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Heinz Hopf

Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician whose pioneering work in algebraic topology, including the introduction of the Hopf fibration and invariant, profoundly shaped modern homotopy theory and geometric structures.^[1]^[2]
Born in Gräbschen near Breslau (present-day Wrocław, Poland) to a family of mixed Jewish-Protestant heritage, Hopf studied at universities in Breslau, Heidelberg, and Berlin, earning his PhD in 1925 from the University of Berlin under Erhard Schmidt for a dissertation on topological-metric connections in manifolds.^[1]^[2]
After habilitation in Göttingen and a visiting stint at Princeton, he joined ETH Zurich in 1931 as professor, succeeding Hermann Weyl, and remained there until retirement, fostering a hub for topological research.^[1]^[2]
In 1931, Hopf defined the fibration S^3 \to S^2, a circle bundle demonstrating non-trivial homotopy groups and the first example of a fiber bundle beyond trivial cases, which extended fixed-point theorems and bridged combinatorial and differential topology.^[1]
His extensions of Lefschetz's fixed-point formula, studies on homology of Lie groups, and co-authorship of foundational texts like Topology I with Pavel Aleksandrov solidified his influence, earning honors such as the presidency of the International Mathematical Union (1955–1958).^[1]^[2]

Early Life and Education

Family Background and Childhood

Heinz Hopf was born on November 19, 1894, in Gräbschen, a suburb of Breslau (now Wrocław, Poland), then part of the German Empire.^[1]^[2] His father, Wilhelm Hopf, originated from a Jewish family and entered the brewing industry by partnering with Heinrich Kirchner, eventually taking over the family brewery in Gräbschen.^[1] Hopf's mother was Elisabeth Kirchner, linking the family through her relation to the brewery's prior ownership.^[1]^[3] From 1901 to 1904, Hopf attended the private school of Dr. Karl Mittelhaus in Breslau, marking the beginning of his formal education in the region.^[2] Limited details survive on his early childhood beyond this institutional start, though the family's brewing enterprise provided a stable, middle-class environment amid the industrializing Prussian Silesia.^[1] No records indicate siblings or other immediate family dynamics influencing his youth.^[1]

Formal Education and Influences

Hopf attended Dr. Karl Mittelhaus's higher boys' school in Breslau from 1901 to 1904, followed by the König-Wilhelm-Gymnasium until 1913, where he demonstrated early mathematical aptitude.^[1]^[2] In April 1913, he enrolled at the Silesian Friedrich Wilhelms University in Breslau to study mathematics, with instructors including Adolf Kneser, Erhard Schmidt, Rudolf Sturm, Max Dehn, and Ernst Steinitz.^[1]^[2] His studies were interrupted in 1914 by service in the German army during World War I, from which he was discharged in 1918; he briefly resumed coursework in Breslau for about a year afterward.^[1] In 1919, Hopf transferred to the University of Heidelberg, studying under Oskar Perron and Paul Stäckel, before moving to the University of Berlin in 1920.^[1] At Berlin, he worked under Erhard Schmidt and Issai Schur, earning his doctorate in 1925 with a dissertation on the topology of manifolds, classifying simply connected Riemannian three-manifolds of constant curvature; Ludwig Bieberbach served as an examiner alongside Schmidt.^[1]^[2] Schmidt profoundly influenced Hopf by guiding him through proofs of the Jordan curve theorem and L.E.J. Brouwer's topological invariance of dimension, fostering his interest in topology.^[1] Early exposure to Kneser's lectures on geometry and analysis further shaped his foundational approach to mathematics.^[2]

Academic Career

Initial Academic Positions

Following his Ph.D. from the University of Berlin in 1925, under the supervision of Erhard Schmidt, Hopf relocated to Göttingen to prepare his habilitation thesis, which he completed there by autumn 1926 and which introduced a proof of the invariance of the Euler characteristic for closed manifolds.^[1]^[2] He subsequently returned to the University of Berlin, where he served as a Privatdozent, delivering lectures and supervising doctoral students, including Hans Freudenthal's 1929 thesis on topological groups.^[2]^[4] In 1927–1928, Hopf held a visiting research position at Princeton University, funded by a Rockefeller Foundation fellowship, during which he collaborated with Pavel Aleksandrov, Solomon Lefschetz, Oswald Veblen, and James Waddell Alexander on foundational problems in algebraic topology.^[1] This period advanced his work on manifold invariants and homology theory. In December 1929, Princeton offered him an assistant professorship, which he declined in favor of continuing his position in Berlin.^[1] Hopf remained a Privatdozent at the University of Berlin until 1931, focusing on research in topology and geometry amid a competitive academic environment, before accepting a full professorship at ETH Zurich to succeed Hermann Weyl.^[2]^[4] This transition marked his elevation from an unsalaried lecturing role dependent on student fees to a tenured chair with institutional support.^[1]

Professorship and Research Leadership at ETH Zurich

Heinz Hopf was appointed full professor of mathematics at ETH Zurich in April 1931, succeeding Hermann Weyl who had departed for the Institute for Advanced Study.^[1] He held this position until his retirement in 1965, during which he became recognized as one of the institution's most distinguished professors.^[5] At ETH Zurich, Hopf continued to lead advancements in algebraic topology, defining the Hopf invariant in 1931 as a key topological measure for mappings between spheres, which provided insights into higher-dimensional structures.^[1] His 1939 examination of the homology groups of compact Lie groups laid foundational groundwork for the development of Hopf algebras, influencing subsequent algebraic structures in geometry and representation theory.^[1] In the early 1940s, he contributed to homological algebra, extending methods for computing invariants in topological spaces.^[1] Hopf's research leadership at ETH fostered a rigorous environment for topological studies, building on his earlier collaborations and inspiring departmental focus on global manifold properties and fixed-point theorems.^[1] His tenure coincided with ETH's emergence as a hub for pure mathematics amid European upheavals, where he prioritized empirical verification through precise homological computations over speculative generalizations.^[6]

International Roles and Administration

Heinz Hopf served as chairman of the selection committee for the first Fields Medals, awarded at the 1936 International Congress of Mathematicians in Oslo. In this role, he led the evaluation of 38 nominations, applying criteria that emphasized youth and potential over established achievement, which resulted in disqualifying prominent mathematicians like Oswald Veblen and Louis Mordell for being over the age limit or too senior.^[7] From 1955 to 1958, Hopf held the presidency of the International Mathematical Union (IMU), guiding the organization during a period of post-war international mathematical collaboration. As IMU president, he was ex officio a member of the Executive Committee of the International Commission on Mathematical Instruction (ICMI), contributing to efforts in global mathematics education coordination.^[1]^[2] In the aftermath of World War II, Hopf played an administrative role in rebuilding European mathematical networks, including organizing visits to institutions like the Mathematical Research Institute at Oberwolfach in 1946 and participating in conferences across Paris, Brussels, Rome, and Oxford to foster renewed international ties.^[1]

Mathematical Contributions

Foundations in Algebraic Topology

Hopf's engagement with algebraic topology stemmed from his studies under Erhard Schmidt in Berlin, where Schmidt's lectures on Brouwer's mapping degree theorem in 1917 sparked his interest in topological invariants.^[1] In his 1925 doctoral dissertation at the University of Berlin, supervised by Ludwig Bieberbach, Hopf classified simply connected closed Riemannian 3-manifolds of constant curvature, connecting metric properties to topological structure.^[1] This work highlighted early links between differential geometry and topology, laying groundwork for using algebraic tools to distinguish manifolds.^[1] His 1926 habilitation thesis advanced fixed-point and index theory by offering a new proof of Lefschetz's theorem: the sum of indices of singularities for a generic vector field on a closed manifold equals the Euler characteristic.^[1] This result reinforced the mapping degree as a fundamental homotopy invariant, extending Brouwer's ideas to higher dimensions and vector fields.^[1] Hopf's approach emphasized algebraic computation of topological features, influencing subsequent developments in characteristic classes. During collaborations with Pavel Aleksandrov in 1927–1928, including at Princeton, Hopf co-defined an intersection product on homology cycles, forming a ring structure that anticipated modern cohomology rings.^[1] Published in their joint topology textbook, this construction provided algebraic machinery for studying manifold intersections combinatorially.^[1] By 1933, Hopf achieved a complete homotopy classification of continuous maps from n-dimensional polytopes to the n-sphere using homology invariants alone, demonstrating that degree suffices for such mappings under certain conditions.^[8] These efforts established key relations between homotopy and homology, with Hopf's later 1942 paper "Fundamentalgruppe und zweite Bettische Gruppe" exploring how the fundamental group acts on the second homology group via free resolutions over group rings, initiating aspects of homological algebra for groups.^[9] This bridged discrete group theory with continuous topology, enabling algebraic descriptions of homotopy influences on homology in aspherical spaces.^[9] Hopf's foundational innovations prioritized empirical verification through explicit computations, fostering causal understanding of topological phenomena via first-principles algebraic reductions. In 1931, Heinz Hopf published a seminal paper demonstrating the existence of non-trivial homotopy classes of maps from the 3-sphere S^3 to the 2-sphere S^2, introducing what is now known as the Hopf fibration.^[1] This fibration, denoted S^1 \to S^3 \to S^2, consists of a surjective submersion where each fiber is a great circle in S^3, providing the first explicit example of a non-trivial principal circle bundle over S^2.^[10] Hopf constructed the map explicitly using complex coordinates on S^3 \subset \mathbb{C}^2, defined by (z_1, z_2) \mapsto (2z_1 \overline{z_2}, |z_1|^2 - |z_2|^2) up to normalization, which projects points along linked circles onto S^2.^[11] Central to Hopf's proof was the introduction of the Hopf invariant, a homotopy invariant for maps f: S^{2n-1} \to S^n that captures linking properties of preimages.^[1] For the fibration map, the Hopf invariant equals 1, proving it is essential and generating \pi_3(S^2) \cong \mathbb{Z}.^[12] This invariant, computed via an integral over linking numbers of fiber components, revealed that homotopy groups of spheres could be non-trivial even when dimensions differ by more than one, challenging prevailing expectations.^[1] Hopf's work extended to related fibrations; in a 1935 paper, he described analogous constructions for odd-dimensional spheres, such as S^3 \to S^7 \to S^4 using quaternions, though these were later generalized beyond his scope.^[13] These results laid foundational groundwork for fiber bundle theory and homotopy classification, influencing subsequent developments in algebraic topology, including Adams' resolution of the Hopf invariant one problem in the 1960s.^[5] The fibration's geometric interpretation as linked circles also connected topology to classical knot theory, with preimages under the Hopf map forming the Hopf link, a fundamental example of linking number 1.^[10]

Work in Geometry and Dynamical Systems

Hopf's contributions to differential geometry centered on the interplay between curvature and topology, particularly for surfaces in Euclidean space. In classical differential geometry, he proved that the standard 2-sphere is the unique compact immersed surface of genus zero in \mathbb{R}^3 with constant nonzero mean curvature, a result established through analysis of the associated holomorphic quadratic differentials and Gauss map properties.^[5] This theorem, often termed Hopf's sphere theorem, implies that any such surface must be a round sphere, relying on the rigidity imposed by constant mean curvature and the topological constraints of the sphere.^[14] He also explored broader connections between topology and curvature, initiating systematic investigations into how topological invariants influence geometric structures under curvature conditions. For instance, Hopf conjectured properties of positively curved Riemannian manifolds, such as the sign conjecture asserting that such compact even-dimensional manifolds without boundary have positive Euler characteristic, though these remain partially unresolved.^[15] His work emphasized causal geometric constraints over purely metric ones, privileging empirical verification through specific examples like spheres and tori. In dynamical systems, Hopf generalized the Poincaré index theorem to higher dimensions, proving in his 1926 habilitation thesis that the sum of the indices of zeros for a generic vector field on a closed orientable manifold equals its Euler characteristic, a topological invariant independent of the field's choice.^[1] This extension, building on Poincaré's two-dimensional case from 1885, applies to gradient fields and rest points, providing a fixed-point formula essential for analyzing stability and bifurcations in flows on manifolds.^[16] The result underpins applications in understanding invariant sets and periodic orbits, as the index sum constrains possible configurations of equilibria in bounded dynamical systems.^[17] Hopf's proof utilized homology groups, linking dynamical behavior directly to algebraic topology for rigorous quantification.^[1]

Legacy and Recognition

Influence on Subsequent Mathematical Developments

Hopf's introduction of the Hopf invariant in 1931 provided a fundamental tool for classifying maps between spheres, significantly advancing homotopy theory and enabling computations of higher homotopy groups, such as \pi_3(S^2), which influenced subsequent developments in stable homotopy and spectral sequences.^[1] His extensions of fixed-point theorems and early emphasis on homology groups helped solidify algebraic topology as an independent discipline, bridging geometric intuition with algebraic rigor and paving the way for modern homological algebra.^[1] The Hopf fibration S^3 \to S^2, discovered in 1931, exemplified a non-trivial fiber bundle structure over spheres, catalyzing the study of principal bundles and their classifications; it directly informed the theory of characteristic classes for vector bundles and played a central role in Adams' resolution of the Hopf invariant one problem, linking to the maximal number of linearly independent vector fields on spheres via dimensions tied to division algebras (1, 2, 4, 8).^[18] This construction extended to K-theory, where it provided topological invariants essential for index theorems, and influenced applications in gauge theories and fluid dynamics.^[18] Hopf's investigations into the cohomology of Lie groups and H-spaces during the 1940s introduced comultiplication structures on cohomology rings, laying the groundwork for Hopf algebras; these topological origins inspired algebraic generalizations, culminating in Milnor and Moore's 1965 structure theorems that unified the field across topology and algebraic groups.^[19] In dynamical systems, his 1942 generalization of the Poincaré-Andronov bifurcation to dimensions n > 2 prefigured center manifold reductions and normal form analyses, influencing stability theory in nonlinear differential equations and later computational methods in chaos.^[20]

Notable Students and Collaborators

Heinz Hopf supervised 49 doctoral students, primarily at ETH Zurich following his appointment there in 1931, though some earlier theses were completed under his guidance at the University of Berlin.^[21] His students contributed significantly to algebraic topology, geometry, and related areas, forming a influential school at ETH that emphasized rigorous foundational work in these fields.^[22] Notable among them was Hans Freudenthal, who completed his 1930 thesis at Berlin and later advanced the study of homotopy groups through the Freudenthal suspension theorem and work on Lie group representations.^[21] Eduard Stiefel earned his 1935 doctorate at ETH Zurich and co-developed Stiefel manifolds, which play a key role in vector bundle theory, while also pioneering numerical methods like the conjugate gradient algorithm.^[21] ^[5] Beno Eckmann, with a 1942 ETH thesis, made foundational contributions to homological algebra and fixed-point theory, extending Hopf's topological ideas.^[21] ^[5] Other prominent students included Friedrich Hirzebruch (1950, Münster under Hopf's supervision), renowned for the Hirzebruch-Riemann-Roch theorem in algebraic geometry; Michel Kervaire (1956, ETH), who investigated the Kervaire invariant problem in surgery theory; Ernst Specker (1949, ETH), influential in set theory and undecidability results with topological applications; Hans Samelson (1941, ETH), who worked on differential topology and Lie groups; and Willi Rinow (1932, Berlin), contributing to differential geometry and manifold theory.^[21] ^[5] Hopf's key collaborator was Pavel S. Aleksandrov, with whom he developed foundational ideas in dimension theory and general topology during joint seminars in Göttingen (1926 onward) and a visiting year at Princeton (1927–1928).^[1] Their partnership culminated in the co-authored textbook Topologie (1935), the first volume of a planned series that systematized combinatorial topology and became a cornerstone reference for subsequent researchers.^[1] ^[23] Hopf's collaborations often extended through correspondence and shared seminars rather than numerous joint papers, reflecting his emphasis on mentoring and collective advancement in topology.^[1]

Enduring Impact and Named Theorems

Hopf's introduction of the Hopf fibration in 1931, a fiber bundle structure S^1 \to S^3 \to S^2, provided the first example of a non-trivial homotopy group \pi_3(S^2) \cong \mathbb{Z}, fundamentally advancing the understanding of higher homotopy groups and inspiring the development of fiber bundle theory and characteristic classes in algebraic topology.^[24] This construction, visualized as a projection of the 3-sphere onto the 2-sphere with circle fibers, revealed deep connections between geometry and topology, influencing subsequent work on principal bundles and gauge theories.^[25] Key theorems named after Hopf include the Hopf degree theorem (1931), which asserts that for smooth maps from a compact, connected, oriented n-manifold to the n-sphere, the topological degree uniquely classifies homotopy classes, providing a foundational result in degree theory and manifold classification.^[26] The Poincaré-Hopf theorem (extended by Hopf in 1927), states that the sum of the indices of singularities of a vector field on a compact oriented manifold equals its Euler characteristic, with applications in dynamical systems and global analysis.^[27] In Riemannian geometry, the Hopf-Rinow theorem (1931, with Willi Rinow) establishes the equivalence of metric completeness, geodesic completeness, and the Heine-Borel property for manifolds, enabling the study of complete spaces via geodesics.^[28] Hopf's Hopf algebra structures, introduced in the 1940s through work on formal groups and Lie algebras, have enduring relevance in representation theory, with later extensions underpinning quantum groups and non-commutative geometry.^[29] His emphasis on algebraic methods in topology, including relations between homotopy, homology, and cohomology, shaped the field's transition to a rigorous algebraic framework, impacting areas from knot theory to modern homotopy type theory.^[4] These contributions, grounded in precise geometric intuitions, continue to inform research in differential topology and beyond, as evidenced by ongoing citations in studies of manifold invariants and bundle classifications.^[25]

Personal Life

Marriage and Immediate Family

Heinz Hopf married Anja von Mickwitz in October 1928.^[1]^[2] His wife, also known as Anna Marie von Mickwitz, was born in 1891 and died in 1967.^[2] The couple had no children.^[1]^[3] Hopf's immediate family included his parents, Wilhelm Hopf and Elisabeth Kirchner. Wilhelm, from a Jewish family, had joined his father-in-law Heinrich Kirchner's brewery in Breslau in 1887 and married Elisabeth, the eldest daughter of the Protestant Kirchner family, in 1892; he converted to Protestantism in 1895.^[1] Hopf had one sibling, an older sister named Hedwig, born in 1893.^[1]

Health, Retirement, and Death

Hopf retired from his professorship in mathematics at ETH Zurich in 1965, concluding a tenure that had begun in 1931.^[4] Following retirement, he maintained an active interest in mathematics, delivering lectures and participating in discussions, while remaining affiliated with ETH as a loyal member of its community.^[5] ^[1] In his later years, Hopf suffered from a prolonged illness.^[4] He died on June 3, 1971, in Zollikon, Switzerland, at the age of 76.^[1] ^[4]

Honors and Awards

Major Prizes and Medals

Heinz Hopf received the Lobachevsky Prize in 1969, awarded by the Soviet Academy of Sciences for his pioneering contributions to geometry, topology, and related fields.^[1]^[30] He was also honored with the Gauss-Weber Medal for his significant advancements in pure mathematics.^[1]

Honorary Degrees and Society Memberships

Hopf received honorary doctorates from Princeton University in 1947, the University of Freiburg, the University of Manchester, the Sorbonne, the Free University of Brussels, and the University of Lausanne.^[1]^[6]^[2] He was elected honorary member of the London Mathematical Society in 1956 and the Swiss Mathematical Society in 1957, among numerous other learned societies worldwide.^[6]^[31]^[1] Hopf served as president of the International Mathematical Union from 1955 to 1958.^[8]^[2]

References

[1]
Heinz Hopf - Biography - MacTutor - University of St Andrews
Heinz Hopf was one of the most distinguished mathematicians of the twentieth century. His work is closely linked with the emergence of algebraic topology.
[2]
heinz hopf - History of ICMI
Heinz Hopf was born on 19 November 1894 in Gräbschen, Germany (now Grabiszyn, Wrocław, Poland); from 1901 until 1904 he attended the school of Dr. Karl ...
[3]
Heinz Wilhelm Hopf (1894 - 1971) - Genealogy - Geni
Jun 29, 2024 · Hopf was born in Gräbschen, Germany (now Grabiszyn, part of Wrocław, Poland), the son of Elizabeth (née Kirchner) and Wilhelm Hopf.Missing: childhood | Show results with:childhood
[4]
[PDF] In memory of Heinz Hopf
Heinz Hopf was born on November 19, 1984 in Breslau. There he spent his youth and began his mathematical studies, interrupted shortly after, during the four ...
[5]
Why Hopf Lectures? – Department of Mathematics | ETH Zurich
Hopf's answer was one of the first results in "modern" homotopy theory. Many ramifications followed, including notions like homology groups of a discrete group, ...
[6]
Heinz Hopf - London Mathematical Society (LMS)
Heinz Hopf, the great Swiss mathematician, died on 3 June, 1971, at the age of 76. He will be deeply mourned by his many friends, colleagues and students, ...
[7]
Fields Medal was never meant for 'the greatest mathematical genius
History repeated itself: Committee chair Heinz Hopf at the Swiss Federal Institute of Technology in Zurich ruled out two nominees because they were already ...
[8]
Hopf, Heinz - Encyclopedia.com
The greater part of Hopf's work was algebraic topology, motivated by vigorous geometric intuitions. Although the number of his papers was relatively small, no ...
[9]
"Heinz Hopf Collected papers" Preface - MacTutor
This is the beginning of homological algebra: Hopf comes into the realm of the Hurewicz homotopy theory and finds algebraic descriptions for the influence of ...
[10]
[PDF] An Elementary Introduction to the Hopf Fibration - Niles Johnson
The Hopf fibration, named after Heinz Hopf who studied it in a 1931 paper [8], is an important object in mathematics and physics.
[11]
[PDF] arXiv:0808.3089v2 [math-ph] 11 Sep 2008
Sep 11, 2008 · A Survey of Hopf Fibrations. Heinz Hopf defined a mapping in his 1931 paper [2] that we now call the Hopf fibration. It was a landmark ...
[12]
[PDF] jt3(S\ H. Hopf, W.K. Clifford, F. Klein
precisely the Hopf fibration, and this must be where Hopf became acquainted with it. ... Heinz Hopf. Translation: Princeton, N.J., 30 Murray Place, Aug 17 ...<|control11|><|separator|>
[13]
[PDF] the hopf fibrations are characterized by being fiberwise
In two papers published in 1931 and 1935, Heinz Hopf discovered several beautiful fi- brations of spheres by great subspheres: fibrations of S2n+1 by great ...
[14]
[PDF] Redalyc.Hopf´s theorem for constant mean curvature spheres in E3 ...
We review Hopf's theorem on immersed spheres with constant mean curvature in the Eu- clidean space and then the recent work of Abresch and Rosenberg, where they ...
[15]
The Hopf Conjectures - Quantum Calculus
Jan 13, 2020 · The Hopf conjectures were first formulated by Hopf in print in 1931. The sign conjecture claims that positive curvature compact Riemannian 2d-manifolds have ...
[16]
Poincaré–Hopf theorem - EPFL Graph Search
The theorem was proven for two dimensions by Henri Poincaré and later generalized to higher dimensions by Heinz Hopf. The Euler characteristic of a closed ...
[17]
[PDF] Perspectives on the legacy of Poincar´e in the field of dynamical ...
Sep 3, 2012 · We introduce a few geometric ideas and re- sults introduced and obtained by Poincaré. This includes the rotation number and the. Poincaré–Hopf ...
[18]
[PDF] Hopf Fibrations and Hurwitz-Radon Numbers - V. Ovsienko
Hopf fibration was defined and studied by the famous. German geometer and topologist Heintz Hopf in 1931. [16]. In a sense, contemporary algebraic topology has ...
[19]
[0901.2460] The beginnings of the theory of Hopf algebras - arXiv
Jan 16, 2009 · Hopf algebras are named from the work of Heinz Hopf in the 1940's. In this note we trace the infancy of the subject back to papers from the 40's ...
[20]
History of dynamical systems - Scholarpedia
Oct 21, 2011 · Perhaps the most influential Russian contribution to dynamical systems ... Hopf, in Germany, had extended the Poincaré-Andronov bifurcation ...<|separator|>
[21]
Heinz Hopf - The Mathematics Genealogy Project
Heinz Hopf Biography MathSciNet Ph.D. Universität Berlin 1925 Germany Dissertation: Über Zusammmenhänge zwischen Topologie und Metrik von Mannigfaltigkeiten.
[22]
Early 20th century – Department of Mathematics | ETH Zurich
Hopf was a great personality, as well as an outstanding teacher and researcher. He succeeded in building a school of algebraic topology of international ...Missing: positions | Show results with:positions
[23]
Mathematics and physics (post-1800) - Detlev Auvermann Rare Books
Topology. ALEKSANDROFF (or ALEKSANDROV), Pavel and Heinz HOPF. Topologie. Berlin, J. Springer, 1935. £1400. First edition, published in the series Grundlagen ...
[24]
https://math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Bosshardt.pdf
[25]
[PDF] Is algebraic topology a respectable field?
In any case, certain papers of Hopf had a decisive influence on the later place of topology within mathematics, and we list them in more detail. 5.1 Hopf, 1925.Missing: dissertation | Show results with:dissertation<|separator|>
[26]
[PDF] the hopf degree theorem - UChicago Math
Aug 26, 2011 · As an immediate result we can use the Hopf Degree Theorem to completely characterize homotopy classes of maps under the given conditions.Missing: Heinz 1933<|control11|><|separator|>
[27]
[PDF] TOPOLOGY FROM THE DIFFERENTIABLE VIEWPOINT
7. 8. Exercises. Vector fields and the Euler number. Framed cobordism; the Pontryagin construction. The Hopf theorem ... A famous theorem due to Heinz Hopf ...
[28]
[PDF] THE HOPF-RINOW THEOREM Contents 1. Introduction 1 2. Tensors ...
Aug 3, 2016 · This paper is an introduction to Riemannian geometry, with an aim towards proving the Hopf-Rinow theorem on complete Riemannian man- ifolds. We ...Missing: Heinz | Show results with:Heinz
[29]
[PDF] Heinz Hopf Prize 2013
Dec 3, 2013 · He also introduced Hopf algebra, which later became fundamentally significant in the theory of quantum groups, amongst other areas.
[30]
N.I. Lobachevsky medal and prize
The International N. I. Lobachevsky Prize was awarded in 1972 for a series of works on the homological theory of dimension. Hopf Heinz (German: Heinz Hopf; ...
[31]
Honorary Members | The Swiss Mathematical Society
Sep 22, 2006 · Heinz Hopf (1957) Walter Saxer (1957) George Pólya (1957) Louis Kollros (1958) Jean Leray (1960) Hassler Whitney (1960) Georges de Rham ...