Gerhard Ringel was a German mathematician who earned his Ph. D. from the University of Bonn in 1951. He was one of the pioneers in graph theory and contributed to the proof of the Heawood conjecture, a mathematical problem linked with the Four Color Theorem. Although born in Kollnbrunn, Ringel was raised in Czechoslovakia and attended Charles University before being drafted into the German Army in 1940 (after Germany had taken control of much of what had been Czechoslovakia. After the war Ringel served for over four years in a Soviet prisoner of war camp. Gerhard Ringel started his academic career as professor at the Free University Berlin. In 1970 he left Germany due to bureaucratic consequences of the German student movement, continued his career at the University of California, Santa Cruz, having been invited there by his coauthor, Professor John W. T. Youngs, he was awarded honorary doctorate degrees from the University of Karlsruhe and the Free University Berlin. Besides his mathematical skills he was a acknowledged entomologist.
His main emphasis lay on breeding butterflies. Prior to his death, he gave his outstanding collection of butterflies to the UCSC Museum of Natural History Collections. Ringel, Gerhard. W. T.. "Solution of the Heawood map-coloring problem". Proc. Natl. Acad. Sci. USA. 60: 438–445. Doi:10.1073/pnas.60.2.438. MR 0228378. PMC 225066. PMID 16591648. Ringel, Gerhard. Map Color Theorem. New York/Berlin: Springer-Verlag. Hartsfield, Nora. Pearls in graph theory. Academic Press, Boston, MA. ISBN 0-12-328552-6. Gerhard Ringel at the Mathematics Genealogy Project Gerhard Ringel in the German National Library catalogue
In mathematics, a group is a set equipped with a binary operation which combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study. Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie groups are the symmetry groups used in the Standard Model of particle physics; the concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s.
After contributions from other fields such as number theory and geometry, the group notion was generalized and established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists study the different ways in which a group can be expressed concretely, both from a point of view of representation theory and of computational group theory. A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed in 2004. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become an active area in group theory; the modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4.
The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of a particular polynomial equation in terms of the symmetry group of its roots. The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois' ideas were rejected by his contemporaries, published only posthumously. More general permutation groups were investigated in particular by Augustin Louis Cauchy. Arthur Cayley's On the theory of groups, as depending on the symbolic equation θn = 1 gives the first abstract definition of a finite group. Geometry was a second field in which groups were used systematically symmetry groups as part of Felix Klein's 1872 Erlangen program. After novel geometries such as hyperbolic and projective geometry had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, Sophus Lie founded the study of Lie groups in 1884; the third field contributing to group theory was number theory.
Certain abelian group structures had been used implicitly in Carl Friedrich Gauss' number-theoretical work Disquisitiones Arithmeticae, more explicitly by Leopold Kronecker. In 1847, Ernst Kummer made early attempts to prove Fermat's Last Theorem by developing groups describing factorization into prime numbers; the convergence of these various sources into a uniform theory of groups started with Camille Jordan's Traité des substitutions et des équations algébriques. Walther von Dyck introduced the idea of specifying a group by means of generators and relations, was the first to give an axiomatic definition of an "abstract group", in the terminology of the time; as of the 20th century, groups gained wide recognition by the pioneering work of Ferdinand Georg Frobenius and William Burnside, who worked on representation theory of finite groups, Richard Brauer's modular representation theory and Issai Schur's papers. The theory of Lie groups, more locally compact groups was studied by Hermann Weyl, Élie Cartan and many others.
Its algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevalley and by the work of Armand Borel and Jacques Tits. The University of Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, led to the classification of finite simple groups, with the final step taken by Aschbacher and Smith in 2004; this project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers. Research is ongoing to simplify the proof of this classification; these days, group theory is still a active mathematical branch, impacting many other fields. One of the most familiar groups is the set of integers Z which consists of the numbers... − 4, − 3, − − 1, 0, 1, 2, 3, 4... together with addition. The following properties of integer addition serve as a model for the group axioms given in the definition below.
For any two integers a and b, the sum a + b is an integer. That is, addition of integers always yields an integer; this property is known as closure under addition. For all integers a, b and c, + c = a +. Expressed in words
Combinatorics is an area of mathematics concerned with counting, both as a means and an end in obtaining results, certain properties of finite structures. It is related to many other areas of mathematics and has many applications ranging from logic to statistical physics, from evolutionary biology to computer science, etc. To understand the scope of combinatorics requires a great deal of further amplification, the details of which are not universally agreed upon. According to H. J. Ryser, a definition of the subject is difficult because it crosses so many mathematical subdivisions. Insofar as an area can be described by the types of problems it addresses, combinatorics is involved with the enumeration of specified structures, sometimes referred to as arrangements or configurations in a general sense, associated with finite systems, the existence of such structures that satisfy certain given criteria, the construction of these structures in many ways, optimization, finding the "best" structure or solution among several possibilities, be it the "largest", "smallest" or satisfying some other optimality criterion.
Leon Mirsky has said: "combinatorics is a range of linked studies which have something in common and yet diverge in their objectives, their methods, the degree of coherence they have attained." One way to define combinatorics is to describe its subdivisions with their problems and techniques. This is the approach, used below. However, there are purely historical reasons for including or not including some topics under the combinatorics umbrella. Although concerned with finite systems, some combinatorial questions and techniques can be extended to an infinite but discrete setting. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory and geometry, as well as in its many application areas. Many combinatorial questions have been considered in isolation, giving an ad hoc solution to a problem arising in some mathematical context. In the twentieth century, however and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right.
One of the oldest and most accessible parts of combinatorics is graph theory, which by itself has numerous natural connections to other areas. Combinatorics is used in computer science to obtain formulas and estimates in the analysis of algorithms. A mathematician who studies combinatorics is called a combinatorialist. Basic combinatorial concepts and enumerative results appeared throughout the ancient world. In the 6th century BCE, ancient Indian physician Sushruta asserts in Sushruta Samhita that 63 combinations can be made out of 6 different tastes, taken one at a time, two at a time, etc. thus computing all 26 − 1 possibilities. Greek historian Plutarch discusses an argument between Chrysippus and Hipparchus of a rather delicate enumerative problem, shown to be related to Schröder–Hipparchus numbers. In the Ostomachion, Archimedes considers a tiling puzzle. In the Middle Ages, combinatorics continued to be studied outside of the European civilization; the Indian mathematician Mahāvīra provided formulae for the number of permutations and combinations, these formulas may have been familiar to Indian mathematicians as early as the 6th century CE.
The philosopher and astronomer Rabbi Abraham ibn Ezra established the symmetry of binomial coefficients, while a closed formula was obtained by the talmudist and mathematician Levi ben Gerson, in 1321. The arithmetical triangle— a graphical diagram showing relationships among the binomial coefficients— was presented by mathematicians in treatises dating as far back as the 10th century, would become known as Pascal's triangle. In Medieval England, campanology provided examples of what is now known as Hamiltonian cycles in certain Cayley graphs on permutations. During the Renaissance, together with the rest of mathematics and the sciences, combinatorics enjoyed a rebirth. Works of Pascal, Jacob Bernoulli and Euler became foundational in the emerging field. In modern times, the works of J. J. Sylvester and Percy MacMahon helped lay the foundation for algebraic combinatorics. Graph theory enjoyed an explosion of interest at the same time in connection with the four color problem. In the second half of the 20th century, combinatorics enjoyed a rapid growth, which led to establishment of dozens of new journals and conferences in the subject.
In part, the growth was spurred by new connections and applications to other fields, ranging from algebra to probability, from functional analysis to number theory, etc. These connections shed the boundaries between combinatorics and parts of mathematics and theoretical computer science, but at the same time led to a partial fragmentation of the field. Enumerative combinatorics is the most classical area of combinatorics and concentrates on counting the number of certain combinatorial objects. Although counting the number of elements in a set is a rather broad mathematical problem, many of the problems that arise in applications have a simple combinatorial description. Fibonacci numbers is the basic example of a problem in enumerative combinatorics; the twelvefold way provides a unified framework for counting permutations and partitions. Analytic combinatorics concerns the enumeration of combinatorial structures using tools from complex analysis and probability theo
Notices of the American Mathematical Society
Notices of the American Mathematical Society is the membership journal of the American Mathematical Society, published monthly except for the combined June/July issue. The first volume appeared in 1953; each issue of the magazine since January 1995 is available in its entirety on the journal web site. Articles are peer-reviewed by an editorial board of mathematical experts. Since 2019, the editor-in-chief is Erica Flapan; the cover features mathematical visualizations. The Notices is the world's most read mathematical journal; as the membership journal of the American Mathematical Society, the Notices is sent to the 30,000 AMS members worldwide, one-third of whom reside outside the United States. By publishing high-level exposition, the Notices provides opportunities for mathematicians to find out what is going on in the field; each issue contains one or two such expository articles that describe current developments in mathematical research, written by professional mathematicians. The Notices carries articles on the history of mathematics, mathematics education, professional issues facing mathematicians, as well as reviews of books and other works.
American Mathematical Monthly, another "most read mathematics journal in the world" Official website
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
In mathematics, genus has a few different, but related, meanings. The most common concept, the genus of an surface, is the number of "holes"; this is made more precise below. The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected, it is equal to the number of handles on it. Alternatively, it can be defined in terms of the Euler characteristic χ, via the relationship χ = 2 − 2g for closed surfaces, where g is the genus. For surfaces with b boundary components, the equation reads χ b. In layman's terms, it's the number of "holes". A doughnut, or torus, has 1 such hole. A sphere has 0; the green surface pictured above has 2 holes of the relevant sort. For instance: The sphere S2 and a disc both have genus zero. A torus has genus one; this is the source of the joke "topologists are people who can't tell their donut from their coffee mug."An explicit construction of surfaces of genus g is given in the article on the fundamental polygon.
Genus of orientable surfaces In simpler terms, the value of an orientable surface's genus is equal to the number of "holes" it has. The non-orientable genus, demigenus, or Euler genus of a connected, non-orientable closed surface is a positive integer representing the number of cross-caps attached to a sphere. Alternatively, it can be defined for a closed surface in terms of the Euler characteristic χ, via the relationship χ = 2 − k, where k is the non-orientable genus. For instance: A projective plane has non-orientable genus one. A Klein bottle has non-orientable genus two; the genus of a knot K is defined as the minimal genus of all Seifert surfaces for K. A Seifert surface of a knot is however a manifold with boundary, the boundary being the knot, i.e. homeomorphic to the unit circle. The genus of such a surface is defined to be the genus of the two-manifold, obtained by gluing the unit disk along the boundary; the genus of a 3-dimensional handlebody is an integer representing the maximum number of cuttings along embedded disks without rendering the resultant manifold disconnected.
It is equal to the number of handles on it. For instance: A ball has genus zero. A solid torus D2 × S1 has genus one; the genus of a graph is the minimal integer n such that the graph can be drawn without crossing itself on a sphere with n handles. Thus, a planar graph has genus 0; the non-orientable genus of a graph is the minimal integer n such that the graph can be drawn without crossing itself on a sphere with n cross-caps. The Euler genus is the minimal integer n such that the graph can be drawn without crossing itself on a sphere with n cross-caps or on a sphere with n/2 handles. In topological graph theory there are several definitions of the genus of a group. Arthur T. White introduced the following concept; the genus of a group G is the minimum genus of a Cayley graph for G. The graph genus problem is NP-complete. There are two related definitions of genus of any projective algebraic scheme X: the arithmetic genus and the geometric genus; when X is an algebraic curve with field of definition the complex numbers, if X has no singular points these definitions agree and coincide with the topological definition applied to the Riemann surface of X.
For example, the definition of elliptic curve from algebraic geometry is connected non-singular projective curve of genus 1 with a given rational point on it. By the Riemann-Roch theorem, an irreducible plane curve of degree d has geometric genus g = 2 − s, where s is the number of singularities when properly counted. Genus can be calculated for the graph spanned by the net of chemical interactions in nucleic acids or proteins. In particular, one may study the growth of the genus along the chain; such a function shows the topological domain structure of biomolecules. Cayley graph Group Arithmetic genus Geometric genus Genus of a multiplicative sequence Genus of a quadratic form Spinor genus