In mathematics, more specifically algebraic topology, the fundamental group ( defined by Henri Poincaré in his article Analysis Situs, published in 1895 ) is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other.

The objections to his work were occasionally fierce: Poincaré referred to Cantor's ideas as a " grave disease " infecting the discipline of mathematics, and Kronecker's public opposition and personal attacks included describing Cantor as a " scientific charlatan ", a " renegade " and a " corrupter of youth.

As Henri Poincaré famously said, mathematics is not the study of objects, but instead, the relations ( isomorphisms for instance ) between them.

Henri Poincaré is regarded as the last mathematician to excel in every field of the mathematics of his time.

On December 22, 2006, the journal Science honored Perelman's proof of the Poincaré conjecture as the scientific " Breakthrough of the Year ", the first time this had been bestowed in the area of mathematics.

Like his contemporary, Henri Poincaré, Picard was much concerned with the training of mathematics, physics, and engineering students.

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic ( or Euler – Poincaré characteristic ) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent.

" On 22 December 2006, the journal Science recognized Perelman's proof of the Poincaré conjecture as the scientific " Breakthrough of the Year ", the first such recognition in the area of mathematics.

Mathematician and physicist Henri Poincaré made important contributions to pure and applied mathematics, and also published books for the general public on mathematical and scientific subjects.

In physics and mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime.

In physics and mathematics, a number of ideas are named after Henri Poincaré:

In mathematics, the Poincaré duality theorem named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds.

An important reason why singularities cause problems in mathematics is that, with a failure of manifold structure, the invocation of Poincaré duality is also disallowed.

In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion ( after Henri Poincaré ) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point.

In mathematics, the representation theory of the Poincaré group is an example of the representation theory of a Lie group that is neither a compact group nor a semisimple group.

Related to the concept of eternal return is the Poincaré recurrence theorem in mathematics.

In mathematics, particularly in dynamical systems, a first recurrence map or Poincaré map, named after Henri Poincaré, is the intersection of a periodic orbit in the state space of a continuous dynamical system with a certain lower dimensional subspace, called the Poincaré section, transversal to the flow of the system.

* Poincaré homology sphere, in mathematics, an example of a homology sphere

What Poincaré and the Pre-Intuitionists shared was the perception of a difference between logic and mathematics which is not a matter of language alone, but of knowledge itself.

In mathematics, a conjecture is an unproven proposition that appears correct.

In mathematics, any number of cases supporting a conjecture, no matter how large, is insufficient for establishing the conjecture's veracity, since a single counterexample would immediately bring down the conjecture.

Euler's conjecture is a disproved conjecture in mathematics related to Fermat's last theorem which was proposed by Leonhard Euler in 1769.

The Collatz conjecture is a conjecture in mathematics named after Lothar Collatz, who first proposed it in 1937.

Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and in all of mathematics.

As with many famous conjectures in mathematics, there are a number of purported proofs of the Goldbach conjecture, none accepted by the mathematical community.

* The television drama Lewis featured a mathematics professor who had won the Fields medal for his work on Goldbach's conjecture.

In mathematics, Tait's conjecture states that " Every 3-connected planar cubic graph has a Hamiltonian cycle ( along the edges ) through all its vertices ".

In mathematics the modularity theorem ( formerly called the Taniyama – Shimura – Weil conjecture and several related names ) states that elliptic curves over the field of rational numbers are related to modular forms.

In mathematics, Lawson's conjecture states that the Clifford torus is the only minimally embedded torus in the 3-sphere S < sup > 3 </ sup >.

Context is also relevant ; in mathematics, the Pólya conjecture is true for numbers less than 906, 150, 257, but fails for this number.

Assuming something to be true for all numbers when it has been shown for over 906 million cases would not generally be considered hasty, but in mathematics a statement remains a conjecture until it is shown to be universally true.

The Hodge conjecture is a problem in mathematics that remains unsolved.

** Roger Frye used experimental mathematics techniques to find the smallest counterexample to Euler's sum of powers conjecture.

* Goldbach's conjecture, one of the oldest unsolved problems in number theory and in all of mathematics.

In mathematics, the Mertens conjecture is the incorrect statement that the Mertens function M ( n ) is bounded by √ n, which implies the Riemann hypothesis.

In mathematics, the Birch and Swinnerton-Dyer conjecture is an open problem in the field of number theory.

In some ways it is the most accessible discipline in pure mathematics for the general public: for instance the Goldbach conjecture is easily stated ( but is yet to be proved or disproved ).

He pursued mathematics as an amateur, his most famous achievement being his confirmation in 1901 of Leonhard Euler's conjecture that no 6 × 6 Graeco-Latin square was possible .< ref >

The participants were Professor H. Bondi, professor of mathematics at King's College, London ; ;

Dr. W. B. Bonnor, reader in mathematics at Queen Elizabeth College, London ; ;

Another possibility, raised in an essay by the Swedish fantasy writer and editor Rickard Berghorn, is that the name Alhazred was influenced by references to two historical authors whose names were Latinized as Alhazen: Alhazen ben Josef, who translated Ptolemy into Arabic ; and Abu ' Ali al-Hasan ibn al-Haytham, who wrote about optics, mathematics and physics.

Associative operations are abundant in mathematics ; in fact, many algebraic structures ( such as semigroups and categories ) explicitly require their binary operations to be associative.

In later life Ampère claimed that he knew as much about mathematics and science when he was eighteen as ever he knew ; but, a polymath, his reading embraced history, travels, poetry, philosophy, and the natural sciences.

He used his time in Bourg to research mathematics, producing Considérations sur la théorie mathématique de jeu ( 1802 ; “ Considerations on the Mathematical Theory of Games ”), a treatise on mathematical probability that he sent to the Paris Academy of Sciences in 1803.

Initially a study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, and it becomes even more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find a specific solution ; this leads into some of the deepest areas in all of mathematics, both conceptually and in terms of technique.

Following Desargues ' thinking, the sixteen-year-old Pascal produced, as a means of proof, a short treatise on what was called the " Mystic Hexagram ", Essai pour les coniques (" Essay on Conics ") and sent it — his first serious work of mathematics — to Père Mersenne in Paris ; it is known still today as Pascal's theorem.

In mathematics, specifically in measure theory, a Borel measure is defined as follows: let X be a locally compact Hausdorff space, and let be the smallest σ-algebra that contains the open sets of X ; this is known as the σ-algebra of Borel sets.

Sets are of great importance in mathematics ; in fact, in modern formal treatments, most mathematical objects ( numbers, relations, functions, etc.

On November 29, 1921, the trustees declared it to be the express policy of the Institute to pursue scientific research of the greatest importance and at the same time " to continue to conduct thorough courses in engineering and pure science, basing the work of these courses on exceptionally strong instruction in the fundamental sciences of mathematics, physics, and chemistry ; broadening and enriching the curriculum by a liberal amount of instruction in such subjects as English, history, and economics ; and vitalizing all the work of the Institute by the infusion in generous measure of the spirit of research.

General category theory, an extension of universal algebra having many new features allowing for semantic flexibility and higher-order logic, came later ; it is now applied throughout mathematics.

This was a major discovery in an important field of mathematics ; construction problems had occupied mathematicians since the days of the Ancient Greeks, and the discovery ultimately led Gauss to choose mathematics instead of philology as a career.

The Langlands program is a far-reaching web of these ideas of ' unifying conjectures ' that link different subfields of mathematics, e. g. number theory and representation theory of Lie groups ; some of these conjectures have since been proved.

* Engineering mathematics ; a series of lectures delivered at Union College, 1917.

The scope of this school was not limited to theological subjects ; science, mathematics and humanities were also taught there.

Academically, Eisenhower's best subject by far was English ; otherwise his performance was average, though he thoroughly enjoyed the typical emphasis of engineering on science and mathematics.

# all of mathematics follows from a correctly chosen finite system of axioms ; and

Hilbert said " Physics is too hard for physicists ", implying that the necessary mathematics was generally beyond them ; the Courant-Hilbert book made it easier for them.

Decision problems typically appear in mathematical questions of decidability, that is, the question of the existence of an effective method to determine the existence of some object or its membership in a set ; many of the important problems in mathematics are undecidable.

High-dimensional spaces occur in mathematics and the sciences for many reasons, frequently as configuration spaces such as in Lagrangian or Hamiltonian mechanics ; these are abstract spaces, independent of the physical space we live in.