MathematicianSovietPrize winnerTopologist

Lazar Lyusternik

Lazar Lyusternik was a Soviet mathematician noted for contributions to topology and differential geometry, including the Lusternik-Schnirelmann theory and the Theorem of Three Geodesics.

1. Overview

Lazar Aronovich Lyusternik was a prominent Soviet mathematician known for his significant contributions to the fields of topology and differential geometry. He is particularly recognized for his innovative application of the variational principle to these areas. Collaborating with Lev Schnirelmann, Lyusternik co-proved the theorem of the three geodesics, a long-standing conjecture by Henri Poincaré. This achievement, along with the development of the influential Lusternik-Schnirelmann theory, earned him the prestigious Stalin Prize in 1946. However, his career was also marked by controversial involvement in politically charged academic purges, including the denunciation of Dmitri Egorov and the persecution of his own mentor, Nikolai Luzin, during the Luzin affair. His mathematical legacy endures through concepts such as the Lusternik-Schnirelmann category and the Lusternik-Petrov theorem, which continue to influence modern mathematics.

2. Early Life and Education

Lazar Aronovich Lyusternik's early life and educational journey laid the foundation for his distinguished mathematical career.

2.1. Birth and Early Life

Lazar Aronovich Lyusternik (also spelled Lusternik, Lusternick, or Ljusternik; Лазарь Аронович ЛюстерникRussian; Łazar LusternikWażar RusternikPolish) was born on December 31, 1899, in Zduńska Wola, a town then located within the Congress Poland region of the Russian Empire. His early years were spent in this environment, which was part of the broader political and social landscape of the Russian Empire. He died on July 22, 1981.

2.2. Education and Academic Mentorship

Lyusternik pursued his higher education in mathematics, where he came under the significant influence of the renowned mathematician Nikolai Luzin. Luzin served as Lyusternik's mentor, guiding his academic development and shaping his foundational understanding of mathematical principles. This mentorship was crucial in Lyusternik's formation as a mathematician.

3. Mathematical Career and Research

Lyusternik's professional life as a mathematician spanned several key Soviet institutions, where he held various academic and research positions while focusing on his primary research interests.

3.1. Academic Positions

Throughout his career, Lyusternik held significant academic and research roles within the Soviet Union. He served as a professor of mathematics at Moscow State University, one of the most prestigious educational institutions in the country. In addition to his professorial duties, Lyusternik was also affiliated with the Steklov Mathematical Institute (part of the Russian Academy of Sciences), where he conducted research from 1934 to 1948. Following this, he transitioned to the Lebedev Institute of Precise Mechanics and Computer Engineering (IPMCE), contributing to its work from 1948 to 1955.

3.2. Research Areas

Lyusternik's primary research interests lay in the fields of topology and differential geometry. A hallmark of his approach was the innovative application of the variational principle to problems within these areas. This methodology allowed him to explore complex geometric and topological structures through the lens of optimization and critical points, leading to profound insights and significant breakthroughs.

4. Key Mathematical Contributions

Lyusternik's most important mathematical achievements include the co-proof of the Theorem of Three Geodesics and the development of the Lusternik-Schnirelmann Theory, which have had a lasting impact on mathematics.

4.1. Theorem of Three Geodesics

One of Lyusternik's most notable achievements was the co-proof of the theorem of the three geodesics with his collaborator, Lev Schnirelmann. This theorem addressed a long-standing conjecture proposed by the influential French mathematician Henri Poincaré. The conjecture posited that every convex body in three-dimensional space has at least three simple closed geodesics. Lyusternik and Schnirelmann's work provided a rigorous proof for this assertion, marking a significant advance in differential geometry. A particularly interesting aspect of this theorem is its application to an ellipsoid with distinct but nearly equal axes; in this specific, critical case, such an ellipsoid is shown to possess exactly three closed geodesics.

4.2. Lusternik-Schnirelmann Theory

Building upon their work on geodesics, Lyusternik and Schnirelmann developed what is now known as the Lusternik-Schnirelmann theory. This theory is deeply rooted in the foundational work of earlier mathematicians, including Henri Poincaré, David Birkhoff, and Marston Morse. It provides powerful tools for studying the global properties of manifolds by relating the number of critical points of a function to the topological structure of the underlying space. The development of this theory has had far-reaching implications, leading to numerous advances in both differential geometry and topology, and remains a fundamental concept in these fields.

4.3. Awards and Recognition

For his groundbreaking work, particularly the development of the Lusternik-Schnirelmann theory, Lyusternik received significant recognition from the Soviet state. In 1946, he was awarded the prestigious Stalin Prize, one of the highest honors for scientific achievement in the Soviet Union. This award underscored the importance and impact of his contributions to mathematics within the Soviet academic and political context.

5. Controversies and Political Involvement

Lyusternik's career was not without controversy, marked by his involvement in politically charged events within the Soviet academic system that had severe consequences for his colleagues and mentors.

5.1. The Luzin Affair

Lyusternik played a significant and controversial role in the political purges that swept through the Soviet scientific community during the 1930s. In 1930, he was among the initiators of the Egorov affair, an event that saw the prominent mathematician Dmitri Egorov denounced as a "counter-revolutionary element." This denunciation led to Egorov's persecution and eventual death. Later, in 1936, Lyusternik became a participant in the notorious political persecution of his own teacher and mentor, Nikolai Luzin, an event widely known as the Luzin affair. This affair involved a public campaign of denunciation and accusations against Luzin, orchestrated by the Soviet authorities and supported by some of his former students, including Lyusternik. Lyusternik's involvement in these events highlights the immense political pressures exerted on academics during the Stalinist era, where conformity to ideological directives often superseded academic integrity and personal loyalties, leading to severe consequences for those targeted.

6. Legacy and Influence

Lyusternik's mathematical work continues to have an enduring impact, influencing subsequent research and mathematicians, with several key concepts named in his honor.

6.1. Mathematical Concepts Named After Him

Lyusternik's lasting contributions to mathematics are reflected in several concepts and theorems that bear his name. Among these is the Lusternik-Schnirelmann category, a fundamental topological invariant that measures the "complexity" of a topological space in terms of the minimum number of open sets required to cover it, where each set is contractible within the space. Another concept is Lyusternik's generalization of the Brunn-Minkowski theorem, which extends a classical result in convex geometry. Additionally, the Lusternik-Petrov theorem is also named in his honor, further solidifying his place in mathematical history.

6.2. Broader Impact on Mathematics

Lyusternik's research significantly advanced the fields of topology and differential geometry. His innovative application of the variational principle provided new methodologies and insights, particularly in the study of geodesics and the global properties of manifolds. The theories and theorems he developed, especially the Lusternik-Schnirelmann theory, have become indispensable tools for mathematicians, influencing subsequent generations of researchers and contributing to the ongoing development of these complex and interconnected areas of mathematics.