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Combinatorial Mirror Symmetry
A purely combinatorial approach to mirror symmetry was suggested by Victor Batyrev using the polar duality for d-dimensional convex polyhedra. The most famous examples of the polar duality provide Platonic solids: e.g., the cube is dual to octahedron, the dodecahedron is dual to icosahedron. There is a natural bijection between the k-dimensional faces of a d-dimensional convex polyhedron P and (d-k-1)-dimensional faces of the dual polyhedron P^* and one has (P^*)^* = P. In Batyrev's combinatorial approach to mirror symmetry the polar duality is applied to special d-dimensional convex lattice polytopes which are called reflexive polytopes. It was observed by Victor Batyrev and Duco van Straten that the method of Philip Candelas et al. for computing the number of rational curves on Calabi–Yau quintic 3-folds can be applied to arbitrary Calabi–Yau complete intersections using the generalized A-hypergeometric functions introduced by Israel Gelfand, Michail Kapranov and And ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mirror Symmetry (string Theory)
In algebraic geometry and theoretical physics, mirror symmetry is a relationship between geometry, geometric objects called Calabi–Yau manifolds. The term refers to a situation where two Calabi–Yau manifolds look very different geometrically but are nevertheless equivalent when employed as extra dimensions of string theory. Early cases of mirror symmetry were discovered by physicists. Mathematicians became interested in this relationship around 1990 when Philip Candelas, Xenia de la Ossa, Paul Green, and Linda Parkes showed that it could be used as a tool in enumerative geometry, a branch of mathematics concerned with counting the number of solutions to geometric questions. Candelas and his collaborators showed that mirror symmetry could be used to count rational curves on a Calabi–Yau manifold, thus solving a longstanding problem. Although the original approach to mirror symmetry was based on physical ideas that were not understood in a mathematically precise way, some of i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fano Variety
In algebraic geometry, a Fano variety, introduced by Gino Fano , is an algebraic variety that generalizes certain aspects of complete intersections of algebraic hypersurfaces whose sum of degrees is at most the total dimension of the ambient projective space. Such complete intersections have important applications in geometry and number theory, because they typically admit rational points, an elementary case of which is the Chevalley–Warning theorem. Fano varieties provide an abstract generalization of these basic examples for which rationality questions are often still tractable. Formally, a Fano variety is a complete variety ''X'' whose anticanonical bundle ''K''X* is ample. In this definition, one could assume that ''X'' is smooth over a field, but the minimal model program has also led to the study of Fano varieties with various types of singularities, such as terminal or klt singularities. Recently techniques in differential geometry have been applied to the study of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Physics
Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories". An alternative definition would also include those mathematics that are inspired by physics, known as physical mathematics. Scope There are several distinct branches of mathematical physics, and these roughly correspond to particular historical parts of our world. Classical mechanics Applying the techniques of mathematical physics to classical mechanics typically involves the rigorous, abstract, and advanced reformulation of Newtonian mechanics in terms of Lagrangian mechanics and Hamiltonian mechanics (including both approaches in the presence of constraints). Both formulations are embodied in analytical mechanics and lead ... [...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] |
Homological Mirror Symmetry
Homological mirror symmetry is a mathematical conjecture made by Maxim Kontsevich. It seeks a systematic mathematical explanation for a phenomenon called mirror symmetry first observed by physicists studying string theory. History In an address to the 1994 International Congress of Mathematicians in Zürich, speculated that mirror symmetry for a pair of Calabi–Yau manifolds ''X'' and ''Y'' could be explained as an equivalence of a triangulated category constructed from the algebraic geometry of ''X'' (the derived category of coherent sheaves on ''X'') and another triangulated category constructed from the symplectic geometry of ''Y'' (the derived Fukaya category). Edward Witten originally described the topological twisting of the N=(2,2) supersymmetric field theory into what he called the A and B model topological string theories. These models concern maps from Riemann surfaces into a fixed target—usually a Calabi–Yau manifold. Most of the mathematical predicti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toric Variety
In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Some authors also require it to be normal. Toric varieties form an important and rich class of examples in algebraic geometry, which often provide a testing ground for theorems. The geometry of a toric variety is fully determined by the combinatorics of its associated fan, which often makes computations far more tractable. For a certain special, but still quite general class of toric varieties, this information is also encoded in a polytope, which creates a powerful connection of the subject with convex geometry. Familiar examples of toric varieties are affine space, projective spaces, products of projective spaces and bundles over projective space. Toric varieties from tori The original motivation to study toric varieties was to study torus embeddings. Given the algebr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conformal Field Theory
A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometimes be exactly solved or classified. Conformal field theory has important applications to condensed matter physics, statistical mechanics, quantum statistical mechanics, and string theory. Statistical and condensed matter systems are indeed often conformally invariant at their thermodynamic or quantum critical points. Scale invariance vs conformal invariance In quantum field theory, scale invariance is a common and natural symmetry, because any fixed point of the renormalization group is by definition scale invariant. Conformal symmetry is stronger than scale invariance, and one needs additional assumptions to argue that it should appear in nature. The basic idea behind its plausibility is that ''local'' scale invariant theories have t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex Operator Algebra
In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string theory. In addition to physical applications, vertex operator algebras have proven useful in purely mathematical contexts such as monstrous moonshine and the geometric Langlands correspondence. The related notion of vertex algebra was introduced by Richard Borcherds in 1986, motivated by a construction of an infinite-dimensional Lie algebra due to Igor Frenkel. In the course of this construction, one employs a Fock space that admits an action of vertex operators attached to elements of a unimodular lattice, lattice. Borcherds formulated the notion of vertex algebra by axiomatizing the relations between the lattice vertex operators, producing an algebraic structure that allows one to construct new Lie algebras by following Frenkel's method. The notion of vertex operator algebra was introduced as a modification of the notion ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polymake
polymake is a software for the algorithmic treatment of convex polyhedra. Albeit primarily a tool to study the combinatorics and the geometry of convex polytopes and polyhedra, it is by now also capable of dealing with simplicial complexes, matroids, polyhedral fans, graphs, tropical objects, toric varieties and other objects. In particular, its capability to compute the convex hull and lattice points of a polytope proved itself to be quite useful for different kinds of research. polymake has been cited in over 300 recent articles indexed by Zentralblatt MATH as can be seen from its entry in the swMATH database. Special features and applications polymake exhibits a few particularities, making it special to work with. Firstly, polymake can be used within a Perl script. Moreover, users can extend polymake and define new objects, properties, rules for computing properties, and algorithms. Secondly, it exhibits an internal client-server scheme to accommodate the usage of Perl for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximilian Kreuzer
Maximilian or Maximillian (Maximiliaan in Dutch and Maximilien in French) is a male name. The name "Max" is considered a shortening of "Maximilian" as well as of several other names. List of people Monarchs *Maximilian I, Holy Roman Emperor (1459–1519) *Maximilian II, Holy Roman Emperor (1527–1576) *Maximilian I, Elector of Bavaria (1573–1651) *Maximilian II Emanuel, Elector of Bavaria (1662–1726) *Maximilian III Joseph, Elector of Bavaria (1727–1777) *Maximilian I Joseph of Bavaria (1756–1825) *Maximilian II of Bavaria (1811–1864) *Prince Maximilian of Baden (1867–1929) *Duke Maximilian Joseph in Bavaria (1808–1888) *Maximilian I of Mexico (1832–1867) Other royalty *Maximilian, Hereditary Prince of Saxony (1759–1838) *Maximilian, Margrave of Baden (1933–2022) Saints *Maximilian of Lorch (died 288), Roman bishop, missionary and martyr *Maximilian of Tebessa (274–295), Roman martyr *Maximilian of Antioch (died ), Christian martyr *Maximilian (died 447) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toric Variety
In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Some authors also require it to be normal. Toric varieties form an important and rich class of examples in algebraic geometry, which often provide a testing ground for theorems. The geometry of a toric variety is fully determined by the combinatorics of its associated fan, which often makes computations far more tractable. For a certain special, but still quite general class of toric varieties, this information is also encoded in a polytope, which creates a powerful connection of the subject with convex geometry. Familiar examples of toric varieties are affine space, projective spaces, products of projective spaces and bundles over projective space. Toric varieties from tori The original motivation to study toric varieties was to study torus embeddings. Given the algebr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calabi–Yau Manifold
In algebraic and differential geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has certain properties, such as Ricci flatness, yielding applications in theoretical physics. Particularly in superstring theory, the extra dimensions of spacetime are sometimes conjectured to take the form of a 6-dimensional Calabi–Yau manifold, which led to the idea of mirror symmetry. Their name was coined by , after , who first conjectured that compact complex manifolds of Kähler type with vanishing first Chern class always admit Ricci-flat Kähler metrics, and , who proved the Calabi conjecture. Calabi–Yau manifolds are complex manifolds that are generalizations of K3 surfaces in any number of complex dimensions (i.e. any even number of real dimensions). They were originally defined as compact Kähler manifolds with a vanishing first Chern class and a Ricci-flat metric, though many other similar but inequivalent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
