Simply connected space
In topology, a topological space is called connected if it is path-connected and every path between two points can be continuously transformed into any other such path while preserving the two endpoints in question. The fundamental group of a topological space is an indicator of the failure for the space to be connected: a path-connected topological space is connected if and only if its fundamental group is trivial. A topological space X is called connected if it is path-connected and any loop in X defined by f: S1 → X can be contracted to a point: there exists a continuous map F: D2 → X such that F restricted to S1 is f. Here, S1 and D2 closed unit disk in the Euclidean plane respectively. An equivalent formulation is this: X is connected if and only if it is path-connected, whenever p: → X and q: → X are two paths with the same start and endpoint p can be continuously deformed into q while keeping both endpoints fixed. Explicitly, there exists a continuous homotopy F: × → X such that F = F = q.
A topological space X is connected if and only if X is path-connected and the fundamental group of X at each point is trivial, i.e. consists only of the identity element. X is connected if and only if for all points x, y ∈ X, the set of morphisms Hom Π in the fundamental groupoid of X has only one element. In complex analysis: an open subset X ⊆ C is connected if and only if both X and its complement in the Riemann sphere are connected; the set of complex numbers with imaginary part greater than zero and less than one, furnishes a nice example of an unbounded, open subset of the plane whose complement is not connected. It is simply connected, it might be worth pointing out that a relaxation of the requirement that X be connected leads to an interesting exploration of open subsets of the plane with connected extended complement. For example, a open set has connected extended complement when each of its connected components are connected. Informally, an object in our space is connected if it consists of one piece and does not have any "holes" that pass all the way through it.
For example, neither a doughnut nor a coffee cup is connected, but a hollow rubber ball is connected. In two dimensions, a circle is not connected, but a disk and a line are. Spaces that are connected but not connected are called non-simply connected or multiply connected; the definition only rules out handle-shaped holes. A sphere is connected, because any loop on the surface of a sphere can contract to a point though it has a "hole" in the hollow center; the stronger condition, that the object has no holes of any dimension, is called contractibility. The Euclidean plane R2 is connected, but R2 minus the origin is not. If n > 2 both Rn and Rn minus the origin are connected. Analogously: the n-dimensional sphere Sn is connected if and only if n > 2. Every convex subset of Rn is connected. A torus, the cylinder, the Möbius strip, the projective plane and the Klein bottle are not connected; every topological vector space is connected. For n ≥ 2, the special orthogonal group SO is not connected and the special unitary group SU is connected.
The one-point compactification of R is not connected. The long line L is connected, but its compactification, the extended long line L* is not. A surface is connected if and only if it is connected and its genus is 0. A universal cover of any space X is a connected space which maps to X via a covering map. If X and Y are homotopy equivalent and X is connected so is Y; the image of a connected set under a continuous function need not be connected. Take for example the complex plane under the exponential map: the image is C -, not connected; the notion of simple connectedness is important in complex analysis because of the following facts: The Cauchy's integral theorem states that if U is a connected open subset of the complex plane C, f: U → C is a holomorphic function f has an antiderivative F on U, the value of every line integral in U with integrand f depends only on the end points u and v of the path, can be computed as F - F. The integral thus does not depend on the particular path connecting u and v.
The Riemann mapping theorem states that any non-empty open connected subset of C is conformally eq
Algebraic varieties are the central objects of study in algebraic geometry. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition.:58Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility; the concept of an algebraic variety is similar to that of an analytic manifold. An important difference is that an algebraic variety may have singular points, while a manifold cannot; the fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial in one variable with complex number coefficients is determined by the set of its roots in the complex plane.
Generalizing this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and algebraic sets. Using the Nullstellensatz and related results, mathematicians have established a strong correspondence between questions on algebraic sets and questions of ring theory; this correspondence is a defining feature of algebraic geometry. An affine variety over an algebraically closed field is conceptually the easiest type of variety to define, which will be done in this section. Next, one can define quasi-projective varieties in a similar way; the most general definition of a variety is obtained by patching together smaller quasi-projective varieties. It is not obvious that one can construct genuinely new examples of varieties in this way, but Nagata gave an example of such a new variety in the 1950s. For an algebraically closed field K and a natural number n, let An be affine n-space over K; the polynomials f in the ring K can be viewed as K-valued functions on An by evaluating f at the points in An, i.e. by choosing values in K for each xi.
For each set S of polynomials in K, define the zero-locus Z to be the set of points in An on which the functions in S vanish, to say Z =. A subset V of An is called an affine algebraic set if V = Z for some S.:2 A nonempty affine algebraic set V is called irreducible if it cannot be written as the union of two proper algebraic subsets.:3 An irreducible affine algebraic set is called an affine variety.:3 Affine varieties can be given a natural topology by declaring the closed sets to be the affine algebraic sets. This topology is called the Zariski topology.:2Given a subset V of An, we define I to be the ideal of all polynomial functions vanishing on V: I =. For any affine algebraic set V, the coordinate ring or structure ring of V is the quotient of the polynomial ring by this ideal.:4 Let k be an algebraically closed field and let Pn be the projective n-space over k. Let f in k be a homogeneous polynomial of degree d, it is not well-defined to evaluate f on points in Pn in homogeneous coordinates.
However, because f is homogeneous, meaning that f = λd f , it does make sense to ask whether f vanishes at a point. For each set S of homogeneous polynomials, define the zero-locus of S to be the set of points in Pn on which the functions in S vanish: Z =. A subset V of Pn is called a projective algebraic set if V = Z for some S.:9 An irreducible projective algebraic set is called a projective variety.:10Projective varieties are equipped with the Zariski topology by declaring all algebraic sets to be closed. Given a subset V of Pn, let I be the ideal generated by all homogeneous polynomials vanishing on V. For any projective algebraic set V, the coordinate ring of V is the quotient of the polynomial ring by this ideal.:10A quasi-projective variety is a Zariski open subset of a projective variety. Notice that every affine variety is quasi-projective. Notice that the complement of an algebraic set in an affine variety is a quasi-projective variety. In classical algebraic geometry, a
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. A feature of quaternions is. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space or equivalently as the quotient of two vectors. Quaternions are represented in the form: a + b i + c j + d k where a, b, c, d are real numbers, i, j, k are the fundamental quaternion units. Quaternions find uses in both pure and applied mathematics, in particular for calculations involving three-dimensional rotations such as in three-dimensional computer graphics, computer vision, crystallographic texture analysis. In practical applications, they can be used alongside other methods, such as Euler angles and rotation matrices, or as an alternative to them, depending on the application. In modern mathematical language, quaternions form a four-dimensional associative normed division algebra over the real numbers, therefore a domain.
In fact, the quaternions were the first noncommutative division algebra. The algebra of quaternions is denoted by H, or in blackboard bold by H, it can be given by the Clifford algebra classifications Cℓ0,2 ≅ Cℓ03,0. The algebra ℍ holds a special place in analysis since, according to the Frobenius theorem, it is one of only two finite-dimensional division rings containing the real numbers as a proper subring, the other being the complex numbers; these rings are Euclidean Hurwitz algebras, of which quaternions are the largest associative algebra. Further extending the quaternions yields the non-associative octonions, the last normed division algebra over the reals; the unit quaternions can be thought of as a choice of a group structure on the 3-sphere S3 that gives the group Spin, isomorphic to SU and to the universal cover of SO. Quaternions were introduced by Hamilton in 1843. Important precursors to this work included Euler's four-square identity and Olinde Rodrigues' parameterization of general rotations by four parameters, but neither of these writers treated the four-parameter rotations as an algebra.
Carl Friedrich Gauss had discovered quaternions in 1819, but this work was not published until 1900. Hamilton knew that the complex numbers could be interpreted as points in a plane, he was looking for a way to do the same for points in three-dimensional space. Points in space can be represented by their coordinates, which are triples of numbers, for many years he had known how to add and subtract triples of numbers. However, Hamilton had been stuck on the problem of division for a long time, he could not figure out. The great breakthrough in quaternions came on Monday 16 October 1843 in Dublin, when Hamilton was on his way to the Royal Irish Academy where he was going to preside at a council meeting; as he walked along the towpath of the Royal Canal with his wife, the concepts behind quaternions were taking shape in his mind. When the answer dawned on him, Hamilton could not resist the urge to carve the formula for the quaternions, i 2 = j 2 = k 2 = i j k = − 1 into the stone of Brougham Bridge as he paused on it.
Although the carving has since faded away, there has been an annual pilgrimage since 1989 called the Hamilton Walk for scientists and mathematicians who walk from Dunsink Observatory to the Royal Canal bridge in remembrance of Hamilton's discovery. On the following day, Hamilton wrote a letter to his friend and fellow mathematician, John T. Graves, describing the train of thought that led to his discovery; this letter was published in a letter to a science magazine. An electric circuit seemed to close, a spark flashed forth. Hamilton called a quadruple with these rules of multiplication a quaternion, he devoted most of the remainder of his life to studying and teaching them. Hamilton's treatment is more geometric than the modern approach, which emphasizes quaternions' algebraic properties, he founded a school of "quaternionists", he tried to popularize quaternions in several books. The last and longest of his books, Elements of Quaternions, was 800 pages long. After Hamilton's death, his student Peter Tait continued promoting quaternions.
At this time, quaternions were a mandatory examination topic in Dublin. Topics in physics and geometry that would now be described using vectors, such as kinematics in space and Maxwell's equations, were described in terms of quaternions. There was a professional research association, the Quaternion Society, devoted to the study of quaternions and other hypercomplex number systems. From the mid-1880s, quaternions began to be displaced by vector analysis, developed by Josiah Willard Gibbs, Oliver Heaviside, Hermann von Helmholtz. Vector analys
ArXiv is a repository of electronic preprints approved for posting after moderation, but not full peer review. It consists of scientific papers in the fields of mathematics, astronomy, electrical engineering, computer science, quantitative biology, mathematical finance and economics, which can be accessed online. In many fields of mathematics and physics all scientific papers are self-archived on the arXiv repository. Begun on August 14, 1991, arXiv.org passed the half-million-article milestone on October 3, 2008, had hit a million by the end of 2014. By October 2016 the submission rate had grown to more than 10,000 per month. ArXiv was made possible by the compact TeX file format, which allowed scientific papers to be transmitted over the Internet and rendered client-side. Around 1990, Joanne Cohn began emailing physics preprints to colleagues as TeX files, but the number of papers being sent soon filled mailboxes to capacity. Paul Ginsparg recognized the need for central storage, in August 1991 he created a central repository mailbox stored at the Los Alamos National Laboratory which could be accessed from any computer.
Additional modes of access were soon added: FTP in 1991, Gopher in 1992, the World Wide Web in 1993. The term e-print was adopted to describe the articles, it began as a physics archive, called the LANL preprint archive, but soon expanded to include astronomy, computer science, quantitative biology and, most statistics. Its original domain name was xxx.lanl.gov. Due to LANL's lack of interest in the expanding technology, in 2001 Ginsparg changed institutions to Cornell University and changed the name of the repository to arXiv.org. It is now hosted principally with eight mirrors around the world, its existence was one of the precipitating factors that led to the current movement in scientific publishing known as open access. Mathematicians and scientists upload their papers to arXiv.org for worldwide access and sometimes for reviews before they are published in peer-reviewed journals. Ginsparg was awarded a MacArthur Fellowship in 2002 for his establishment of arXiv; the annual budget for arXiv is $826,000 for 2013 to 2017, funded jointly by Cornell University Library, the Simons Foundation and annual fee income from member institutions.
This model arose in 2010, when Cornell sought to broaden the financial funding of the project by asking institutions to make annual voluntary contributions based on the amount of download usage by each institution. Each member institution pledges a five-year funding commitment to support arXiv. Based on institutional usage ranking, the annual fees are set in four tiers from $1,000 to $4,400. Cornell's goal is to raise at least $504,000 per year through membership fees generated by 220 institutions. In September 2011, Cornell University Library took overall administrative and financial responsibility for arXiv's operation and development. Ginsparg was quoted in the Chronicle of Higher Education as saying it "was supposed to be a three-hour tour, not a life sentence". However, Ginsparg remains on the arXiv Scientific Advisory Board and on the arXiv Physics Advisory Committee. Although arXiv is not peer reviewed, a collection of moderators for each area review the submissions; the lists of moderators for many sections of arXiv are publicly available, but moderators for most of the physics sections remain unlisted.
Additionally, an "endorsement" system was introduced in 2004 as part of an effort to ensure content is relevant and of interest to current research in the specified disciplines. Under the system, for categories that use it, an author must be endorsed by an established arXiv author before being allowed to submit papers to those categories. Endorsers are not asked to review the paper for errors, but to check whether the paper is appropriate for the intended subject area. New authors from recognized academic institutions receive automatic endorsement, which in practice means that they do not need to deal with the endorsement system at all. However, the endorsement system has attracted criticism for restricting scientific inquiry. A majority of the e-prints are submitted to journals for publication, but some work, including some influential papers, remain purely as e-prints and are never published in a peer-reviewed journal. A well-known example of the latter is an outline of a proof of Thurston's geometrization conjecture, including the Poincaré conjecture as a particular case, uploaded by Grigori Perelman in November 2002.
Perelman appears content to forgo the traditional peer-reviewed journal process, stating: "If anybody is interested in my way of solving the problem, it's all there – let them go and read about it". Despite this non-traditional method of publication, other mathematicians recognized this work by offering the Fields Medal and Clay Mathematics Millennium Prizes to Perelman, both of which he refused. Papers can be submitted in any of several formats, including LaTeX, PDF printed from a word processor other than TeX or LaTeX; the submission is rejected by the arXiv software if generating the final PDF file fails, if any image file is too large, or if the total size of the submission is too large. ArXiv now allows one to store and modify an incomplete submission, only finalize the submission when ready; the time stamp on the article is set. The standard access route is through one of several mirrors. Sev
Kunihiko Kodaira was a Japanese mathematician known for distinguished work in algebraic geometry and the theory of complex manifolds, as the founder of the Japanese school of algebraic geometers. He was awarded a Fields Medal in 1954. Kodaira was born in Tokyo, he graduated from the University of Tokyo in 1938 with a degree in mathematics and graduated from the physics department at the University of Tokyo in 1941. During the war years he worked in isolation, but was able to master Hodge theory as it stood, he obtained his Ph. D. from the University of Tokyo in 1949, with a thesis entitled Harmonic fields in Riemannian manifolds. He was involved in cryptographic work from about 1944, while holding an academic post in Tokyo. In 1949 he travelled to the Institute for Advanced Study in Princeton, New Jersey at the invitation of Hermann Weyl, he was subsequently appointed Associate Professor at Princeton University in 1952 and promoted to Professor in 1955. At this time the foundations of Hodge theory were being brought in line with contemporary technique in operator theory.
Kodaira became involved in exploiting the tools it opened up in algebraic geometry, adding sheaf theory as it became available. This work was influential, for example on Friedrich Hirzebruch. In a second research phase, Kodaira wrote a long series of papers in collaboration with Donald C. Spencer, founding the deformation theory of complex structures on manifolds; this gave the possibility of constructions of moduli spaces, since in general such structures depend continuously on parameters. It identified the sheaf cohomology groups, for the sheaf associated with the holomorphic tangent bundle, that carried the basic data about the dimension of the moduli space, obstructions to deformations; this theory is still foundational, had an influence on the scheme theory of Grothendieck. Spencer continued this work, applying the techniques to structures other than complex ones, such as G-structures. In a third major part of his work, Kodaira worked again from around 1960 through the classification of algebraic surfaces from the point of view of birational geometry of complex manifolds.
This resulted in a typology of seven kinds of two-dimensional compact complex manifolds, recovering the five algebraic types known classically. He provided detailed studies of elliptic fibrations of surfaces over a curve, or in other language elliptic curves over algebraic function fields, a theory whose arithmetic analogue proved important soon afterwards; this work included a characterisation of K3 surfaces as deformations of quartic surfaces in P4, the theorem that they form a single diffeomorphism class. Again, this work has proved foundational.. Kodaira left Princeton University and the Institute for Advanced Study in 1961, served as chair at the Johns Hopkins University and Stanford University. In 1967, returned to the University of Tokyo, he was awarded a Wolf Prize in 1984/5. He died in Kofu on 26 July 1997. Morrow, James. J. ISBN 978-0-691-08158-8, MR 0366598 Kodaira, Baily, Walter L. ed. Kunihiko Kodaira: collected works, II, Iwanami Shoten, Tokyo. J. ISBN 978-0-691-08163-2, MR 0366599 Kodaira, Baily, Walter L. ed. Kunihiko Kodaira: collected works, III, Iwanami Shoten, Tokyo.
J. ISBN 978-0-691-08164-9, MR 0366600 Kodaira, Complex manifolds and deformation of complex structures, Classics in Mathematics, New York: Springer-Verlag, ISBN 978-3-540-22614-7, MR 0815922, review by Andrew J. Sommese Kodaira, Complex analysis, Cambridge Studies in Advanced Mathematics, 107, Cambridge University Press, ISBN 978-0-521-80937-5, MR 2343868 Bochner–Kodaira–Nakano identity Spectral theory of ordinary differential equations Kodaira vanishing theorem Kodaira–Spencer mapping Kodaira dimension Kodaira embedding theorem Enriques–Kodaira classification Kodaira's classification of singular fibers O'Connor, John J.. Donald C. Spencer, "Kunihiko Kodaira", Notices of the AMS, 45: 388–389. Friedrich Hirzebruch, "Kunihiko Kodaira: Mathematician and Teacher", Notices of the American Mathematical Society, 45: 1456–1462. "Special Issue to Honor Professor Kunihiko Kodaira on his 85th birthday", Asian Journal of Mathematics, 4, 2000
Complex projective space
In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex lines through the origin of a complex Euclidean space. Formally, a complex projective space is the space of complex lines through the origin of an -dimensional complex vector space; the space is denoted variously as Pn or CPn. When n = 1, the complex projective space CP1 is the Riemann sphere, when n = 2, CP2 is the complex projective plane. Complex projective space was first introduced by von Staudt as an instance of what was known as the "geometry of position", a notion due to Lazare Carnot, a kind of synthetic geometry that included other projective geometries as well. Subsequently, near the turn of the 20th century it became clear to the Italian school of algebraic geometry that the complex projective spaces were the most natural domains in which to consider the solutions of polynomial equations – algebraic varieties.
In modern times, both the topology and geometry of complex projective space are well understood and related to that of the sphere. Indeed, in a certain sense the -sphere can be regarded as a family of circles parametrized by CPn: this is the Hopf fibration. Complex projective space carries a metric, called the Fubini–Study metric, in terms of which it is a Hermitian symmetric space of rank 1. Complex projective space has many applications in both mathematics and quantum physics. In algebraic geometry, complex projective space is the home of projective varieties, a well-behaved class of algebraic varieties. In topology, the complex projective space plays an important role as a classifying space for complex line bundles: families of complex lines parametrized by another space. In this context, the infinite union of projective spaces, denoted CP∞, is the classifying space K. In quantum physics, the wave function associated to a pure state of a quantum mechanical system is a probability amplitude, meaning that it has unit norm, has an inessential overall phase: that is, the wave function of a pure state is a point in the projective Hilbert space of the state space.
The notion of a projective plane arises out of the idea of perspection in geometry and art: that it is sometimes useful to include in the Euclidean plane an additional "imaginary" line that represents the horizon that an artist, painting the plane, might see. Following each direction from the origin, there is a different point on the horizon, so the horizon can be thought of as the set of all directions from the origin; the Euclidean plane, together with its horizon, is called the real projective plane, the horizon is sometimes called a line at infinity. By the same construction, projective spaces can be considered in higher dimensions. For instance, the real projective 3-space is a Euclidean space together with a plane at infinity that represents the horizon that an artist would see; these real projective spaces can be constructed in a more rigorous way as follows. Here, let Rn+1 denote the real coordinate space of n+1 dimensions, regard the landscape to be painted as a hyperplane in this space.
Suppose that the eye of the artist is the origin in Rn+1. Along each line through his eye, there is a point of the landscape or a point on its horizon, thus the real projective space is the space of lines through the origin in Rn+1. Without reference to coordinates, this is the space of lines through the origin in an -dimensional real vector space. To describe the complex projective space in an analogous manner requires a generalization of the idea of vector and direction. Imagine that instead of standing in a real Euclidean space, the artist is standing in a complex Euclidean space Cn+1 and the landscape is a complex hyperplane. Unlike the case of real Euclidean space, in the complex case there are directions in which the artist can look which do not see the landscape. However, in a complex space, there is an additional "phase" associated with the directions through a point, by adjusting this phase the artist can guarantee that he sees the landscape; the "horizon" is the space of directions, but such that two directions are regarded as "the same" if they differ only by a phase.
The complex projective space is the landscape with the horizon attached "at infinity". Just like the real case, the complex projective space is the space of directions through the origin of Cn+1, where two directions are regarded as the same if they differ by a phase. Complex projective space is a complex manifold that may be described by n + 1 complex coordinates as Z = ∈ C n + 1, ≠ where the tuples differing by an overall rescaling are identified: ( Z
In mathematics, genus has a few different, but related, meanings. The most common concept, the genus of an surface, is the number of "holes"; this is made more precise below. The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected, it is equal to the number of handles on it. Alternatively, it can be defined in terms of the Euler characteristic χ, via the relationship χ = 2 − 2g for closed surfaces, where g is the genus. For surfaces with b boundary components, the equation reads χ b. In layman's terms, it's the number of "holes". A doughnut, or torus, has 1 such hole. A sphere has 0; the green surface pictured above has 2 holes of the relevant sort. For instance: The sphere S2 and a disc both have genus zero. A torus has genus one; this is the source of the joke "topologists are people who can't tell their donut from their coffee mug."An explicit construction of surfaces of genus g is given in the article on the fundamental polygon.
Genus of orientable surfaces In simpler terms, the value of an orientable surface's genus is equal to the number of "holes" it has. The non-orientable genus, demigenus, or Euler genus of a connected, non-orientable closed surface is a positive integer representing the number of cross-caps attached to a sphere. Alternatively, it can be defined for a closed surface in terms of the Euler characteristic χ, via the relationship χ = 2 − k, where k is the non-orientable genus. For instance: A projective plane has non-orientable genus one. A Klein bottle has non-orientable genus two; the genus of a knot K is defined as the minimal genus of all Seifert surfaces for K. A Seifert surface of a knot is however a manifold with boundary, the boundary being the knot, i.e. homeomorphic to the unit circle. The genus of such a surface is defined to be the genus of the two-manifold, obtained by gluing the unit disk along the boundary; the genus of a 3-dimensional handlebody is an integer representing the maximum number of cuttings along embedded disks without rendering the resultant manifold disconnected.
It is equal to the number of handles on it. For instance: A ball has genus zero. A solid torus D2 × S1 has genus one; the genus of a graph is the minimal integer n such that the graph can be drawn without crossing itself on a sphere with n handles. Thus, a planar graph has genus 0; the non-orientable genus of a graph is the minimal integer n such that the graph can be drawn without crossing itself on a sphere with n cross-caps. The Euler genus is the minimal integer n such that the graph can be drawn without crossing itself on a sphere with n cross-caps or on a sphere with n/2 handles. In topological graph theory there are several definitions of the genus of a group. Arthur T. White introduced the following concept; the genus of a group G is the minimum genus of a Cayley graph for G. The graph genus problem is NP-complete. There are two related definitions of genus of any projective algebraic scheme X: the arithmetic genus and the geometric genus; when X is an algebraic curve with field of definition the complex numbers, if X has no singular points these definitions agree and coincide with the topological definition applied to the Riemann surface of X.
For example, the definition of elliptic curve from algebraic geometry is connected non-singular projective curve of genus 1 with a given rational point on it. By the Riemann-Roch theorem, an irreducible plane curve of degree d has geometric genus g = 2 − s, where s is the number of singularities when properly counted. Genus can be calculated for the graph spanned by the net of chemical interactions in nucleic acids or proteins. In particular, one may study the growth of the genus along the chain; such a function shows the topological domain structure of biomolecules. Cayley graph Group Arithmetic genus Geometric genus Genus of a multiplicative sequence Genus of a quadratic form Spinor genus