Secondary calculus and cohomological physics

Last updated

In mathematics, secondary calculus is a proposed expansion of classical differential calculus on manifolds, to the "space" of solutions of a (nonlinear) partial differential equation. It is a sophisticated theory at the level of jet spaces and employing algebraic methods.

Contents

Secondary calculus

Secondary calculus acts on the space of solutions of a system of partial differential equations (usually nonlinear equations). When the number of independent variables is zero (i.e. the equations are all algebraic) secondary calculus reduces to classical differential calculus.

All objects in secondary calculus are cohomology classes of differential complexes growing on diffieties. The latter are, in the framework of secondary calculus, the analog of smooth manifolds.

Cohomological physics

Cohomological physics was born with Gauss's theorem, describing the electric charge contained inside a given surface in terms of the flux of the electric field through the surface itself. Flux is the integral of a differential form and, consequently, a de Rham cohomology class. It is not by chance that formulas of this kind, such as the well known Stokes formula, though being a natural part of classical differential calculus, have entered in modern mathematics from physics.

Classical analogues

All the constructions in classical differential calculus have an analog in secondary calculus. For instance, higher symmetries of a system of partial differential equations are the analog of vector fields on differentiable manifolds. The Euler operator, which associates to each variational problem the corresponding Euler–Lagrange equation, is the analog of the classical differential associating to a function on a variety its differential. The Euler operator is a secondary differential operator of first order, even if, according to its expression in local coordinates, it looks like one of infinite order. More generally, the analog of differential forms in secondary calculus are the elements of the first term of the so-called C-spectral sequence, and so on.

The simplest diffieties are infinite prolongations of partial differential equations, which are subvarieties of infinite jet spaces. The latter are infinite dimensional varieties that can not be studied by means of standard functional analysis. On the contrary, the most natural language in which to study these objects is differential calculus over commutative algebras. Therefore, the latter must be regarded as a fundamental tool of secondary calculus. On the other hand, differential calculus over commutative algebras gives the possibility to develop algebraic geometry as if it were differential geometry.

Theoretical physics

Recent developments of particle physics, based on quantum field theories and its generalizations, have led to understand the deep cohomological nature of the quantities describing both classical and quantum fields. The turning point was the discovery of the famous BRST transformation. For instance, it was understood that observables in field theory are classes in horizontal de Rham cohomology which are invariant under the corresponding gauge group and so on. This current in modern theoretical physics is actually growing[ citation needed ] and it is called Cohomological Physics.

It is relevant that secondary calculus and cohomological physics, which developed for twenty years independently from each other, arrived at the same results. Their confluence took place at the international conference Secondary Calculus and Cohomological Physics (Moscow, August 24–30, 1997).

Prospects

A large number of modern mathematical theories harmoniously converges in the framework of secondary calculus, for instance: commutative algebra and algebraic geometry, homological algebra and differential topology, Lie group and Lie algebra theory, differential geometry, etc.

See also

Related Research Articles

<span class="mw-page-title-main">Differential geometry</span> Branch of mathematics dealing with functions and geometric structures on differentiable manifolds

Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries.

Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of spaces that are locally presented by noncommutative algebras of functions, possibly in some generalized sense. A noncommutative algebra is an associative algebra in which the multiplication is not commutative, that is, for which does not always equal ; or more generally an algebraic structure in which one of the principal binary operations is not commutative; one also allows additional structures, e.g. topology or norm, to be possibly carried by the noncommutative algebra of functions.

<span class="mw-page-title-main">Lists of mathematics topics</span>

Lists of mathematics topics cover a variety of topics related to mathematics. Some of these lists link to hundreds of articles; some link only to a few. The template to the right includes links to alphabetical lists of all mathematical articles. This article brings together the same content organized in a manner better suited for browsing. Lists cover aspects of basic and advanced mathematics, methodology, mathematical statements, integrals, general concepts, mathematical objects, and reference tables. They also cover equations named after people, societies, mathematicians, journals, and meta-lists.

In mathematics, the derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, geometry, etc.

In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first integrals, that its motion is confined to a submanifold of much smaller dimensionality than that of its phase space.

The Mathematics Subject Classification (MSC) is an alphanumerical classification scheme that has collaboratively been produced by staff of, and based on the coverage of, the two major mathematical reviewing databases, Mathematical Reviews and Zentralblatt MATH. The MSC is used by many mathematics journals, which ask authors of research papers and expository articles to list subject codes from the Mathematics Subject Classification in their papers. The current version is MSC2020.

Shlomo Zvi Sternberg was an American mathematician known for his work in geometry, particularly symplectic geometry and Lie theory.

In mathematics the differential calculus over commutative algebras is a part of commutative algebra based on the observation that most concepts known from classical differential calculus can be formulated in purely algebraic terms. Instances of this are:

  1. The whole topological information of a smooth manifold is encoded in the algebraic properties of its -algebra of smooth functions as in the Banach–Stone theorem.
  2. Vector bundles over correspond to projective finitely generated modules over via the functor which associates to a vector bundle its module of sections.
  3. Vector fields on are naturally identified with derivations of the algebra .
  4. More generally, a linear differential operator of order k, sending sections of a vector bundle to sections of another bundle is seen to be an -linear map between the associated modules, such that for any elements :

Graduate Studies in Mathematics (GSM) is a series of graduate-level textbooks in mathematics published by the American Mathematical Society (AMS). The books in this series are published in hardcover and e-book formats.

<span class="mw-page-title-main">Jerry Kazdan</span> American mathematician

Jerry Lawrence Kazdan is an American mathematician noted for his work in differential geometry and the study of partial differential equations. His contributions include the Berger–Kazdan comparison theorem, which was a key step in the proof of the Blaschke conjecture and the classification of Wiedersehen manifolds. His best-known work, done in collaboration with Frank Warner, dealt with the problem of prescribing the scalar curvature of a Riemannian metric.

In mathematics, a diffiety is a geometrical object which plays the same role in the modern theory of partial differential equations that algebraic varieties play for algebraic equations, that is, to encode the space of solutions in a more conceptual way. The term was coined in 1984 by Alexandre Mikhailovich Vinogradov as portmanteau from differential variety.

In mathematics, the Lagrangian theory on fiber bundles is globally formulated in algebraic terms of the variational bicomplex, without appealing to the calculus of variations. For instance, this is the case of classical field theory on fiber bundles.

In algebraic geometry, graded manifolds are extensions of the concept of manifolds based on ideas coming from supersymmetry and supercommutative algebra. Both graded manifolds and supermanifolds are phrased in terms of sheaves of graded commutative algebras. However, graded manifolds are characterized by sheaves on smooth manifolds, while supermanifolds are constructed by gluing of sheaves of supervector spaces.

Mathematics is a broad subject that is commonly divided in many areas that may be defined by their objects of study, by the used methods, or by both. For example, analytic number theory is a subarea of number theory devoted to the use of methods of analysis for the study of natural numbers.

<span class="mw-page-title-main">NASU Institute of Mathematics</span>

Institute of Mathematics of the National Academy of Sciences of Ukraine is a government-owned research institute in Ukraine that carries out basic research and trains highly qualified professionals in the field of mathematics. It was founded on 13 February 1934.

Robert C. Hermann was an American mathematician and mathematical physicist. In the 1960s Hermann worked on elementary particle physics and quantum field theory, and published books which revealed the interconnections between vector bundles on Riemannian manifolds and gauge theory in physics, before these interconnections became "common knowledge" among physicists in the 1970s.

<span class="mw-page-title-main">Alexander Varchenko</span>

Alexander Nikolaevich Varchenko is a Soviet and Russian mathematician working in geometry, topology, combinatorics and mathematical physics.

<span class="mw-page-title-main">Alexandre Mikhailovich Vinogradov</span> Russian-Italian mathematician (1938–2019)

Alexandre Mikhailovich Vinogradov was a Russian and Italian mathematician. He made important contributions to the areas of differential calculus over commutative algebras, the algebraic theory of differential operators, homological algebra, differential geometry and algebraic topology, mechanics and mathematical physics, the geometrical theory of nonlinear partial differential equations and secondary calculus.

The Colloquium Lecture of the American Mathematical Society is a special annual session of lectures.

References

      This page is based on this Wikipedia article
      Text is available under the CC BY-SA 4.0 license; additional terms may apply.
      Images, videos and audio are available under their respective licenses.