Evgeny Golod

Evgeny Golod was a Soviet and Russian mathematician noted for foundational work in algebra and number theory, especially the Golod–Shafarevich theorem and its consequences for class field towers and infinite groups. His research influenced developments in Algebraic number theory, Group theory, Homological algebra, and the study of pro-p groups, affecting problems posed by figures such as Emil Artin, Kummer, and Helmut Hasse. Golod collaborated with leading mathematicians of the Soviet school and his results continue to be cited in modern work on Galois theory, Iwasawa theory, and the construction of exotic examples in Combinatorial group theory.

Early life and education

Golod was born in Moscow in 1935 and grew up during the late period of the Soviet Union where scientific institutions like Moscow State University and the Steklov Institute of Mathematics played central roles in mathematical training. He studied at Moscow State University under the supervision of Igor Shafarevich, interacting with contemporaries from the Soviet mathematical community including members of the Egorov Circle and researchers connected to the Keldysh Mathematical School. His doctoral work addressed problems in class field theory related to questions raised by David Hilbert and later reformulated by Artin and Tate, situating him within the lineage of the Russian algebraists who worked on explicit constructions in algebraic number fields.

Mathematical career and contributions

Golod's early publications investigated relations between algebraic structure and cohomological invariants, building on techniques from Jean-Pierre Serre and the cohomology of groups. He developed methods combining power series, dimension counting, and homological algebra that were later recognized as tools in analyzing pro-p groups and group algebras over fields of characteristic p. His approach connected with prior work of Noether, Emmy Noether, and the structural investigations of Alexander Grothendieck in algebraic geometry, while also resonating with combinatorial techniques used by Max Dehn and Otto Schreier in group presentations. Golod's papers influenced research on presentations of associative algebras, the growth of graded algebras, and the construction of algebras with prescribed homological dimensions, themes that appear in the studies of Anatoly Maltsev, Israel Gelfand, and Israel Kleiner.

Major results and the Golod–Shafarevich theorem

Golod's most famous result, obtained in collaboration with Igor Shafarevich, produced what is now known as the Golod–Shafarevich theorem. The theorem gives sufficient conditions, expressed in terms of generators and relations or cohomological data, for a pro-p group or an algebra to be infinite or to have exponential growth. This resolved long-standing questions related to the class field tower problem originally formulated by Hilbert and refined by Emil Artin and Helmut Hasse, exhibiting infinite class field towers and providing counterexamples to expectations influenced by Kronecker and Weber. The Golod–Shafarevich construction has been applied to produce infinite finitely generated torsion groups, linking to work by Gromov on growth of groups and to examples built by Burnside and later by Novikov and Adian. The theorem also informed progress in Iwasawa theory and informed constructions in the theory of p-adic Lie groups, intersecting with research of John Coates and Ralph Greenberg.

Academic positions and collaborations

Golod held positions at major Soviet and Russian institutions, including Moscow State University and the Steklov Institute of Mathematics, where he worked alongside colleagues such as Igor Shafarevich, Yuri Manin, and Israel Gelfand. He maintained collaborations and correspondence with international mathematicians in France, Germany, and the United States, connecting to scholars like Jean-Pierre Serre, Serge Lang, and Alexander Lubotzky. His doctoral students and collaborators continued research on cohomological methods, linking to later generations working with Mikhail Gromov, Efim Zelmanov, and researchers in Combinatorial group theory and Number theory who pursued extensions of Golod's techniques to new contexts, including applications in Geometric group theory and the study of profinite groups.

Honors and legacy

Golod received recognition from the mathematical community in the Soviet Union and internationally for his influential contributions; his theorem remains a central tool referenced in surveys of algebraic number theory and group theory by authors such as Jean-Pierre Serre and Serge Lang. The Golod–Shafarevich criterion is taught in graduate courses on Galois theory and appears in monographs on profinite groups and class field theory authored by specialists like John Neukirch and Serge Lang. Golod's legacy persists in contemporary research on torsion phenomena, growth in algebras, and explicit constructions that challenge intuitive conjectures, inspiring ongoing work by mathematicians including Alexander Lubotzky, Efim Zelmanov, Jean-Pierre Serre, and Mikhail Kapranov.

Category:Russian mathematicians Category:1935 births Category:2023 deaths