Emmy Noether (1882–1935) stands out as one of the most brilliant mathematicians of her era, despite facing systemic barriers that kept women on the fringes of academic life. She was a trailblazer in the creation of abstract algebra and established a foundational theorem that reshaped modern physics. Being Jewish, she was forced to escape Nazi Germany in the 1930s and tragically passed away only a few years after relocating to the United States.

“Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began.”

These were the words of Albert Einstein, written in a 1935 letter to the New York Times shortly after Noether’s death. His admiration echoed a broader consensus: many of her peers and successors regard her as one of the most extraordinary minds in mathematics.

But what exactly did Einstein mean by “creative mathematical genius”? Noether’s primary field was algebra, which focuses on expressing mathematical ideas through symbols and abstract structures. For instance, rather than viewing the equations $$2x = 10$$ and $$3x = 9$$ as entirely separate problems, one can identify them as part of a broader category of linear equations with the general form $$ax = b$$. The solution $$x = \frac{b}{a}$$ then applies universally to any such equation, provided values for $a$ and $b$ are known.

Emmy Noether (circa 1900–1910)

More broadly, algebra serves as a language to articulate the underlying relationships between mathematical entities. These relationships often reveal consistent patterns and structures that, much like musical harmonies, can be intuitively felt by those deeply attuned to them. Just as composers adhere to musical principles, mathematicians rely on the rigour of logical frameworks to derive new insights. Einstein himself described pure mathematics as the “poetry of logical ideas.”

In 1921, Noether released her seminal work, Ideal Theory in Rings, a paper that became a cornerstone of contemporary algebra. Yet her most widely recognized contribution is undoubtedly Noether’s Theorem — a revolutionary link between the physical universe and the abstract concept of symmetry.

The power of symmetry

In physics, certain quantities are preserved over time — these are called conservation laws. Energy, for instance, cannot be conjured from nothing nor can it vanish into thin air. When you kick a ball, the energy from your foot transfers into the ball’s motion. Momentum — calculated as mass times velocity — is another conserved quantity. After a kick, your foot may stop moving, but its momentum persists in the ball. These principles are fundamental to the way we understand physical systems.

However, when Einstein introduced his groundbreaking general theory of relativity in 1915, a dilemma emerged. Renowned mathematicians like David Hilbert and Felix Klein noticed that this new framework didn’t seem to preserve energy in the familiar way other theories did.

The issue was deeply rooted in mathematics and required someone with expertise in invariants — mathematical entities that remain constant under transformation. Emmy Noether, already renowned for her algebraic prowess, was the ideal person to tackle this challenge, and she did so at Hilbert’s invitation.

With extraordinary clarity, Noether demonstrated how conservation laws are embedded within the mathematical language of physics. She showed that the invariance of physical equations under spatial translations corresponds to the conservation of momentum. Similarly, the fact that equations remain unchanged over time explains the conservation of energy. Whether you’re on Earth, the Moon, or in another galaxy, and whether it’s today or a century from now — the governing equations of physics remain consistent.

Mathematically, an object that doesn’t change under a particular transformation is said to exhibit symmetry. For example, a butterfly’s image retains its form when reflected down its centerline — a classic case of mirror symmetry. Similarly, physical laws that hold true across time and space exhibit symmetry under those transformations. Noether’s theorem elegantly connects this notion of symmetry with the laws that govern conserved quantities in physics.

Though she was a mathematician, Noether managed to forge a profound connection between these symmetrical properties and the core laws of physics. This link exists because symmetry — a central idea in group theory — is expressible using algebraic structures. Thus, Noether, through her deep understanding of algebra, was able to unveil one of the most significant insights in theoretical physics.

Emmy Noether and the power of symmetry — Discover how her insights created a lasting bond between physics and the mathematics of symmetry.

Against the odds

Noether’s intellectual legacy has profoundly influenced both mathematics and physics. Leading thinkers such as Einstein, Hilbert, and Klein recognized the depth of her contributions. Yet her path in academia was riddled with obstacles.

When she began her university education in 1900, women were still denied formal degrees. She had to request special permission to attend lectures. Only after a policy change in 1904 was she allowed to graduate, eventually earning a doctorate with distinction in 1907. Despite this, she was barred from official academic positions for much of her career, often working without proper salary or title.

In the 1930s, as Germany’s academic institutions became entangled with Nazi ideology, Noether was among the first faculty members expelled from the University of Göttingen. Her Jewish heritage and liberal beliefs made her a target. She later secured temporary posts at American universities, but her time in the United States was short. She died of ovarian cancer while in exile, still in her early fifties, at the height of her intellectual powers.

Emmy Noether: Against the odds — Learn more about her extraordinary personal journey and the barriers she overcame as both a woman and a Jewish mathematician.

This article is part of a collaborative project with the Isaac Newton Institute for Mathematical Sciences (INI), an international research hub located next to our own campus at the University of Cambridge.