Julius Plücker was a German mathematician and physicist. He made fundamental contributions to the field of analytical geometry and was a pioneer in the investigations of cathode rays that led eventually to the discovery of the electron. He also vastly extended the study of Lamé curves.
In mathematics, the Weil conjectures were highly influential proposals by André Weil (1949). They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory.
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures. Étale cohomology theory can be used to construct ℓ-adic cohomology, which is an example of a Weil cohomology theory in algebraic geometry. This has many applications, such as the proof of the Weil conjectures and the construction of representations of finite groups of Lie type.
Carl Ludwig Siegel was a German mathematician specialising in analytic number theory. He is known for, amongst other things, his contributions to the Thue–Siegel–Roth theorem in Diophantine approximation, Siegel's method, Siegel's lemma and the Siegel mass formula for quadratic forms. He was named as one of the most important mathematicians of the 20th century.
The mathematical term perverse sheaves refers to a certain abelian category associated to a topological space X, which may be a real or complex manifold, or a more general topologically stratified space, usually singular. This concept was introduced in the thesis of Zoghman Mebkhout, gaining more popularity after the (independent) work of Joseph Bernstein, Alexander Beilinson, and Pierre Deligne (1982) as a formalisation of the Riemann-Hilbert correspondence, which related the topology of singular spaces and the algebraic theory of differential equations. It was clear from the outset that perverse sheaves are fundamental mathematical objects at the crossroads of algebraic geometry, topology, analysis and differential equations. They also play an important role in number theory, algebra, and representation theory. The properties characterizing perverse sheaves already appeared in the 75's paper of Kashiwara on the constructibility of solutions of holonomic D-modules.
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.
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, vanishing cycles are studied in singularity theory and other parts of algebraic geometry. They are those homology cycles of a smooth fiber in a family which vanish in the singular fiber.
In mathematics, the Frobenius endomorphism is defined in any commutative ring R that has characteristic p, where p is a prime number. Namely, the mapping φ that takes r in R to rp is a ring endomorphism of R.
In mathematics, a constructible sheaf is a sheaf of abelian groups over some topological space X, such that X is the union of a finite number of locally closed subsets on each of which the sheaf is a locally constant sheaf. It is a generalization of constructible topology in classical algebraic geometry.
In algebraic geometry, the Fourier–Deligne transform, or ℓ-adic Fourier transform, or geometric Fourier transform, is an operation on objects of the derived category of ℓ-adic sheaves over the affine line. It was introduced by Pierre Deligne on November 29, 1976 in a letter to David Kazhdan as an analogue of the usual Fourier transform. It was used by Gérard Laumon to simplify Deligne's proof of the Weil conjectures.
Christopher Deninger is a German mathematician at the University of Münster. Deninger's research focuses on arithmetic geometry, including applications to L-functions.
In algebraic geometry, the Grothendieck trace formula expresses the number of points of a variety over a finite field in terms of the trace of the Frobenius endomorphism on its cohomology groups. There are several generalizations: the Frobenius endomorphism can be replaced by a more general endomorphism, in which case the points over a finite field are replaced by its fixed points, and there is also a more general version for a sheaf over the variety, where the cohomology groups are replaced by cohomology with coefficients in the sheaf.
Gerrit Bol was a Dutch mathematician, who specialized in geometry. He is known for introducing Bol loops in 1937, and Bol’s conjecture on sextactic points.
Helmut Röhrl or Rohrl was a German mathematician.
Günter Harder is a German mathematician, specializing in arithmetic geometry and number theory.
Eberhard Freitag is a German mathematician, specializing in complex analysis and especially modular forms.
Uwe Jannsen is a German mathematician, specializing in algebra, algebraic number theory, and algebraic geometry.
In mathematics, a Siegel modular variety or Siegel moduli space is an algebraic variety that parametrizes certain types of abelian varieties of a fixed dimension. More precisely, Siegel modular varieties are the moduli spaces of principally polarized abelian varieties of a fixed dimension. They are named after Carl Ludwig Siegel, the 20th-century German number theorist who introduced the varieties in 1943.
Eberhard Hermann Erich Zeidler was a German mathematician, who worked primarily in the field of non-linear functional analysis.
