Karol Borsuk | |
---|---|

Born | |

Died | January 24, 1982 76) | (aged

Nationality | Polish |

Alma mater | Warsaw University |

Known for | Borsuk's conjecture Borsuk–Ulam theorem |

Scientific career | |

Fields | Mathematics |

Doctoral advisor | Stefan Mazurkiewicz |

Notable students |

**Karol Borsuk** (May 8, 1905 – January 24, 1982) was a Polish mathematician. His main interest was topology, while he obtained significant results also in functional analysis.

Borsuk introduced the theory of * absolute retracts * (ARs) and * absolute neighborhood retracts * (ANRs), and the cohomotopy groups, later called Borsuk–Spanier cohomotopy groups. He also founded shape theory. He has constructed various beautiful examples of topological spaces, e.g. an acyclic, 3-dimensional continuum which admits a fixed point free homeomorphism onto itself; also 2-dimensional, contractible polyhedra which have no free edge. His topological and geometric conjectures and themes stimulated research for more than half a century; in particular, his open problems stimulated the infinite-dimensional topology.

Borsuk received his master's degree and doctorate from Warsaw University in 1927 and 1930, respectively; his Ph.D. thesis advisor was Stefan Mazurkiewicz. He was a member of the Polish Academy of Sciences from 1952. Borsuk's students included Samuel Eilenberg, Włodzimierz Holsztyński, Jan Jaworowski, Krystyna Kuperberg, Włodzimierz Kuperberg, Hanna Patkowska, and Andrzej Trybulec.

*Geometria analityczna w n wymiarach*(1950) (translated to English as*Multidimensional Analytic Geometry*, Polish Scientific Publishers, 1969)*Podstawy geometrii*(1955)*Foundations of Geometry*(1960) with Wanda Szmielew, North Holland publisher^{ [1] }*Theory of Retracts*(1967), PWN, Warszawa.*Theory of Shape*(1975)*Collected papers*vol. I, (1983), PWN, Warszawa.

- Bing–Borsuk conjecture
- Borsuk's conjecture
- Borsuk–Ulam theorem
- Zygmunt Janiszewski
- Stanislaw Ulam
- Scottish Café
- Animal Husbandry, an educational dice game published by Borsuk at his own expense in 1943 during the German occupation of Warsaw. The original game was lost during the Warsaw uprising in August 1944. Very few copies survived outside Warsaw and one was given back to the Borsuk family. The game is now published by Granna under the name of "Super Farmer".

In mathematics, the **Borsuk–Ulam theorem** states that every continuous function from an *n*-sphere into Euclidean *n*-space maps some pair of antipodal points to the same point. Here, two points on a sphere are called antipodal if they are in exactly opposite directions from the sphere's center.

In mathematics, the **Poincaré conjecture** is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space.

**Algebraic topology** is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.

**Kazimierz Kuratowski** was a Polish mathematician and logician. He was one of the leading representatives of the Warsaw School of Mathematics.

**Samuel Eilenberg** was a Polish-American mathematician who co-founded category theory and homological algebra.

**Discrete geometry** and **combinatorial geometry** are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.

**Krystyna M. Kuperberg** is a Polish-American mathematician who currently works as a professor of mathematics at Auburn University, where she was formerly an Alumni Professor of Mathematics.

* Fundamenta Mathematicae* is a peer-reviewed scientific journal of mathematics with a special focus on the foundations of mathematics, concentrating on set theory, mathematical logic, topology and its interactions with algebra, and dynamical systems. Originally it only covered topology, set theory, and foundations of mathematics: it was the first specialized journal in the field of mathematics. It is published by the Mathematics Institute of the Polish Academy of Sciences.

In topology, a branch of mathematics, a **retraction** is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a **retract** of the original space. A **deformation retraction** is a mapping that captures the idea of *continuously shrinking* a space into a subspace.

**Andrzej Wojciech Trybulec**, was a Polish mathematician and computer scientist noted for work on the Mizar system.

The **Borsuk problem in geometry**, for historical reasons incorrectly called **Borsuk's conjecture**, is a question in discrete geometry. It is named after Karol Borsuk.

**Włodzimierz Kuperberg** is a professor of mathematics at Auburn University, with research interests in geometry and topology.

The mathematical discipline of **topological combinatorics** is the application of topological and algebro-topological methods to solving problems in combinatorics.

**Shape theory** is a branch of topology, which provides a more global view of the topological spaces than homotopy theory. The two coincide on compacta dominated homotopically by finite polyhedra. Shape theory associates with the Čech homology theory while homotopy theory associates with the singular homology theory.

**Gunnar E. Carlsson** is an American mathematician, working in algebraic topology. He is known for his work on the Segal conjecture, and for his work on applied algebraic topology, especially topological data analysis. He is a Professor Emeritus in the Department of Mathematics at Stanford University. He is the founder and president of the predictive technology company Ayasdi.

**Jan W. Jaworowski** was a Polish and American mathematician, topologist.

**Wanda Szmielew** née **Montlak** was a Polish mathematical logician who first proved the decidability of the first-order theory of abelian groups.

In mathematics, the **Bing–Borsuk conjecture** states that every -dimensional homogeneous absolute neighborhood retract space is a topological manifold. The conjecture has been proved for dimensions 1 and 2, and it is known that the 3-dimensional version of the conjecture implies the Poincaré conjecture.

* Using the Borsuk–Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry* is a graduate-level mathematics textbook in topological combinatorics. It describes the use of results in topology, and in particular the Borsuk–Ulam theorem, to prove theorems in combinatorics and discrete geometry. It was written by Czech mathematician Jiří Matoušek, and published in 2003 by Springer-Verlag in their Universitext series (ISBN 978-3-540-00362-5).

**Yuri Mikhailovich Smirnov** was a Soviet and Russian mathematician, specializing in topology.

- ↑ Freudenthal, H. (1961). "Review:
*Foundations of geometry, Euclidean and Bolyai–Lobachevskian geometry, projective geometry*. By K. Borsuk and Wanda Szmielew. Revised English translation" (PDF).*Bull. Amer. Math. Soc*.**67**(4): 342–344. doi: 10.1090/s0002-9904-1961-10606-x .

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