Mathematics

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

Burnside problem

The Burnside problem, posed by William Burnside in 1902 and one of the oldest and most influential questions in group theory, asks whether a finitely generated group in which every element has finite order must be a finite group. Evgeny Golod and Igor Shafarevich provided a counter-example in 1964; the problem has many variants that differ in the additional conditions imposed on the orders of the group elements. Initial work pointed towards the affirmative answer. For example, if a group G is finitely generated and the order of each element of G is a divisor of 4 G is finite. Moreover, A. I. Kostrikin was able to prove in 1958 that among the finite groups with a given number of generators and a given prime exponent, there exists a largest one; this provides a solution for the restricted Burnside problem for the case of prime exponent. Issai Schur had showed in 1911 that any finitely generated periodic group, a subgroup of the group of invertible n × n complex matrices was finite; the general answer to Burnside problem turned out to be negative.

In 1964, Golod and Shafarevich constructed an infinite group of Burnside type without assuming that all elements have uniformly bounded order. In 1968, Pyotr Novikov and Sergei Adian's supplied a negative solution to the bounded exponent problem for all odd exponents larger than 4381. In 1982, A. Yu. Ol'shanskii found some striking counterexamples for sufficiently large odd exponents, supplied a simpler proof based on geometric ideas; the case of exponents turned out to be much harder to settle. In 1992, S. V. Ivanov announced the negative solution for sufficiently large exponents divisible by a large power of 2. Joint work of Ol'shanskii and Ivanov established a negative solution to an analogue of Burnside problem for hyperbolic groups, provided the exponent is sufficiently large. By contrast, when the exponent is small and different from 2,3,4 and 6 little is known. A group G is called periodic; every finite group is periodic. There exist defined groups such as the p∞-group which are infinite periodic groups.

General Burnside problem. If G is a finitely generated, periodic group is G finite? This question was answered in the negative in 1964 by Evgeny Golod and Igor Shafarevich, who gave an example of an infinite p-group, finitely generated. However, the orders of the elements of this group are not a priori bounded by a single constant. Part of the difficulty with the general Burnside problem is that the requirements of being finitely generated and periodic give little information about the possible structure of a group. Therefore, we pose more requirements on G. Consider a periodic group G with the additional property that there exists a least integer n such that for all g in G, gn = 1. A group with this property is said to be periodic with bounded exponent n, or just a group with exponent n. Burnside problem for groups with bounded exponent asks: Burnside problem. If G is a finitely generated group with exponent n, is G finite? It turns out that this problem can be restated as a question about the finiteness of groups in a particular family.

The free Burnside group of rank m and exponent n, denoted B, is a group with m distinguished generators x1... xm in which the identity xn = 1 holds for all elements x, and, the "largest" group satisfying these requirements. More the characteristic property of B is that, given any group G with m generators g1... gm and of exponent n, there is a unique homomorphism from B to G that maps the ith generator xi of B into the ith generator gi of G. In the language of group presentations, free Burnside group B has m generators x1... xm and the relations xn = 1 for each word x in x1... xm, any group G with m generators of exponent n is obtained from it by imposing additional relations. The existence of the free Burnside group and its uniqueness up to an isomorphism are established by standard techniques of group theory, thus if G is any finitely generated group of exponent n G is a homomorphic image of B, where m is the number of generators of G. Burnside problem can now be restated as follows: Burnside problem II.

For which positive integers m, n is the free Burnside group B finite? The full solution to Burnside problem in this form is not known. Burnside considered some easy cases in his original paper: B is the cyclic group of order n. B is the direct product of m copies of the cyclic group of order 2 and hence finite; the following additional results are known: B, B, B are finite for all m. The particular case of B remains open: as of 2005 it was not known whether this group is finite; the breakthrough in Burnside problem was achieved by Pyotr Novikov and Sergei Adian in 1968. Using a complicated combinatorial argument, they demonstrated that for every odd number n with n > 4381, there exist infinite, finitely generated groups of exponent n. Adian improved the bound on the odd exponent to 665; the case of exponent turned out to be more difficult. It was only in 1994 that Sergei Vasilievich Ivanov was able to prove an analogue of Novikov–Adian theorem: for any m > 1 and an n ≥ 248, n divisible by 29, the group B i

Group algebra

In mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra, such that representations of the algebra are related to representations of the group. As such, they are similar to the group ring associated to a discrete group. In particular, these all use convolution to extend the group Hopf algebra from the set of functions with finite support to more useful classes of functions. For the purposes of functional analysis, in particular of harmonic analysis, one wishes to carry over the group ring construction to topological groups G. In case G is a locally compact Hausdorff group, G carries an unique left-invariant countably additive Borel measure μ called a Haar measure. Using the Haar measure, one can define a convolution operation on the space Cc of complex-valued continuous functions on G with compact support. To define the convolution operation, let f and g be two functions in Cc. For t in G, define = ∫ G f g d μ; the fact that f * g is continuous is immediate from the dominated convergence theorem.

Support ⊆ Support ⋅ Support where the dot stands for the product in G. Cc has a natural involution defined by: f ∗ = f ¯ Δ where Δ is the modular function on G. With this involution, it is a *-algebra. Theorem. With the norm: ‖ f ‖ 1:= ∫ G | f | d μ, Cc becomes an involutive normed algebra with an approximate identity; the approximate identity can be indexed on a neighborhood basis of the identity consisting of compact sets. Indeed, if V is a compact neighborhood of the identity, let fV be a non-negative continuous function supported in V such that ∫ V f V d μ = 1. V is an approximate identity. A group algebra has an identity, as opposed to just an approximate identity, if and only if the topology on the group is the discrete topology. Note that for discrete groups, Cc is the same thing as the complex group ring C; the importance of the group algebra is that it captures the unitary representation theory of G as shown in the following Theorem. Let G be a locally compact group. If U is a continuous unitary representation of G on a Hilbert space H π U = ∫ G f U d μ is a non-degenerate bounded *-representation of the normed algebra Cc.

The map U ↦ π U is a bijection between the set of continuous unitary representations of G and non-degenerate bounded *-representations of Cc. This bijection respects unitary strong containment. In particular, πU is only if U is irreducible. Non-degeneracy of a representation π of Cc on a Hilbert space Hπ means, dense in Hπ, it is a standard theorem of measure theory that the completion of Cc in the L1 norm is isomorphic to the space L1 of equivalence classes of functions which are integrable with respect to the Haar measure, where, as usual, two functions are regarded as equivalent if and only if they differ only on a set of Haar measure zero. Theorem. L1 is a Banach *-algebra with the convolution product and involution defined above and with the L1 norm. L1 has a bounded approximate identity. Let C be the group ring of a discrete group G. For a locally compact group G, the group C*-algebra C* of G is defined to be the C*-enveloping algebra of L1, i.e. the completion of Cc with respect to the largest C*-norm: ‖ f ‖ C ∗:= sup π ‖ π ‖, where π ranges over all non-degenerate *-re

Efim Zelmanov

Efim Isaakovich Zelmanov is a Russian-American mathematician, known for his work on combinatorial problems in nonassociative algebra and group theory, including his solution of the restricted Burnside problem. He was awarded a Fields Medal at the International Congress of Mathematicians in Zürich in 1994. Zelmanov was born into a Jewish family in Khabarovsk, Soviet Union, he entered Novosibirsk State University in 1972. He obtained doctoral degree at Novosibirsk State University in 1980, a higher degree at Leningrad State University in 1985, he had a position in Novosibirsk until 1987. In 1990 he moved to the United States, he was at the University of Chicago in 1994/5 at Yale University. As of 2011, he is a professor at the University of California, San Diego and a Distinguished Professor at the Korea Institute for Advanced Study. Zelmanov was elected a member of the U. S. National Academy of Sciences in 2001, becoming, at the age of 47, the youngest member of the mathematics section of the academy.

He is an elected member of the American Academy of Arts and Sciences and a foreign member of the Korean Academy of Science and Engineering and of the Spanish Royal Academy of Sciences. In 2012 he became a fellow of the American Mathematical Society. Zelmanov gave invited talks at the International Congress of Mathematicians in Warsaw and Zurich, he was awarded Honorary Doctor degrees from the University of Alberta, Shevchenko National University of Kyiv, the Universidad Internacional Menéndez Pelayo in Santander and the University of Lincoln, UK. Zelmanov's early work was on Jordan algebras in the case of infinite dimensions, he was able to show. He showed that the Engel identity for Lie algebras implies nilpotence, in the case of infinite dimensions. Efim Zelmanov at the Mathematics Genealogy Project O'Connor, John J.. The Work of Efim Zelmanov by Kapil Hari Paranjape