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# Glossary cards — Chapter ᚜(Epilogue): The homeward view

*Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#group-T28, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.*

[Guide for AI readers](index.md) · [The chapter](ch28.md)

*Seven prizes as seven numbers: analytic number theory, elliptic curves, topology, complexity, algebraic geometry, fluid dynamics, and quantum field theory. Each card: its symbol, type, one-line headline, the longer note, the reference entries it cites and the cards it links to.*

### the reversed feather mark — `᚜` {#g-og-feather-rev}

*Ogham mark*

Not a letter but punctuation: the reversed ogham feather mark closes the text.

U+169C OGHAM REVERSED FEATHER MARK is classed as close punctuation and is the partner of the feather mark; the Unicode Standard gives the two as marking the beginning and the end of ogham text, particularly in manuscript ogham. Like its partner it is not one of the twenty-six letters, so it has no name-meaning, no word-ogham, and no plant. And like its partner it spends no letter, which is how the piece comes to have twenty-five lettered chapters standing between the two marks, with Ruis outside them on the left.

Cites: [2.13](ch02-refs.md#r-2.13), [2.15](ch02-refs.md#r-2.15), [2.17](ch02-refs.md#r-2.17)

See also: [the feather mark](ch02-cards.md#g-og-feather)

### de Bruijn–Newman constant — `Λ_{dBN}` {#g-dbn}

*Status open*

A definite real number nobody knows: it is 0 exactly when the Riemann Hypothesis is true, and otherwise lies in (0, 0.2].

The de Bruijn–Newman constant. Nicolaas de Bruijn introduced in 1950 a one-parameter heat-flow deformation of the Riemann ξ function and showed that for parameters ≥ ½ all its zeros are real; Charles Newman proved in 1976 that there is a single real constant Λ such that the zeros are real exactly when the parameter is ≥ Λ, and conjectured Λ ≥ 0, remarking that the Riemann Hypothesis, if true, is only barely so. The Riemann Hypothesis is equivalent to Λ ≤ 0. Brad Rodgers and Terence Tao proved Λ ≥ 0 (arXiv 2018), and Platt and Trudgian’s verification of the hypothesis up to height 3·10¹² gives Λ ≤ 0.2 (preprint 2020, published 2021). So Λ is a definite real in [0, 0.2] that is exactly 0 if and only if the hypothesis holds; its status is open, which is its color, and because it may be zero it can never sit in a denominator.

Cites: [28.3](ch28-refs.md#r-28.3), [28.4](ch28-refs.md#r-28.4)

See also: [the Riemann Hypothesis](ch18-cards.md#g-rh), [the Riemann xi function](ch18-cards.md#g-xifn)

### the L-function of the rank-one curve at one — `L(E₃₇, 1)` {#g-lE37}

*Integer or rational*

Exactly 0: the curve’s root number is −1; LMFDB records analytic rank 1 and algebraic rank 1, so the rank equality predicted by Birch and Swinnerton-Dyer holds for this curve.

The L-function of the elliptic curve of conductor 37, y² + y = x³ − x (LMFDB 37.a1), the smallest-conductor curve of rank one, evaluated at s = 1. Its root number is −1, so L(E, s) vanishes at s = 1 to odd order and L(E₃₇, 1) is exactly 0. LMFDB records analytic rank 1 and algebraic rank 1 for the curve, so the equality of analytic and algebraic ranks predicted by Birch and Swinnerton-Dyer holds here. The value is the integer 0, which is its color, and it is the zero that makes the whole epilogue vanish.

Cites: [19.8](ch19-refs.md#r-19.8), [28.5](ch28-refs.md#r-28.5)

See also: [the curve of conductor 37](ch28-cards.md#g-E37), [the L-function of an elliptic curve](ch19-cards.md#g-LfuncE), [a Dirichlet L-function](ch18-cards.md#g-dirichletL), [the rank of an elliptic curve](ch28-cards.md#g-rankE)

### the Poincaré conjecture, as a bit — `[every simply connected closed 3-manifold is S³]` {#g-poincare}

*Integer or rational*

Conjectured in 1904, proven by Perelman in 2002–2003, the only Millennium Problem solved; the bracket is 1.

The Poincaré conjecture as an Iverson bracket: 1 if every simply connected closed 3-manifold is the 3-sphere, 0 otherwise. Henri Poincaré posed it in 1904 after finding a counterexample to his own earlier version; Grigori Perelman proved it in 2002–2003 using Hamilton’s Ricci flow, in three papers posted online and never sent to a journal, and declined the 2006 Fields Medal and the 2010 Clay prize. The bracket is the integer 1, the only Millennium bit whose value is known.

Cites: [28.6](ch28-refs.md#r-28.6)

See also: [Iverson bracket](ch08-cards.md#g-iverson), [Ricci flow](ch28-cards.md#g-ricci), [a closed three-manifold](ch28-cards.md#g-threemfld)

### the P versus NP question, as a bit — `[P = NP]` {#g-pnp}

*Integer or rational*

A definite integer, 0 or 1, that nobody knows: 1 if every problem whose answer can be checked quickly can also be solved quickly, 0 if not. Most complexity theorists expect 0.

The P versus NP question as an Iverson bracket. Stephen Cook formulated it in 1971, Leonid Levin independently in work published in 1973, and Richard Karp showed in 1972 that twenty-one natural problems are all equally hard, which is why the question matters: P is what can be solved quickly, NP what can be checked quickly, and [P = NP] is 1 if they coincide and 0 if not. It is a definite integer nobody knows, so it is colored as an integer rather than as an open real; finite either way, it multiplies in safely and can never sit in a denominator. Most complexity theorists expect 0, in which case this brace joins L(E₃₇, 1) in killing the product; if 1, it is a factor of one and changes nothing.

Cites: [28.17](ch28-refs.md#r-28.17), [28.7](ch28-refs.md#r-28.7)

See also: [Iverson bracket](ch08-cards.md#g-iverson), [the L-function of the rank-one curve at one](ch28-cards.md#g-lE37)

### the Hodge conjecture, as a bit — `[every Hodge class is algebraic]` {#g-hodge}

*Integer or rational*

A definite integer, 0 or 1, that nobody knows: 1 if, on every smooth projective complex variety, every rational cohomology class of type (p, p) comes from algebraic subvarieties, 0 if some class does not.

The Hodge conjecture as an Iverson bracket. W. V. D. Hodge stated it in 1950: on a smooth projective complex variety, every rational cohomology class of type (p, p) is a rational combination of classes of algebraic subvarieties. Atiyah and Hirzebruch showed in work published in 1962 that the version with integer coefficients is false, which is why the bracket must say rational. The natural alternative, a count of the Hodge classes that are not algebraic taken over all varieties, could be infinite if the conjecture fails, and the piece forbids an infinity beside a zero; the bracket is the only shape that obeys the rules. A definite integer, 0 or 1.

Cites: [28.21](ch28-refs.md#r-28.21), [28.8](ch28-refs.md#r-28.8)

See also: [cohomology](ch12-cards.md#g-cohomology), [a Hodge class](ch28-cards.md#g-hodgeclass), [Iverson bracket](ch08-cards.md#g-iverson), [a smooth projective variety](ch28-cards.md#g-smoothproj)

### the Navier–Stokes blow-up rate — `1/T_{∗}` {#g-nsrate}

*Status open*

A definite real number, zero or positive, that nobody knows: the least upper bound of the reciprocal smooth lifetimes of unforced three-dimensional flows started from data of enstrophy at most one, and 0 exactly when no such flow ever breaks down.

The Navier–Stokes blow-up rate, a number built for this book from the Clay problem. Claude-Louis Navier (1822) and George Stokes (1845) wrote the equations of viscous flow; Jean Leray proved in 1934 that smooth solutions exist for a short time and weak solutions exist forever, and the prize asks whether smooth solutions can always continue for all time in three dimensions. Definition: take every smooth, divergence-free, rapidly decaying starting velocity on ℝ³ with unit viscosity and enstrophy at most one (the integral of the squared velocity gradient); let T(u₀) be how long the smooth solution from u₀ lasts; set 1/T∗ := sup 1/T(u₀), reading 1/∞ as 0. Leray’s local theory gives every such flow a lifetime of at least a universal c > 0, so the number lies in [0, 1/c] and is finite. The equations’ scaling symmetry shrinks enstrophy while stretching time, so any blow-up anywhere rescales to one inside the unit ball, and, by this book’s own argument, 1/T∗ = 0 if and only if the Clay statement of existence and smoothness holds on ℝ³; without the unit-enstrophy normalization the same scaling would push the earliest blow-up time to 0 and the rate to infinity, which is why the definition must be stated carefully. Written as a supremum of reciprocals so that T∗, which may be infinite, is never divided by. In September 2026 a finite-time singularity was announced for a forced flow and Clay marked the problem’s page active; the official problem is four separate statements, and that result bears on the forced ones, while this number is a supremum over flows with no force term. Status open, like its twin Λ_dBN.

Cites: [28.10](ch28-refs.md#r-28.10), [28.22](ch28-refs.md#r-28.22), [28.26](ch28-refs.md#r-28.26), [28.9](ch28-refs.md#r-28.9)

See also: [de Bruijn–Newman constant](ch28-cards.md#g-dbn), [enstrophy](ch28-cards.md#g-enstrophy), [a weak solution of Navier–Stokes](ch28-cards.md#g-weaksol)

### the complete Yang–Mills problem, as a bit — `[a non-trivial quantum Yang–Mills theory exists on ℝ⁴ and has Δ>0]` {#g-ymbit}

*Integer or rational*

A definite integer, 0 or 1, that nobody knows: it includes both the rigorous existence of a non-trivial theory on ℝ⁴ for every compact simple gauge group and a positive mass gap, so it is never undefined.

The official Clay problem asks for a non-trivial quantum Yang–Mills theory on ℝ⁴ for every compact simple gauge group, satisfying rigorous axioms, and for a mass gap Δ>0. A raw Δ presupposes the theory’s construction; without it a quotient such as Δ/m_{p} would be undefined. The Iverson bracket is total: 1 if the complete statement is true, 0 otherwise. Its value remains unknown, but it is always a finite integer and never appears in a denominator.

Cites: [28.11](ch28-refs.md#r-28.11), [28.12](ch28-refs.md#r-28.12)

See also: [Iverson bracket](ch08-cards.md#g-iverson), [proton mass](ch28-cards.md#g-mp), [the Yang–Mills mass gap](ch28-cards.md#g-ymgap)

### the Yang–Mills mass gap — `Δ` {#g-ymgap}

*Status open*

The mass of the lightest particle of a pure Yang–Mills theory in four dimensions; the conjecture is that it is positive, and lattice computations put it near 1.7 GeV for SU(3).

The Yang–Mills mass gap. Chen Ning Yang and Robert Mills wrote the non-abelian gauge theory in 1954. The Clay problem asks for a rigorous construction of a non-trivial quantum Yang–Mills theory for every compact simple gauge group and a proof of a positive gap Δ. A numerical Δ presupposes that construction, so the book keeps Δ in prose and represents the complete prize problem with a total Iverson bit. Lattice glueball computations remain physical motivation, not a proof of the Clay statement. Status open.

Cites: [28.11](ch28-refs.md#r-28.11), [28.19](ch28-refs.md#r-28.19), [28.26](ch28-refs.md#r-28.26)

See also: [the lightest glueball](ch28-cards.md#g-glueball), [Iverson bracket](ch08-cards.md#g-iverson)

### proton mass — `m_{p}` {#g-mp}

*Physics: measured*

The measured rest mass of the proton, 938.272 MeV/c² (CODATA 2022). It enters the formula only through the proton-to-electron mass ratio mp/me; against the Yang–Mills mass gap Δ it is a comparison in prose, never a divisor, since no Δ has been shown to exist.

The proton’s rest mass is 938.27208943(29) MeV/c² in CODATA 2022. The book does not divide an unconstructed mass gap by this value; the proton remains a useful physical comparison in prose. Physics: measured.

Cites: [28.13](ch28-refs.md#r-28.13)

See also: [the Yang–Mills mass gap](ch28-cards.md#g-ymgap)

### the critical line — `Re(s) = ½` {#g-critline}

*Function or operator*

Where the Riemann Hypothesis says every nontrivial zero of ζ lies; more than 10¹² of them have been checked and all are there.

The critical line Re(s) = ½ in the complex plane, on which Riemann conjectured in 1859 that every nontrivial zero of ζ(s) lies. Platt and Trudgian rigorously verified (preprint 2020, published 2021) that every zero up to height 3·10¹² is on it. The de Bruijn–Newman constant is 0 exactly when all the rest lie on it too. A locus of the framework, not a value.

Cites: [18.17](ch18-refs.md#r-18.17), [18.28](ch18-refs.md#r-18.28), [28.14](ch28-refs.md#r-28.14), [28.26](ch28-refs.md#r-28.26)

See also: [the complex plane](ch21-cards.md#g-cplane), [de Bruijn–Newman constant](ch28-cards.md#g-dbn), [the Riemann Hypothesis](ch18-cards.md#g-rh), [Riemann zeta function](ch18-cards.md#g-zeta)

### the rank of an elliptic curve — `rank(E)` {#g-rankE}

*Function or operator*

How many independent rational points of infinite order a curve has; the Birch and Swinnerton-Dyer conjecture ties it to the L-function.

The rank of an elliptic curve over ℚ, the number of independent rational points of infinite order in its Mordell–Weil group, which Mordell proved finitely generated in 1922. The curve of conductor 37 has rank 1. The Birch and Swinnerton-Dyer conjecture says the rank equals the order of vanishing of L(E, s) at s = 1. An integer-valued invariant of the framework.

Cites: [19.19](ch19-refs.md#r-19.19), [28.15](ch28-refs.md#r-28.15)

See also: [the curve of conductor 37](ch28-cards.md#g-E37), [the L-function of an elliptic curve](ch19-cards.md#g-LfuncE), [the Birch and Swinnerton-Dyer conjecture](ch28-cards.md#g-bsdconj), [the Mordell–Weil theorem](ch19-cards.md#g-mordellweil)

### the curve of conductor 37 — `y² + y = x³ − x` {#g-E37}

*Non-real object*

The smallest-conductor elliptic curve of rank one, LMFDB 37.a1, whose L-function vanishes at 1.

The elliptic curve y² + y = x³ − x, Cremona’s 37a1 and LMFDB’s 37.a1: conductor 37, discriminant 37, rank 1, no torsion, generated by the point (0, 0). It is the first curve in conductor order whose rank is positive; LMFDB lists its invariants for the Birch and Swinnerton-Dyer conjecture, and it is the curve this epilogue brings back from Chapter ᚒ(U). A non-real object.

Cites: [19.8](ch19-refs.md#r-19.8), [28.15](ch28-refs.md#r-28.15), [28.5](ch28-refs.md#r-28.5)

See also: [the L-function of an elliptic curve](ch19-cards.md#g-LfuncE), [the Birch and Swinnerton-Dyer conjecture](ch28-cards.md#g-bsdconj), [the rank of an elliptic curve](ch28-cards.md#g-rankE)

### Ricci flow — `∂g/∂t = −2 Ric` {#g-ricci}

*Function or operator*

Hamilton’s 1982 equation that lets a shape’s metric evolve like heat; Perelman drove it through its singularities to prove Poincaré.

Ricci flow, the equation ∂g/∂t = −2 Ric(g) introduced by Richard Hamilton in 1982, which evolves the metric of a manifold so that curvature spreads out like heat. Hamilton’s program was to use the flow, which usually deforms a manifold towards a rounder shape, to prove the conjecture for every simply connected closed 3-manifold; Perelman’s three papers of 2002–2003 controlled the singularities with his entropy and surgery and completed it, proving both the Poincaré conjecture and Thurston’s geometrization. An operator of the framework.

Cites: [28.16](ch28-refs.md#r-28.16), [28.26](ch28-refs.md#r-28.26)

See also: [the Poincaré conjecture, as a bit](ch28-cards.md#g-poincare), [a closed three-manifold](ch28-cards.md#g-threemfld)

### the hardest problems in NP — `NP-complete` {#g-npcomplete}

*Function or operator*

Problems, such as satisfiability, to which every problem in NP reduces in polynomial time: solve one quickly and P = NP.

The NP-complete problems: those in NP to which every other NP problem reduces in polynomial time, so a fast algorithm for any one of them would give P = NP. Cook’s 1971 result established Boolean satisfiability as NP-complete; Karp used it in 1972 to show twenty other natural problems NP-complete, including graph coloring, knapsack, and the Hamiltonian circuit. P is the class solvable in polynomial time, NP the class checkable in polynomial time; the bracket [P = NP] is 1 exactly when the two coincide. A class of the framework.

Cites: [28.17](ch28-refs.md#r-28.17), [28.26](ch28-refs.md#r-28.26)

See also: [the P versus NP question, as a bit](ch28-cards.md#g-pnp)

### a Hodge class — `H^{p,p}(X) ∩ H^{2p}(X, ℚ)` {#g-hodgeclass}

*Function or operator*

On a smooth projective complex variety, a rational cohomology class of type (p, p): algebraic subvarieties always produce such classes, and the conjecture says every one of them is a rational combination of those.

A Hodge class on a smooth projective complex variety X is a rational cohomology class in H^(2p)(X, ℚ) whose complexification has Hodge type (p, p). The fundamental class of an algebraic subvariety of codimension p is always of type (p, p), and the Hodge conjecture says every rational (p, p)-class is a rational combination of such fundamental classes. An object of the framework.

Cites: [28.18](ch28-refs.md#r-28.18)

See also: [cohomology](ch12-cards.md#g-cohomology), [the Hodge conjecture, as a bit](ch28-cards.md#g-hodge), [a smooth projective variety](ch28-cards.md#g-smoothproj)

### a weak solution of Navier–Stokes — `Leray–Hopf solution` {#g-weaksol}

*Function or operator*

A flow that satisfies the equations in an averaged sense and, from finite-energy data, exists forever (Leray 1934); whether it is also smooth is the prize.

A weak, or Leray–Hopf, solution of the Navier–Stokes equations: a divergence-free velocity field of finite energy that satisfies the equations when tested against smooth functions and obeys the energy inequality. Jean Leray proved in 1934 that such a solution exists for all time from finite-energy data on ℝ³. Smooth solutions have been proved to exist only for a short time, or for all time when the data are small; the Clay problem asks whether the weak solutions are in fact smooth, or whether some smooth flow blows up. An object of the framework.

Cites: [28.9](ch28-refs.md#r-28.9)

### enstrophy — `∫|∇u|² dx` {#g-enstrophy}

*Function or operator*

The integral of the squared velocity gradient, the quantity whose growth signals a flow tearing itself apart; the rate 1/T∗ is defined over flows with enstrophy at most one.

Enstrophy, the integral of the squared velocity gradient, equal for divergence-free flows to the integral of the squared vorticity. Its rate of growth controls whether a Navier–Stokes solution stays smooth: Leray showed in 1934 that a smooth solution lasts at least a time proportional to one over the square of the initial enstrophy (with the viscosity set to 1), and that the enstrophy must blow up if the solution does. The blow-up rate 1/T∗ is normalized to starting flows of enstrophy at most one for exactly this reason. A quantity of the framework.

Cites: [28.10](ch28-refs.md#r-28.10), [28.9](ch28-refs.md#r-28.9)

See also: [the Navier–Stokes blow-up rate](ch28-cards.md#g-nsrate)

### the lightest glueball — `m(0^{++}) ≈ 1.7 GeV` {#g-glueball}

*Status open*

The lightest particle of pure SU(3) gauge theory on the lattice, about 1.7 GeV (1730 ± 50 ± 80 and 1710 ± 50 ± 80 MeV); the numerical face of the mass gap Δ.

The scalar glueball, the lightest state of pure SU(3) Yang–Mills theory, a bound state of gluons with no quarks. Lattice computations give its mass as 1730 ± 50 ± 80 MeV (Morningstar and Peardon 1999) and 1710 ± 50 ± 80 MeV (Chen et al. 2006), so the lightest glueball weighs about 1.8 times the proton (m_p, about 938 MeV), a comparison of the book’s own. No glueball has been unambiguously identified in experiment, where glueballs mix with ordinary mesons, which is why the value is a computation rather than a measurement. Status open, like Δ itself.

Cites: [28.19](ch28-refs.md#r-28.19), [28.26](ch28-refs.md#r-28.26)

See also: [proton mass](ch28-cards.md#g-mp), [the Yang–Mills mass gap](ch28-cards.md#g-ymgap)

### the Millennium Prize Problems — `Clay Mathematics Institute, 2000` {#g-clay}

*Function or operator*

Seven problems announced in Paris on 24 May 2000, a million dollars each; one has been solved and the prize declined.

The Millennium Prize Problems, announced by the Clay Mathematics Institute at the Collège de France on 24 May 2000, a century after Hilbert’s Paris address, with a million-dollar prize for each: the Birch and Swinnerton-Dyer conjecture, the Hodge conjecture, Navier–Stokes existence and smoothness, P versus NP, the Poincaré conjecture, the Riemann Hypothesis, and Yang–Mills existence and mass gap. Perelman solved Poincaré in 2002–2003 and declined the prize in 2010; the other six are open. An event of the framework.

Cites: [28.1](ch28-refs.md#r-28.1), [28.6](ch28-refs.md#r-28.6)

See also: [the Birch and Swinnerton-Dyer conjecture](ch28-cards.md#g-bsdconj), [the lightest glueball](ch28-cards.md#g-glueball), [the Hodge conjecture, as a bit](ch28-cards.md#g-hodge), [the P versus NP question, as a bit](ch28-cards.md#g-pnp), [the Poincaré conjecture, as a bit](ch28-cards.md#g-poincare), [the Riemann Hypothesis](ch18-cards.md#g-rh)

### Hilbert’s problems of 1900 — `Hilbert’s 23 problems` {#g-hilbert23}

*Function or operator*

The list David Hilbert launched in Paris in August 1900 that shaped a century; the Riemann Hypothesis is the only problem on both Hilbert’s list and the Clay Millennium list.

Hilbert’s twenty-three problems: ten of them were read out at the International Congress of Mathematicians in Paris on 8 August 1900, and all twenty-three were printed soon after. The first was the continuum hypothesis (Chapter ᚚ(P)), the eighth contains the Riemann Hypothesis and Goldbach’s conjecture, the seventh asked whether 2^√2 is transcendental (Chapter ᚂ(L)), the tenth whether Diophantine equations are decidable (answered no by Matiyasevich in 1970). The Riemann Hypothesis is the only problem on both Hilbert’s list and the Clay list. An event of the framework.

Cites: [28.1](ch28-refs.md#r-28.1), [28.2](ch28-refs.md#r-28.2), [28.20](ch28-refs.md#r-28.20), [28.30](ch28-refs.md#r-28.30)

See also: [the continuum hypothesis](ch27-cards.md#g-ch), [Gelfond–Schneider constant](ch04-cards.md#g-gelfond), [the Riemann Hypothesis](ch18-cards.md#g-rh)

### a closed three-manifold — `M³, ∂M = ∅` {#g-threemfld}

*Function or operator*

A space that looks like ordinary three-dimensional space at every point, with no edge and no room to run off to infinity.

A three-manifold is a space in which every point has a neighborhood that looks like ordinary three-dimensional space; closed adds that it is compact and has no boundary—no edge to fall off and nowhere to escape to. The three-sphere S³ is the model example, the three-dimensional counterpart of the surface of a ball. Poincaré’s question was whether being closed and having every loop contractible is enough to force a three-manifold to be the three-sphere, and the answer, after a century, is yes: Perelman’s proof settled the one Clay problem that has been solved. A class of spaces, uncolored.

Cites: [28.23](ch28-refs.md#r-28.23)

See also: [the Poincaré conjecture, as a bit](ch28-cards.md#g-poincare)

### a smooth projective variety — `X ⊂ ℙⁿ(ℂ)` {#g-smoothproj}

*Function or operator*

The common zeros of homogeneous polynomials in complex projective space, with no singular points—the setting in which the Hodge conjecture is stated.

A projective variety is the set of common zeros, inside complex projective space, of a family of homogeneous polynomials; smooth, or non-singular, means it has no corners or self-crossings, so it is a complex manifold as well as an algebraic object. Living in projective space makes it compact, and that is what keeps its cohomology finite and well behaved. This double nature—defined by equations, studied by topology—is what the Hodge conjecture trades on: it asks which classes in the cohomology of such a variety are combinations of the classes cut out by its own subvarieties. A class of spaces, uncolored.

Cites: [28.24](ch28-refs.md#r-28.24)

See also: [cohomology](ch12-cards.md#g-cohomology), [the Hodge conjecture, as a bit](ch28-cards.md#g-hodge)

### gauge symmetry — `ψ ↦ e^(iθ(x))ψ` {#g-gauge}

*Function or operator*

A symmetry you may apply differently at every point of space and time, and which the physics cannot see. Weyl named it in 1918, after cask sizes.

A gauge symmetry is a transformation that may be chosen independently at every point of spacetime and that leaves the Lagrangian unchanged; insisting on such a symmetry is what forces a force field into existence. Hermann Weyl introduced both the idea and the word in 1918, trying to fold electromagnetism into general relativity by letting the scale of length vary from point to point—Eichinvarianz, after the standard sizes of casks, the same root as railway gauge. The scale idea failed, but once quantum mechanics arrived Weyl, Fock, and London replaced the scale factor with a phase, and the surviving U(1) symmetry is electromagnetism. Yang–Mills theory is the same construction with a nonabelian group, and the Clay problem asks for it to be built rigorously for every compact simple gauge group. A principle, uncolored.

Cites: [28.25](ch28-refs.md#r-28.25)

### the Birch and Swinnerton-Dyer conjecture — `rank(E) = ord_{s=1} L(E, s)` {#g-bsdconj}

*Status open*

The Birch and Swinnerton-Dyer conjecture proposes that an elliptic curve’s rank, the number of independent rational points of infinite order it has, equals the order of vanishing of its L-function at s = 1.

Bryan Birch and Peter Swinnerton-Dyer formulated the conjecture in the early 1960s from numerical computations of curves’ point counts modulo many primes, tying the purely algebraic rank, from the Mordell–Weil theorem’s finitely generated group of rational points, to the analytic behavior of L(E, s), an Euler product built from those same point counts. It is one of the seven Millennium Prize Problems, and the strongest evidence for it comes from ranks 0 and 1, including the curve of conductor 37 whose L-function vanishes to order exactly 1 at s = 1, matching its rank of 1. No proof yet covers curves of rank greater than 1, and the general statement remains open.

Cites: [28.27](ch28-refs.md#r-28.27)

See also: [the curve of conductor 37](ch28-cards.md#g-E37), [the L-function of an elliptic curve](ch19-cards.md#g-LfuncE), [the Millennium Prize Problems](ch28-cards.md#g-clay), [the Mordell–Weil theorem](ch19-cards.md#g-mordellweil)