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Latin and Greek letters are used in mathematics, science, engineering, and other areas where mathematical notation is used as symbols for constants, special functions, and also conventionally for variables representing certain quantities.
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Latin and Greek letters are used in mathematics, science, engineering, and other areas where mathematical notation is used as symbols for constants, special functions, and also conventionally for variables representing certain quantities.
| Name | Digits |
|---|---|
| Double-struck | 𝟘 𝟙 𝟚 𝟛 𝟜 𝟝 𝟞 𝟟 𝟠 𝟡 |
| א | Cardinality of infinite sets |
|---|---|
| ב | Cardinality of infinite sets |
| ג | Gimel function |
| ת | Tav (number) |
| Л | Lobachevsky function [1] |
|---|---|
| Ш | Tate–Shafarevich group |
| ш | Shuffle product |
| よ | Yoneda embedding [2] |
|---|---|
| サ | Satake compactification [3] [4] |
| Å | Angstrom |
|---|---|
| ∀ | Universal quantification |
| Đ | Dispersity |
| ∂ | Partial derivative |
| ð | Spin-weighted partial derivative |
| ∃ | Existential quantification |
| Reduced Planck constant | |
| Ø | Empty set |
| Integral |
| ∇ | Del operator Gradient Divergence Curl |
|---|---|
| ∈ | Element (mathematics) |
| ƛ | Reduced wavelength |
| ∐ | Coproduct |
In mathematics, in particular algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spaces frequently arise as solutions to classification problems: If one can show that a collection of interesting objects can be given the structure of a geometric space, then one can parametrize such objects by introducing coordinates on the resulting space. In this context, the term "modulus" is used synonymously with "parameter"; moduli spaces were first understood as spaces of parameters rather than as spaces of objects. A variant of moduli spaces is formal moduli. Bernhard Riemann first used the term "moduli" in 1857.
Pierre René, Viscount Deligne is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Prize, and 1978 Fields Medal.
David Bryant Mumford is an American mathematician known for his work in algebraic geometry and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded the National Medal of Science. He is currently a University Professor Emeritus in the Division of Applied Mathematics at Brown University.
In mathematics, an algebraic stack is a vast generalization of algebraic spaces, or schemes, which are foundational for studying moduli theory. Many moduli spaces are constructed using techniques specific to algebraic stacks, such as Artin's representability theorem, which is used to construct the moduli space of pointed algebraic curves and the moduli stack of elliptic curves. Originally, they were introduced by Alexander Grothendieck to keep track of automorphisms on moduli spaces, a technique which allows for treating these moduli spaces as if their underlying schemes or algebraic spaces are smooth. After Grothendieck developed the general theory of descent, and Giraud the general theory of stacks, the notion of algebraic stacks was defined by Michael Artin.
In algebraic geometry, a moduli space of (algebraic) curves is a geometric space whose points represent isomorphism classes of algebraic curves. It is thus a special case of a moduli space. Depending on the restrictions applied to the classes of algebraic curves considered, the corresponding moduli problem and the moduli space is different. One also distinguishes between fine and coarse moduli spaces for the same moduli problem.
In mathematics, a moduli scheme is a moduli space that exists in the category of schemes developed by Alexander Grothendieck. Some important moduli problems of algebraic geometry can be satisfactorily solved by means of scheme theory alone, while others require some extension of the 'geometric object' concept.
In mathematics, a Drinfeld module is roughly a special kind of module over a ring of functions on a curve over a finite field, generalizing the Carlitz module. Loosely speaking, they provide a function field analogue of complex multiplication theory. A shtuka is a sort of generalization of a Drinfeld module, consisting roughly of a vector bundle over a curve, together with some extra structure identifying a "Frobenius twist" of the bundle with a "modification" of it.
In mathematics, topological modular forms (tmf) is the name of a spectrum that describes a generalized cohomology theory. In concrete terms, for any integer n there is a topological space , and these spaces are equipped with certain maps between them, so that for any topological space X, one obtains an abelian group structure on the set of homotopy classes of continuous maps from X to . One feature that distinguishes tmf is the fact that its coefficient ring, (point), is almost the same as the graded ring of holomorphic modular forms with integral cusp expansions. Indeed, these two rings become isomorphic after inverting the primes 2 and 3, but this inversion erases a lot of torsion information in the coefficient ring.
In mathematics a stack or 2-sheaf is, roughly speaking, a sheaf that takes values in categories rather than sets. Stacks are used to formalise some of the main constructions of descent theory, and to construct fine moduli stacks when fine moduli spaces do not exist.
In algebraic geometry, a stable curve is an algebraic curve that is asymptotically stable in the sense of geometric invariant theory.
In mathematics, the Hodge bundle, named after W. V. D. Hodge, appears in the study of families of curves, where it provides an invariant in the moduli theory of algebraic curves. Furthermore, it has applications to the theory of modular forms on reductive algebraic groups and string theory.
Alex Eskin is an American mathematician. He is the Arthur Holly Compton Distinguished Service Professor in the Department of Mathematics at the University of Chicago. His research focuses on rational billiards and geometric group theory.
In mathematics, a nilcurve is a pointed stable curve over a finite field with an indigenous bundle whose p-curvature is square nilpotent. Nilcurves were introduced by Mochizuki (1996) as a central concept in his theory of p-adic Teichmüller theory.
In algebraic geometry, a quotient stack is a stack that parametrizes equivariant objects. Geometrically, it generalizes a quotient of a scheme or a variety by a group: a quotient variety, say, would be a coarse approximation of a quotient stack.
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 algebraic geometry, the Behrend function of a scheme X, introduced by Kai Behrend, is a constructible function
Barbara Fantechi is an Italian mathematician and Professor at the International School for Advanced Studies. Her research area is algebraic geometry. She is a member of the Accademia dei Lincei.
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.
Abelian varieties are a natural generalization of elliptic curves, including algebraic tori in higher dimensions. Just as elliptic curves have a natural moduli space over characteristic 0 constructed as a quotient of the upper-half plane by the action of , there is an analogous construction for abelian varieties using the Siegel upper half-space and the symplectic group .
In mathematics, and in particular algebraic geometry, K-stability is an algebro-geometric stability condition for projective algebraic varieties and complex manifolds. K-stability is of particular importance for the case of Fano varieties, where it is the correct stability condition to allow the formation of moduli spaces, and where it precisely characterises the existence of Kähler–Einstein metrics.
