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Behrend Function
In algebraic geometry, the Behrend function of a scheme ''X'', introduced by Kai Behrend, is a constructible function (mathematics), constructible function :\nu_X: X \to \mathbb such that if ''X'' is a quasi-projective proper moduli scheme carrying a symmetric obstruction theory, then the weighted Euler characteristic :\chi(X, \nu_X) = \sum_ n \, \chi(\) is the degree of the virtual fundamental class :[X]^ of ''X'', which is an element of the zeroth Chow group of ''X''. Modulo some solvable technical difficulties (e.g., what is the Chow group of a stack?), the definition extends to moduli stacks such as the moduli stack of stable sheaves (the Donaldson–Thomas theory) or that of stable maps (the Gromov–Witten theory). References *. Geometry {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects. The fundamental objects of study in algebraic geometry are algebraic variety, algebraic varieties, which are geometric manifestations of solution set, solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are line (geometry), lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscate of Bernoulli, lemniscates and Cassini ovals. These are plane algebraic curves. A point of the plane lies on an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of points of special interest like singular point of a curve, singular p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kai Behrend
Kai Behrend is a German mathematician. He is a professor at the University of British Columbia in Vancouver, British Columbia, Canada. His work is in algebraic geometry and he has made important contributions in the theory of algebraic stacks, Gromov–Witten invariants and Donaldson–Thomas theory (cf. Behrend function In algebraic geometry, the Behrend function of a scheme ''X'', introduced by Kai Behrend, is a constructible function (mathematics), constructible function :\nu_X: X \to \mathbb such that if ''X'' is a quasi-projective proper moduli scheme carrying ....) He is also known for Behrend's formula, the generalization of the Grothendieck–Lefschetz trace formula to algebraic stacks. He is the recipient of the 2001 Coxeter–James Prize, the 2011 Jeffery–Williams Prize, and the 2015 CRM-Fields-PIMS Prize. He was elected to the 2018 class of fellows of the American Mathematical Society. Selected publications * * * * References External linksThe p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constructible Function (mathematics)
In complexity theory, a time-constructible function is a function ''f'' from natural numbers to natural numbers with the property that ''f''(''n'') can be constructed from ''n'' by a Turing machine A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algori ... in the time of order ''f''(''n''). The purpose of such a definition is to exclude functions that do not provide an upper bound on the runtime of some Turing machine. Time-constructible Let the Turing machine be defined in the standard way, with an alphabet that includes the symbols 0, 1. It has a standard input tape containing zeros except for an input string. Let 1^n denote a string composed of n ones. That is, it's the unary representation. Let , n, be the binary representation. There are two different definitions of a time-construc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Obstruction Theory
In algebraic geometry, given a Deligne–Mumford stack ''X'', a perfect obstruction theory for ''X'' consists of: # a perfect two-term complex E = ^ \to E^0/math> in the derived category D(\text(X)_) of quasi-coherent étale sheaves on ''X'', and # a morphism \varphi\colon E \to \textbf_X, where \textbf_X is the cotangent complex of ''X'', that induces an isomorphism on h^0 and an epimorphism on h^. The notion was introduced by for an application to the intersection theory on moduli stacks; in particular, to define a virtual fundamental class. Examples Schemes Consider a regular embedding I \colon Y \to W fitting into a cartesian square : \begin X & \xrightarrow & V \\ g \downarrow & & \downarrow f \\ Y & \xrightarrow & W \end where V,W are smooth. Then, the complex :E^\bullet = ^*N_^ \to j^*\Omega_V/math> (in degrees -1, 0) forms a perfect obstruction theory for ''X''. The map comes from the composition :g^*N_^\vee \to g^*i^*\Omega_W =j^*f^*\Omega_W \to j^*\Omega_V Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Virtual Fundamental Class
In mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ..., specifically enumerative geometry and symplectic geometry, the virtual fundamental class \text \in H_*(X) of a (typically very singular) space X (or a stack) is a generalization of the classical fundamental class of a smooth manifold which has better behavior with respect to the enumerative problems being considered. In this way, there exists a cycle with can be used for answering specific enumerative problems, such as the number of degree d rational curves on a quintic threefold. For example, in Gromov–Witten theory, the Kontsevich moduli spaces\overline_(X,\beta)for X a smooth complex projective variety (or a symplectic manifold) \beta \in H_2(X) a curve class, could have wild singularities such aspg 503 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chow Group
In algebraic geometry, the Chow groups (named after Wei-Liang Chow by ) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties (so-called algebraic cycles) in a similar way to how simplicial or cellular homology groups are formed out of subcomplexes. When the variety is smooth, the Chow groups can be interpreted as cohomology groups (compare Poincaré duality) and have a multiplication called the intersection product. The Chow groups carry rich information about an algebraic variety, and they are correspondingly hard to compute in general. Rational equivalence and Chow groups For what follows, define a variety over a field k to be an integral scheme of finite type over k. For any scheme X of finite type over k, an algebraic cycle on X means a finite linear combination of subvarieties of X with integer coefficients. (Here and below, subvarieties are unde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chow Group Of A Stack
In algebraic geometry, the Chow group of a stack is a generalization of the Chow group of a variety or scheme to stacks. For a quotient stack X = /G/math>, the Chow group of ''X'' is the same as the ''G''-equivariant Chow group of ''Y''. A key difference from the theory of Chow groups of a variety is that a cycle is allowed to carry non-trivial automorphisms and consequently intersection-theoretic operations must take this into account. For example, the degree of a 0-cycle on a stack need not be an integer but is a rational number (due to non-trivial stabilizers). Definitions develops the basic theory (mostly over Q) for the Chow group of a (separated) Deligne–Mumford stack. There, the Chow group is defined exactly as in the classical case: it is the free abelian group generated by integral closed substacks modulo rational equivalence. If a stack ''X'' can be written as the quotient stack X = /G/math> for some quasi-projective variety ''Y'' with a linearized action of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Donaldson–Thomas Theory
In mathematics, specifically algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ..., Donaldson–Thomas theory is the theory of Donaldson–Thomas invariants. Given a compact moduli space of sheaves on a Calabi–Yau threefold, its Donaldson–Thomas invariant is the virtual number of its points, i.e., the integral of the cohomology class 1 against the virtual fundamental class. The Donaldson–Thomas invariant is a holomorphic analogue of the Casson invariant. The invariants were introduced by . Donaldson–Thomas invariants have close connections to Gromov–Witten invariants of algebraic three-folds and the theory of stable pairs due to Rahul Pandharipande and Thomas. Donaldson–Thomas theory is physically motivated by certain BPS states that occur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stable Map
In mathematics, specifically in symplectic topology and algebraic geometry, one can construct the moduli space of stable maps, satisfying specified conditions, from Riemann surfaces into a given symplectic manifold. This moduli space is the essence of the Gromov–Witten invariants, which find application in enumerative geometry and type IIA string theory. The idea of stable map was proposed by Maxim Kontsevich around 1992 and published in . Because the construction is lengthy and difficult, it is carried out here rather than in the Gromov–Witten invariants article itself. Smooth pseudoholomorphic curves Fix a closed symplectic manifold X with symplectic form \omega. Let g and n be natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...s (including zero) and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
