Robert Hermann (1931–2020): The Architect of Interdisciplinary Geometry
Robert Hermann was a mathematical polymath whose career spanned the second half of the 20th century. A "maverick" in the truest sense, Hermann spent his life demonstrating that the abstract tools of differential geometry and Lie theory were not merely intellectual playthings for pure mathematicians, but the essential language of modern physics, control theory, and engineering.
1. Biography: A Journey Through the Academic Vanguard
Robert Hermann was born on April 28, 1931, in Brooklyn, New York. His mathematical journey began during a period of immense growth for American science. He attended Brown University for his undergraduate studies before moving to Princeton University, then the global epicenter of mathematical research.
At Princeton, Hermann worked under the tutelage of Donald Spencer, a giant in the field of differential geometry. He earned his Ph.D. in 1955 with a dissertation focused on the theory of connections in fiber bundles. Following his doctorate, Hermann held a series of prestigious positions:
- 1955–1957: Research Associate at the University of Chicago and Harvard.
- 1960s: Faculty positions at Northwestern University and Rutgers University.
- The Independent Path: By the mid-1970s, Hermann grew disillusioned with the rigid silos of traditional academia. He sought a more flexible environment to pursue his "Interdisciplinary Mathematics" vision. This led him to establish Math Sci Press in Brookline, Massachusetts, where he served as an independent scholar, publisher, and researcher for the remainder of his life.
2. Major Contributions: Bridging Abstract Geometry and Reality
Hermann’s work was defined by "geometric thinking." He believed that physical and engineering systems possessed underlying geometric structures that, if understood, could solve complex problems.
Geometric Control Theory
Hermann was a founding father of this field. He applied Lie group theory and differential geometry to linear and non-linear systems. He famously showed that the "controllability" of a system could be determined by the properties of Lie brackets of vector fields—a result now fundamental to robotics and aerospace engineering.
The Geometry of Physics
Long before "Gauge Theory" became a household term in physics, Hermann was translating the work of physicists into the language of fiber bundles and manifolds. He was among the first to provide a rigorous mathematical framework for Yang-Mills fields and General Relativity.
Integrable Systems (Solitons)
In the 1970s, Hermann made significant contributions to the study of the Korteweg-de Vries (KdV) equation. Working with Wahlquist and Estabrook, he developed the Wahlquist-Estabrook procedure, a geometric method for finding "pseudopotentials" and Bäcklund transformations for non-linear partial differential equations.
The Hermann-Martin Theorem
In collaboration with Clyde Martin, he developed a geometric approach to the "pole placement" problem in linear systems theory, linking algebraic geometry to feedback control.
3. Notable Publications: A Prolific Output
Hermann was an incredibly prolific author, writing over 30 books and 100 research papers. His most significant works include:
- Differential Geometry and the Calculus of Variations (1968): A foundational text that introduced a generation of mathematicians to the geometric underpinnings of optimization.
- Lie Groups for Physicists (1966): One of the first accessible bridges between advanced group theory and particle physics.
- Interdisciplinary Mathematics Series (1973–2000s): This was Hermann’s magnum opus—a series of over 25 volumes published through his own Math Sci Press. These volumes covered everything from "Gauge Fields and Cartan-Ehresmann Connections" to "Algebraic Topics in Systems Theory."
- Geometry, Physics, and Systems (1973): A seminal work that argued for the unification of these disciplines through differential geometry.
4. Awards & Recognition
While Hermann’s "outsider" status as an independent publisher meant he was sometimes removed from the standard academic prize circuit, his peers recognized his brilliance:
- Guggenheim Fellowship (1966): Awarded for his work in mathematics.
- Fellow of the American Physical Society (APS): A rare honor for a mathematician, acknowledging his profound impact on theoretical physics.
- National Science Foundation (NSF) Grants: He was a frequent recipient of research funding during his tenure at Rutgers and Northwestern.
5. Impact & Legacy
Hermann’s legacy is found in the "geometric turn" taken by 21st-century engineering and physics.
Control Theory
Every time a modern drone or autonomous vehicle uses non-linear control algorithms, it is utilizing the geometric foundations Hermann helped build.
Mathematical Physics
He was a precursor to the modern era of "string theory" and "quantum geometry," where the distinction between a physicist and a geometer is almost non-existent.
Education
His books, though often dense, served as the primary resource for researchers looking to "cross over" from one field to another. He democratized advanced mathematics by making it relevant to applied scientists.
6. Collaborations
Hermann was a deeply collaborative thinker who thrived on interacting with people outside his primary discipline.
- Clyde Martin: His most significant long-term collaborator in the field of systems and control theory.
- H.D. Wahlquist and F.B. Estabrook: Collaborators in the study of non-linear waves and general relativity.
- The Physics Community: He spent significant time at CERN and the Argonne National Laboratory, working directly with physicists to translate their experimental observations into geometric models.
7. Lesser-Known Facts
The "Maverick" Publisher
Hermann founded Math Sci Press because he was frustrated with the high costs and slow speeds of commercial publishers like Springer or Academic Press. He wanted to get ideas into the hands of researchers quickly and cheaply.
The "Hermann Style"
His books were often written in a "stream-of-consciousness" mathematical style. He would frequently include personal asides, philosophical musings on the state of science, and critiques of the academic establishment within his technical proofs.
A Lifelong Student of Sophus Lie
Hermann was obsessed with the 19th-century mathematician Sophus Lie. He spent much of his later career translating and re-interpreting Lie’s original works, believing that modern mathematicians had forgotten the "true" intuitive power of Lie’s ideas.
An Early Advocate for Computers
Despite being a "pure" geometer, he was an early proponent of using symbolic computation (like MACSYMA, the precursor to Mathematica) to handle the grueling calculations involved in Lie theory.
Robert Hermann passed away in February 2020, leaving behind a vast body of work that continues to serve as a map for those navigating the intersection of abstract thought and physical reality. He remains a towering figure for any researcher who refuses to be confined by the boundaries of a single department.