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In mathematics, the Langlands program is a set of conjectures about connections between number theory and geometry. It was proposed by Robert Langlands. It seeks to relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. It is the biggest project in mathematical research. It was described by Edward Frenkel as "grand unified theory of mathematics."
Faltings's theorem is a result in arithmetic geometry, according to which a curve of genus greater than 1 over the field of rational numbers has only finitely many rational points. This was conjectured in 1922 by Louis Mordell, and known as the Mordell conjecture until its 1983 proof by Gerd Faltings. The conjecture was later generalized by replacing by any number field.
The modularity theorem states that elliptic curves over the field of rational numbers are related to modular forms in a particular way. Andrew Wiles and Richard Taylor proved the modularity theorem for semistable elliptic curves, which was enough to imply Fermat's Last Theorem. Later, a series of papers by Wiles's former students Brian Conrad, Fred Diamond and Richard Taylor, culminating in a joint paper with Christophe Breuil, extended Wiles's techniques to prove the full modularity theorem in 2001.
In mathematics, Helmut Hasse's local–global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each different prime number. This is handled by examining the equation in the completions of the rational numbers: the real numbers and the p-adic numbers. A more formal version of the Hasse principle states that certain types of equations have a rational solution if and only if they have a solution in the real numbers and in the p-adic numbers for each prime p.
In mathematics, a rational variety is an algebraic variety, over a given field K, which is birationally equivalent to a projective space of some dimension over K. This means that its function field is isomorphic to
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups can act non-trivially.
In the mathematical area of group theory, Artin groups, also known as Artin–Tits groups or generalized braid groups, are a family of infinite discrete groups defined by simple presentations. They are closely related with Coxeter groups. Examples are free groups, free abelian groups, braid groups, and right-angled Artin–Tits groups, among others.
Don Bernard Zagier is an American-German mathematician whose main area of work is number theory. He is currently one of the directors of the Max Planck Institute for Mathematics in Bonn, Germany. He was a professor at the Collège de France in Paris from 2006 to 2014. Since October 2014, he is also a Distinguished Staff Associate at the International Centre for Theoretical Physics (ICTP).
In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties.
In mathematics, the Artin approximation theorem is a fundamental result of Michael Artin in deformation theory which implies that formal power series with coefficients in a field k are well-approximated by the algebraic functions on k.
In mathematics, Serre's modularity conjecture, introduced by Jean-Pierre Serre, states that an odd, irreducible, two-dimensional Galois representation over a finite field arises from a modular form. A stronger version of this conjecture specifies the weight and level of the modular form. The conjecture in the level 1 case was proved by Chandrashekhar Khare in 2005, and a proof of the full conjecture was completed jointly by Khare and Jean-Pierre Wintenberger in 2008.
This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of proposed conjectures, which can be related at various levels of generality.
In mathematics, a Frey curve or Frey–Hellegouarch curve is the elliptic curve associated with an ABC triple . This relates properties of solutions of equations to elliptic curves. This curve was popularized in its application to Fermat’s Last Theorem where one investigates a (hypothetical) solution of Fermat's equation
In mathematics, real algebraic geometry is the sub-branch of algebraic geometry studying real algebraic sets, i.e. real-number solutions to algebraic equations with real-number coefficients, and mappings between them.
In algebraic geometry, a supersingular K3 surface is a K3 surface over a field k of characteristic p > 0 such that the slopes of Frobenius on the crystalline cohomology H2(X,W(k)) are all equal to 1. These have also been called Artin supersingular K3 surfaces. Supersingular K3 surfaces can be considered the most special and interesting of all K3 surfaces.
Zlil Sela is an Israeli mathematician working in the area of geometric group theory. He is a Professor of Mathematics at the Hebrew University of Jerusalem. Sela is known for the solution of the isomorphism problem for torsion-free word-hyperbolic groups and for the solution of the Tarski conjecture about equivalence of first-order theories of finitely generated non-abelian free groups.
In mathematics, the Schneider–Lang theorem is a refinement by Lang (1966) of a theorem of Schneider (1949) about the transcendence of values of meromorphic functions. The theorem implies both the Hermite–Lindemann and Gelfond–Schneider theorems, and implies the transcendence of some values of elliptic functions and elliptic modular functions.
Christopher Deninger is a German mathematician at the University of Münster. Deninger's research focuses on arithmetic geometry, including applications to L-functions.
In mathematics, Artin's criteria are a collection of related necessary and sufficient conditions on deformation functors which prove the representability of these functors as either Algebraic spaces or as Algebraic stacks. In particular, these conditions are used in the construction of the moduli stack of elliptic curves and the construction of the moduli stack of pointed curves.
In commutative algebra and algebraic geometry, Popescu's theorem, introduced by Dorin Popescu, states:
References
- Popescu, Dorin (1986). "General Néron desingularization and approximation". Nagoya Mathematical Journal . 104: 85–115. doi: 10.1017/S0027763000022698 .
- Rotthaus, Christel (1987). "On the approximation property of excellent rings". Inventiones Mathematicae . 88: 39–63. Bibcode:1987InMat..88...39R. doi:10.1007/BF01405090.
- Artin, M (1969). "Algebraic approximation of structures over complete local rings". Publications Mathématiques de l'IHÉS. 36: 23–58. doi:10.1007/BF02684596. ISSN 0073-8301.
- Artin, M (1968). "On the solutions of analytic equations". Inventiones Mathematicae. 5 (4): 277–291. Bibcode:1968InMat...5..277A. doi:10.1007/BF01389777. ISSN 0020-9910.
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