In mathematics, a **genus of a multiplicative sequence** is a ring homomorphism from the ring of smooth compact manifolds up to the equivalence of bounding a smooth manifold with boundary (i.e., up to suitable cobordism) to another ring, usually the rational numbers, having the property that they are constructed from a sequence of polynomials in characteristic classes that arise as coefficients in formal power series with good multiplicative properties.

A **genus** assigns a number to each manifold *X* such that

- (where is the disjoint union);
- ;
- if
*X*is the boundary of a manifold with boundary.

The manifolds and manifolds with boundary may be required to have additional structure; for example, they might be oriented, spin, stably complex, and so on (see list of cobordism theories for many more examples). The value is in some ring, often the ring of rational numbers, though it can be other rings such as or the ring of modular forms.

The conditions on can be rephrased as saying that is a ring homomorphism from the cobordism ring of manifolds (with additional structure) to another ring.

Example: If is the signature of the oriented manifold *X*, then is a genus from oriented manifolds to the ring of integers.

A sequence of polynomials in variables is called **multiplicative** if

implies that

If is a formal power series in *z* with constant term 1, we can define a multiplicative sequence

by

- ,

where is the *k*th elementary symmetric function of the indeterminates . (The variables will often in practice be Pontryagin classes.)

The genus of compact, connected, smooth, oriented manifolds corresponding to *Q* is given by

where the are the Pontryagin classes of *X*. The power series *Q* is called the **characteristic power series** of the genus . A theorem of René Thom, which states that the rationals tensored with the cobordism ring is a polynomial algebra in generators of degree 4*k* for positive integers *k*, implies that this gives a bijection between formal power series *Q* with rational coefficients and leading coefficient 1, and genera from oriented manifolds to the rational numbers.

The **L genus** is the genus of the formal power series

where the numbers are the Bernoulli numbers. The first few values are:

(for further *L*-polynomials see ^{ [1] } or OEIS: A237111 ). Now let *M* be a closed smooth oriented manifold of dimension 4*n* with Pontrjagin classes . Friedrich Hirzebruch showed that the *L* genus of *M* in dimension 4*n* evaluated on the fundamental class of , denoted , is equal to , the signature of *M* (i.e., the signature of the intersection form on the 2*n*th cohomology group of *M*):

- .

This is now known as the ** Hirzebruch signature theorem ** (or sometimes the **Hirzebruch index theorem**).

The fact that is always integral for a smooth manifold was used by John Milnor to give an example of an 8-dimensional PL manifold with no smooth structure. Pontryagin numbers can also be defined for PL manifolds, and Milnor showed that his PL manifold had a non-integral value of , and so was not smoothable.

Since projective K3 surfaces are smooth complex manifolds of dimension two, their only non-trivial Pontryagin class is in . It can be computed as -48 using the tangent sequence and comparisons with complex chern classes. Since , we have its signature. This can be used to compute its intersection form as a unimodular lattice since it has , and using the classification of unimodular lattices.^{ [2] }

The **Todd genus** is the genus of the formal power series

with as before, Bernoulli numbers. The first few values are

The Todd genus has the particular property that it assigns the value 1 to all complex projective spaces (i.e. ), and this suffices to show that the Todd genus agrees with the arithmetic genus for algebraic varieties as the arithmetic genus is also 1 for complex projective spaces. This observation is a consequence of the Hirzebruch–Riemann–Roch theorem, and in fact is one of the key developments that led to the formulation of that theorem.

The **Â genus** is the genus associated to the characteristic power series

(There is also an Â genus which is less commonly used, associated to the characteristic series .) The first few values are

The Â genus of a spin manifold is an integer, and an even integer if the dimension is 4 mod 8 (which in dimension 4 implies Rochlin's theorem) – for general manifolds, the Â genus is not always an integer. This was proven by Hirzebruch and Armand Borel; this result both motivated and was later explained by the Atiyah–Singer index theorem, which showed that the Â genus of a spin manifold is equal to the index of its Dirac operator.

By combining this index result with a Weitzenbock formula for the Dirac Laplacian, André Lichnerowicz deduced that if a compact spin manifold admits a metric with positive scalar curvature, its Â genus must vanish. This only gives an obstruction to positive scalar curvature when the dimension is a multiple of 4, but Nigel Hitchin later discovered an analogous -valued obstruction in dimensions 1 or 2 mod 8. These results are essentially sharp. Indeed, Mikhail Gromov, H. Blaine Lawson, and Stephan Stolz later proved that the Â genus and Hitchin's -valued analog are the only obstructions to the existence of positive-scalar-curvature metrics on simply-connected spin manifolds of dimension greater than or equal to 5.

A genus is called an **elliptic genus** if the power series satisfies the condition

for constants and . (As usual, *Q* is the characteristic power series of the genus.)

One explicit expression for *f*(*z*) is

where

and *sn* is the Jacobi elliptic function.

Examples:

- . This is the L-genus.
- . This is the Â genus.
- . This is a generalization of the L-genus.

The first few values of such genera are:

Example (Elliptic genus for quaternionic projective plane) :

Example (Elliptic genus for octonionic projective plane (Cayley plane)):

The **Witten genus** is the genus associated to the characteristic power series

where σ_{L} is the Weierstrass sigma function for the lattice *L*, and *G* is a multiple of an Eisenstein series.

The Witten genus of a 4*k* dimensional compact oriented smooth spin manifold with vanishing first Pontryagin class is a modular form of weight 2*k*, with integral Fourier coefficients.

- ↑ McTague, Carl (2014) "Computing Hirzebruch L-Polynomials".
- ↑ Huybrechts, Daniel. "14.1 Existence, uniqueness, and embeddings of lattices".
*Lectures on K3 Surfaces*(PDF). p. 285.

A **hydrogen atom** is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively charged proton and a single negatively charged electron bound to the nucleus by the Coulomb force. **Atomic hydrogen** constitutes about 75% of the baryonic mass of the universe.

In mechanics and geometry, the **3D rotation group**, often denoted **SO(3)**, is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition. By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance, and orientation. Every non-trivial rotation is determined by its axis of rotation and its angle of rotation. Composing two rotations results in another rotation; every rotation has a unique inverse rotation; and the identity map satisfies the definition of a rotation. Owing to the above properties, the set of all rotations is a group under composition. Rotations are not commutative, making it a nonabelian group. Moreover, the rotation group has a natural structure as a manifold for which the group operations are smoothly differentiable; so it is in fact a Lie group. It is compact and has dimension 3.

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In number theory, the study of **Diophantine approximation** deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria.

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In quantum mechanics, the **canonical commutation relation** is the fundamental relation between canonical conjugate quantities. For example,

In general relativity, the **Gibbons–Hawking–York boundary term** is a term that needs to be added to the Einstein–Hilbert action when the underlying spacetime manifold has a boundary.

The **Voigt effect** is a magneto-optical phenomenon which rotates and elliptizes linearly polarised light sent into an optically active medium. Unlike many other magneto-optical effects such as the Kerr or Faraday effect which are linearly proportional to the magnetization, the Voigt effect is proportional to the square of the magnetization and can be seen experimentally at normal incidence. There are several denominations for this effect in the literature: the *Cotton–Mouton effect*, the *Voigt effect*, and *magnetic-linear birefringence*. This last denomination is closer in the physical sense, where the Voigt effect is a magnetic birefringence of the material with an index of refraction parallel and perpendicular ) to the magnetization vector or to the applied magnetic field.

An ** a priori probability** is a probability that is derived purely by deductive reasoning. One way of deriving

In 1927, a year after the publication of the Schrödinger equation, Hartree formulated what are now known as the **Hartree equations** for atoms, using the concept of *self-consistency* that Lindsay had introduced in his study of many electron systems in the context of Bohr theory. Hartree assumed that the nucleus together with the electrons formed a spherically symmetric field. The charge distribution of each electron was the solution of the Schrödinger equation for an electron in a potential , derived from the field. Self-consistency required that the final field, computed from the solutions was self-consistent with the initial field and he called his method the **self-consistent field** method.

**Multipole radiation** is a theoretical framework for the description of electromagnetic or gravitational radiation from time-dependent distributions of distant sources. These tools are applied to physical phenomena which occur at a variety of length scales - from gravitational waves due to galaxy collisions to gamma radiation resulting from nuclear decay. Multipole radiation is analyzed using similar multipole expansion techniques that describe fields from static sources, however there are important differences in the details of the analysis because multipole radiation fields behave quite differently from static fields. This article is primarily concerned with electromagnetic multipole radiation, although the treatment of gravitational waves is similar.

**Lagrangian field theory** is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.

The **Pomeranchuk instability** is an instability in the shape of the Fermi surface of a material with interacting fermions, causing Landau’s Fermi liquid theory to break down. It occurs when a Landau parameter in Fermi liquid theory has a sufficiently negative value, causing deformations of the Fermi surface to be energetically favourable. It is named after the Soviet physicist Isaak Pomeranchuk.

In two-dimensional conformal field theory, **Virasoro conformal blocks** are special functions that serve as building blocks of correlation functions. On a given punctured Riemann surface, Virasoro conformal blocks form a particular basis of the space of solutions of the conformal Ward identites. Zero-point blocks on the torus are characters of representations of the Virasoro algebra; four-point blocks on the sphere reduce to hypergeometric functions in special cases, but are in general much more complicated. In two dimensions as in other dimensions, conformal blocks play an essential role in the conformal bootstrap approach to conformal field theory.

The **two-dimensional critical Ising model** is the critical limit of the Ising model in two dimensions. It is a two-dimensional conformal field theory whose symmetry algebra is the Virasoro algebra with the central charge . Correlation functions of the spin and energy operators are described by the minimal model. While the minimal model has been exactly solved, the solution does not cover other observables such as connectivities of clusters.

- Friedrich Hirzebruch
*Topological Methods in Algebraic Geometry*ISBN 3-540-58663-6 Text of the original German version: http://hirzebruch.mpim-bonn.mpg.de/120/6/NeueTopologischeMethoden_2.Aufl.pdf - Friedrich Hirzebruch, Thomas Berger, Rainer Jung
*Manifolds and Modular Forms*ISBN 3-528-06414-5 - Milnor, Stasheff,
*Characteristic classes*, ISBN 0-691-08122-0 - A.F. Kharshiladze (2001) [1994], "Pontryagin class",
*Encyclopedia of Mathematics*, EMS Press - "Elliptic genera",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994]

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