documentation. He would send letters, publish articles, engage in further correspondence with respondents, and then compile all these writings to highlight what he perceived as contradictions. He frequently mailed these files to mathematicians and other interested parties globally. Some of these files were later published in his books Challenges and The File: Case Study in Correction (1977-1979). His detailed critique of Nobel laureate David Baltimore was published in the journal Ethics and Behavior in January 1993 and also included in Challenges. Lang also opposed Yale University's decision to hire Daniel Kevles, a historian of science, due to his disagreement with Kevles' analysis in The Baltimore Case.

4.3. HIV/AIDS Denialism

In the final twelve years of his life, Serge Lang became a prominent figure in the HIV/AIDS denialist movement. He publicly challenged the scientific consensus that HIV causes AIDS, arguing that existing data did not sufficiently support this conclusion. Lang actively protested Yale University's research into HIV/AIDS and voiced opposition to the appointment of Michael Merson, former Global AIDS Program Director at the World Health Organization, as Yale's Dean of Public Health. His involvement in this movement included conducting a flawed analysis of Duesberg's grant failings and questioning the entire NIH review process, leading to significant controversy on the Yale campus. A portion of his book Challenges is devoted to this issue, reflecting his persistent advocacy against the established scientific understanding of HIV/AIDS.

5. Awards and Recognition

Serge Lang received significant awards and honors for his contributions to mathematics and mathematical exposition, acknowledging his impact on both research and education.

5.1. Mathematical Honors

In 1960, Lang was awarded the sixth Frank Nelson Cole Prize in Algebra by the American Mathematical Society for his paper "Unramified class field theory over function fields in several variables," published in Annals of Mathematics (Series 2, volume 64 (1956), pages 285-325). He also received the Leroy P. Steele Prize for Mathematical Exposition in 1999 from the American Mathematical Society. His Steele Prize citation specifically noted that "Lang's Algebra changed the way graduate algebra is taught...It has affected all subsequent graduate-level algebra books," underscoring the profound impact of his pedagogical works.

6. Publications

Serge Lang was a prolific author of mathematical texts, encompassing a wide range of subjects from undergraduate calculus to advanced graduate-level research monographs. His published works include:

  • Pregraduate-level textbooks
    • A First Course in Calculus (1964, fifth edition 1986). The 1964 first edition was reprinted as Short Calculus: The Original Edition of "A First Course in Calculus" (2002).
    • Introduction to Linear Algebra (1970, second edition 1986).
    • Calculus of Several Variables (1973, third edition 1987). Originally published as A Second Course in Calculus (1965).
    • Linear Algebra (1966, third edition 1987). A solutions manual for this book was published by Rami Shakarchi in 1996.
    • Basic Mathematics (1971, reprint 1988).
    • Geometry: A High School Course (1988), co-authored with Gene Murrow.
    • Undergraduate Analysis (1997, second edition). The first edition (1983) was the second edition of Analysis I (1968). A problems and solutions book for Undergraduate Analysis was published by Rami Shakarchi in 1998.
    • Complex Analysis (1977, fourth edition 1999). A problems and solutions book for Complex Analysis was published by Rami Shakarchi in 1999.
    • Undergraduate Algebra (1990, third edition 2005). The 1990 first edition was a second edition of Algebraic Structures (1967).
    • Graduate-level textbooks
      • Introduction to Transcendental Numbers (1966).
      • Introduction to Algebraic Geometry (1959, third printing 1972).
      • Frobenius Distributions in GL2-Extensions (1976), co-authored with Hale Trotter.
      • Elliptic Curves: Diophantine Analysis (1978).
      • Modular Units (1981), co-authored with Daniel S. Kubert.
      • Introduction to Algebraic and Abelian Functions (1972, second edition 1982).
      • Abelian Varieties (1959, reprint 1983).
      • Complex Multiplication (1983).
      • Fundamentals of Diophantine Geometry (1983). This was the second edition of Diophantine Geometry (1962).
      • Riemann-Roch Algebra (1985), co-authored with William Fulton.
      • SL2(R) (1975, reprint 1985).
      • Elliptic Functions (1973, second edition 1987), with an appendix by John Tate.
      • Introduction to Complex Hyperbolic Spaces (1987).
      • Introduction to Arakelov Theory (1988).
      • Cyclotomic Fields I and II (combined second edition of 1978/1980 original, 1990), with an appendix by Karl Rubin.
      • Topics in Nevanlinna Theory (1990), co-authored with William Cherry, with an appendix by Zhuan Ye.
      • Real and Functional Analysis (1993, third edition). Previously published as Analysis II (1968) and Real Analysis (1983).
      • Basic Analysis of Regularized Series and Products (1993), co-authored with Jay Jorgenson.
      • Algebraic Number Theory (1970, second edition 1994). The first edition was the second edition of Algebraic Numbers (1964).
      • Introduction to Diophantine Approximations (1966, second edition 1995).
      • Introduction to Modular Forms (1976, corrected reprint 1995), with appendixes by D. Zagier and Walter Feit.
      • Topics in Cohomology of Groups (1996), translated from the 1967 French original, with Chapter X based on letters by John Tate.
      • Survey of Diophantine Geometry (1997).
      • Fundamentals of Differential Geometry (1999, fourth edition). Previously published as Introduction to Differentiable Manifolds (1962), Differential Manifolds (1972), and Differential and Riemannian Manifolds (1995). Lang also published a distinct second edition of Introduction to Differentiable Manifolds (2002) as a companion volume.
      • Spherical Inversion on SLn(R) (2001), co-authored with Jay Jorgenson.
      • Algebra (1965, revised third edition 2002).
      • Posn(R) and Eisenstein Series (2005), co-authored with Jay Jorgenson.
      • The Heat Kernel and Theta Inversion on SL2(C) (2008), co-authored with Jay Jorgenson.
      • Heat Eisenstein Series on SLn(C) (2009), co-authored with Jay Jorgenson.
      • Other
        • The File: Case Study in Correction (1977-1979) (1981).
        • The Beauty of Doing Mathematics: Three Public Dialogues (1985), translated from French.
        • Math!: Encounters with High School Students (1985).
        • Challenges (1998).
        • Math Talks for Undergraduates (1999).
        • Collected Papers. I. 1952-1970 (2000).
        • Collected Papers. II. 1971-1977 (2000).
        • Collected Papers. III. 1978-1990 (2000).
        • Collected Papers. IV. 1990-1996 (2000).
        • Collected Papers. V. 1993-1999 (2001).

7. Legacy and Assessment

Serge Lang's legacy is dual-faceted, marked by his immense contributions to mathematics and his controversial role as a public figure. In mathematics, his impact is undeniable; his textbooks, particularly Algebra, revolutionized graduate-level education and influenced generations of mathematicians. His research advanced fields such as number theory, algebraic geometry, and Diophantine geometry, leaving a lasting mark on theoretical mathematics.

However, his legacy is also shaped by his outspoken activism and later-life controversial views. While his opposition to the Vietnam War and his critiques of academic integrity were seen by many as principled stands against injustice and misinformation, his later embrace of HIV/AIDS denialism drew widespread condemnation from the scientific community and public health advocates. This stance, which challenged established scientific consensus, casts a shadow on his otherwise distinguished career and highlights the complex interplay between intellectual rigor and public responsibility. Lang's unwavering commitment to what he perceived as truth, even when it led him into conflict with mainstream views, defines his unique and often polarizing public persona.