In mathematics, **topological Galois theory** is a mathematical theory which originated from a topological proof of Abel's impossibility theorem found by V. I. Arnold and concerns the applications of some topological concepts to some problems in the field of Galois theory. It connects many ideas from algebra to ideas in topology. As described in Khovanskii's book: "According to this theory, the way the Riemann surface of an analytic function covers the plane of complex numbers can obstruct the representability of this function by explicit formulas. The strongest known results on the unexpressibility of functions by explicit formulas have been obtained in this way."

**Algebraic geometry** is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros.

In mathematics, a **field** is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics.

In mathematics, a **group** is a set equipped with a binary operation that combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity, identity 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, and help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study.

**Sir Michael Francis Atiyah** was a British-Lebanese mathematician specialising in geometry.

**Vladimir Igorevich Arnold** was a Soviet and Russian mathematician. While he is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable systems, he made important contributions in several areas including dynamical systems theory, algebra, catastrophe theory, topology, algebraic geometry, symplectic geometry, differential equations, classical mechanics, hydrodynamics and singularity theory, including posing the ADE classification problem, since his first main result—the solution of Hilbert's thirteenth problem in 1957 at the age of 19. He co-founded two new branches of mathematics—KAM theory, and topological Galois theory.

In mathematics and abstract algebra, **group theory** studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.

In mathematics, a **topological group** is a group *G* together with a topology on *G* such that both the group's binary operation and the function mapping group elements to their respective inverses are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a topological structure. Thus, one may perform algebraic operations, because of the group structure, and one may talk about continuous functions, because of the topology.

In mathematics, **Galois theory** provides a connection between field theory and group theory. Using Galois theory, certain problems in field theory can be reduced to group theory, which is in some sense simpler and better understood. It has been used to solve classic problems including showing that two problems of antiquity cannot be solved as they were stated ; showing that there is no quintic formula; and showing which polygons are constructible.

In algebra, the **Abel–Ruffini theorem** states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. The theorem is named after Paolo Ruffini, who made an incomplete proof in 1799, and Niels Henrik Abel, who provided a proof in 1824.

In mathematics, especially in order theory, a **Galois connection** is a particular correspondence (typically) between two partially ordered sets (posets). The same notion can also be defined on preordered sets or classes; this article presents the common case of posets. Galois connections generalize the correspondence between subgroups and subfields investigated in Galois theory. They find applications in various mathematical theories.

In mathematics, **class field theory** is the branch of algebraic number theory concerned with the abelian extensions of number fields, global fields of positive characteristic, and local fields. The theory had its origins in the proof of quadratic reciprocity by Gauss at the end of 18th century. These ideas were developed over the next century, giving rise to a set of conjectures by Hilbert that were subsequently proved by Takagi and Artin. These conjectures and their proofs constitute the main body of class field theory.

In mathematics, **monodromy** is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As the name implies, the fundamental meaning of *monodromy* comes from "running round singly". It is closely associated with covering maps and their degeneration into ramification; the aspect giving rise to monodromy phenomena is that certain functions we may wish to define fail to be *single-valued* as we "run round" a path encircling a singularity. The failure of monodromy can be measured by defining a **monodromy group**: a group of transformations acting on the data that encodes what does happen as we "run round" in one dimension. Lack of monodromy is sometimes called *polydromy*.

In differential geometry, the **Atiyah–Singer index theorem**, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the **analytical index** is equal to the **topological index**. It includes many other theorems, such as the Chern–Gauss–Bonnet theorem and Riemann–Roch theorem, as special cases, and has applications to theoretical physics.

In algebra, the **theory of equations** is the study of algebraic equations, which are equations defined by a polynomial. The main problem of the theory of equations was to know when an algebraic equation has an algebraic solution. This problem was completely solved in 1830 by Évariste Galois, by introducing what is now called Galois theory.

In mathematics, a **duality**, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often by means of an involution operation: if the dual of *A* is *B*, then the dual of *B* is *A*. Such involutions sometimes have fixed points, so that the dual of *A* is *A* itself. For example, Desargues' theorem is self-dual in this sense under the *standard duality in projective geometry*.

**John Griggs Thompson** is a mathematician at the University of Florida noted for his work in the field of finite groups. He was awarded the Fields Medal in 1970, the Wolf Prize in 1992 and the 2008 Abel Prize.

In mathematics, **non-abelian class field theory** is a catchphrase, meaning the extension of the results of class field theory, the relatively complete and classical set of results on abelian extensions of any number field *K*, to the general Galois extension *L*/*K*. While class field theory was essentially known by 1930, the corresponding non-abelian theory has never been formulated in a definitive and accepted sense.

In mathematics, an **algebraic number field***F* is a finite degree field extension of the field of rational numbers **Q**. Thus *F* is a field that contains **Q** and has finite dimension when considered as a vector space over **Q**.

**Askold Georgievich Khovanskii** is a Russian and Canadian mathematician currently a professor of mathematics at the University of Toronto, Canada. His areas of research are algebraic geometry, commutative algebra, singularity theory, differential geometry and differential equations. His research is in the development of the theory of toric varieties and Newton polyhedra in algebraic geometry. He is also the inventor of the theory of fewnomials.

- Arnold, V. I.
*Abel's Theorem in Problems and Solutions*. - Khovanskii, A. G.
*Topological Galois Theory*. - Burda, Y.
*Topological Methods in Galois Theory*(PDF).

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