User:Nomalias/sandbox

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https://en.wikipedia.org/wiki/Spectral_gap_(physics) https://en.wikipedia.org/wiki/Duality_gap +

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https://en.wikipedia.org/wiki/Doubly_periodic_function https://en.wikipedia.org/wiki/Fundamental_pair_of_periods

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$\operatorname {div} \mathbf {F} =\nabla \cdot \mathbf {F} =\operatorname {tr} (\mathbf {J} (f))$ +
$\Delta f=\nabla ^{2}f=\nabla \cdot \nabla f=\operatorname {tr} {\big (}H(f){\big )}$ + +

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Group diagrams

https://commons.wikimedia.org/wiki/Category:Mathematical_diagrams https://commons.wikimedia.org/wiki/Category:Commutative_diagrams https://commons.wikimedia.org/wiki/Category:Group_theory https://commons.wikimedia.org/wiki/File:Projective-representation-lifting.svg +] + + + + + +

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groups: https://en.wikipedia.org/wiki/Bézout's_identity
Rings: https://en.wikipedia.org/wiki/Chinese_remainder_theorem

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https://en.wikipedia.org/wiki/Weierstrass_transform#Generalizations https://en.wikipedia.org/wiki/Shift_operator https://en.wikipedia.org/wiki/Operator_(physics)#The_exponential_map +

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$\pi _{k}(x)\sim {\frac {x(\log \log x)^{k-1}}{(k-1)!\log x}}\qquad \qquad (1)$ +
$\pi _{2}(x)\sim 1.32{\frac {x}{(\log x)^{2}}}$ +

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$p_{n}\approx n\log n+n(\log \log n-1),$ $\sum _{n\leq x}\sigma _{0}(n)\approx x\log x+x(2\gamma -1)$
$\lim _{n\to \infty }{\frac {1}{\log n}}\prod _{p\leq n}{\frac {p}{p-1}}=e^{\gamma },$ $\limsup _{n\rightarrow \infty }{\frac {\sigma (n)}{n\,\log \log n}}=e^{\gamma }$ + + +

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https://en.wikipedia.org/wiki/Chern-Simons_theory#HOMFLY_and_Jones_polynomials https://en.wikipedia.org/wiki/Weyl_character_formula#The_SU(2)_case

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https://en.wikipedia.org/wiki/Representation_theorem

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https://en.wikipedia.org/wiki/Vanishing_theorem

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https://en.wikipedia.org/wiki/Analytic_torsion https://en.wikipedia.org/wiki/Crooks_fluctuation_theorem

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$\langle A\rangle =\operatorname {tr} (A\rho )$ + + + +

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$\operatorname {E} [\exp(-tX)]=...$

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$\int \exp(-{\frac {(X-\operatorname {E} [X])^{2}}{2\operatorname {E} [(X-\operatorname {E} [X])^{2}]}})=(\operatorname {det} (\operatorname {E} [(X-\operatorname {E} [X])^{2}))^{\frac {1}{2}}$

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https://en.wikipedia.org/wiki/Agoh-Giuga_conjecture https://en.wikipedia.org/wiki/Daniel_da_Silva_(mathematician) https://en.wikipedia.org/wiki/Euler's_identity#Generalizations

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https://en.wikipedia.org/wiki/Crenel_function https://en.wikipedia.org/wiki/Montgomery's_pair_correlation_conjecture

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https://en.wikipedia.org/wiki/Cyclotomic_polynomial https://proofwiki.org/wiki/Reciprocals_of_Odd_Numbers_adding_to_1

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https://en.wikipedia.org/wiki/Chebyshev_function#The_Riemann_hypothesis https://en.wikipedia.org/wiki/Explicit_formulae_(L-function)#Weil's_Explicit_Formula https://en.wikipedia.org/wiki/Hilbert–Pólya_conjecture https://mathoverflow.net/questions/62816/the-guinand-weil-explicit-formula-without-entire-function-theory?rq=1

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https://en.wikipedia.org/wiki/Cassini_and_Catalan_identities

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https://en.wikipedia.org/wiki/Generating_function https://en.wikipedia.org/wiki/Table_of_Newtonian_series

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$\zeta (1-n,a)=-{\frac {B_{n}(a)}{n}}\!$ for $n\geq 1\!$+
$\zeta (2n)={\frac {(-1)^{n+1}B_{2n}(2\pi )^{2n}}{2(2n)!}}$
$\zeta (-n)=(-1)^{n}{\frac {B_{n+1}}{n+1}}$

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Hodge duality->Poincaré duality->Grothendieck local duality->Serre duality

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kunneth theorem + + [+

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https://en.wikipedia.org/wiki/Dedekind_psi_function https://mathoverflow.net/questions/14083/modular-forms-and-the-riemann-hypothesis

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$p(x)={\frac {1}{2\pi }}\int _{\mathbf {R} }e^{itx}P(t)\,dt={\frac {1}{2\pi }}\int _{\mathbf {R} }e^{itx}{\overline {\varphi _{X}(t)}}\,dt.$ +

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+ + + ${\widehat {f}}(x)={\frac {1}{2\pi }}\int _{-\infty }^{+\infty }{\widehat {\varphi }}(t)\psi _{h}(t)e^{-itx}\,dt={\frac {1}{2\pi }}\int _{-\infty }^{+\infty }{\frac {1}{n}}\sum _{j=1}^{n}e^{it(x_{j}-x)}\psi (ht)\,dt={\frac {1}{nh}}\sum _{j=1}^{n}{\frac {1}{2\pi }}\int _{-\infty }^{+\infty }e^{-i(ht){\frac {x-x_{j}}{h}}}\psi (ht)\,d(ht)={\frac {1}{nh}}\sum _{j=1}^{n}K{\Big (}{\frac {x-x_{j}}{h}}{\Big )}$

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$\varphi _{Z_{n}}(t)=(\varphi _{Y_{1}}({\frac {t}{\sqrt {n}}}))^{n}=(e^{-{\frac {1}{2}}({\frac {t}{\sqrt {n}}})^{2}})^{n}=e^{-{\frac {t^{2}}{2}}}$

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${\tilde {K}}(p;T)={\tilde {G}}_{\varepsilon }(p)^{T/\varepsilon }=(e^{-{\frac {1}{2}}({\sqrt {\varepsilon }}p)^{2}})^{T/\varepsilon }=e^{-{\frac {Tp^{2}}{2}}}$

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$\psi _{t}(y)=\int \psi _{0}(x)K(x-y;t)\,dx=\int \psi _{0}(x)\int _{x(0)=x}^{x(t)=y}e^{iS}\,Dx,$

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$K(x,y;T)=\int _{x(0)=x}^{x(T)=y}\prod _{t}\exp \left(-{\tfrac {1}{2}}\left({\frac {x(t+\varepsilon )-x(t)}{\varepsilon }}\right)^{2}\varepsilon \right)\,Dx,$

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"Index theory"

$\sum _{i}(-1)^{i}{\text{Tr}}({\text{Frob}},H_{c}^{i}(X,{\bf {{Q}_{\ell }))=|X({\bf {F}}_{q})|}}$
$\sum _{i}(-1)^{i}\mathrm {Tr} (f_{*}|H_{k}(X,\mathbb {Q} ))=\sum _{x\in \mathrm {Fix} (f)}\mathrm {index} _{x}f$
$\sum _{i}(-1)^{i}\dim H_{k}(X,\mathbb {Q} )=\sum _{x\in \mathrm {Sing} (v)}\mathrm {index} _{x}v$
+ + + + +

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$\sum (-1)^{\gamma }C^{\gamma }\,=\chi (M)$
+
$\sum _{i}\operatorname {index} _{x_{i}}(v)=\chi (M)\,$ +
${\textrm {Tr}}[(-1)^{F}e^{-\beta H}]=\sum _{p\in \mathbb {Z} }(-1)^{p}b_{p}=\chi (M)\ .$ + +

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$\sigma (n)=\sum _{i\in \mathbb {Z} }(-1)^{i+1}\left(\sigma (n{-}{\frac {1}{2}}(3i^{2}{-}i))+\delta (n,{\frac {1}{2}}(3i^{2}{-}i))\,n\right)=\sigma (n{-}1)+\sigma (n{-}2)-\sigma (n{-}5)-\sigma (n{-}7)+\sigma (n{-}12)+\sigma (n{-}15)+\cdots$ +

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Genus Stuff

Riemann-Roch_theorem: $\ell ({\mathcal {K}}_{X}-D)=\dim H^{0}(X,\omega _{X}\otimes {\mathcal {L}}(D)^{\vee }),H^{0}(X,\omega _{X}\otimes {\mathcal {L}}(D)^{\vee })$ Line bundle-Riemann surface Vector Bundle-Complex manifold Quotient stack sheaf-Orbifold Chain-complex sheaf-Scheme Arithmetic

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Degenerancy theory

covering map manifold

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Poincaré–Hopf theoremHairy ball theorem

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Banach fixed point theorem(existence and uniqueness) Brouwer_fixed-point_theorem(existence) Fixed point degree

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There are several fixed-point theorems which come in three equivalent variants: an algebraic topology variant, a combinatorial variant and a set-covering variant. Each variant can be proved separately using totally different arguments, but each variant can also be reduced to the other variants in its row. Additionally, each result in the top row can be deduced from the one below it in the same column.

Algebraic topology	Combinatorics	Set covering
Brouwer fixed-point theorem	Sperner's lemma	Knaster–Kuratowski–Mazurkiewicz lemma
Borsuk–Ulam theorem	Tucker's lemma	Lusternik–Schnirelmann theorem

+ +

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Hall's marrriage theorem equivalences

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Cayley's theorem equivalencesWagner-preston theorem (Cayley' theorem on inver semigroup)+

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group theory: definitions basics factsnon basic facts

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$|X^{P}|\equiv |X|\mod p\quad {\text{(P p-group)}}$ +
$a^{p}\equiv a\mod p\quad {\text{(p prime)}}$[1]

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Thompson order formula

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$|X/G|=\sum _{x\in X}{\frac {1}{|G\cdot x|}}$	$|X/G||G|=\sum _{x\in X}|G_{x}|$	$|X/G||G|=\sum _{g\in G}|X^{g}|$
$|X|/|G|=\sum _{x\in X/G}{\frac {1}{|G_{x}|}}$	$|X|=\sum _{x\in X/G}|G\cdot x|$	$|X|=|X^{G}|+\sum _{x\in X/G,|X/G|>1}|G\cdot x|$

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$|G|=|G/G_{x}||G_{x}|=|G_{x}\backslash G||G_{x}|$+
$|G|=|G/H||H|=|H\backslash G||H|$+
$G/Z(G)\cong Inn(G)$
$Aut(G)/Inn(G)\cong Out(G)$

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Centralizer-Normalizer Orbit stabilizer coset-index

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Fundamental theorem of abelian groups Fundamental theorem of cyclic groups Fundamental theorem of free groups Jordan–Hölder theorem Finitely generated abelian group

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$\varphi (n)=\sum _{d\mid n}\mu \left(d\right)\cdot {\frac {n}{d}}$	$n=\sum _{d\mid n}\varphi (d)$
$J_{k}(n)=\sum _{d|n}\mu \left(d\right)\cdot {\frac {n^{k}}{d}}$	$n^{k}=\sum _{d|n}J_{k}(d)$
$\Lambda (n)=-\sum _{d\mid n}\mu (d)\log(d)\$	$\log(n)=\sum _{d\mid n}\Lambda (d)$
$M_{k}(n)\ =\ \sum \nolimits _{d|n}N_{k}(d)\,\mu ({\tfrac {n}{d}})$	$N_{k}(n)\ =\ \sum \nolimits _{d|n}M_{k}(d)$
$\pi _{0}(x)=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\Pi _{0}(x^{1/n})$	$\Pi _{0}(x)=\sum _{n=1}^{\infty }{\frac {1}{n}}\pi _{0}(x^{1/n})$
$\operatorname {R} (x)=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\operatorname {li} (x^{1/n})$	$\operatorname {li} (x)=\sum _{n=1}^{\infty }{\frac {\operatorname {R} (x^{1/n})}{n}}$
$P(s)=\sum _{n>0}\mu (n){\frac {\log \zeta (ns)}{n}}$	$\log \zeta (s)=\sum _{n>0}{\frac {P(ns)}{n}}$

${\frac {1}{\zeta (s)}}=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{s}}}$

$\zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}$

$\Pi _{0}(x)=\operatorname {li} (x)-\sum _{\rho }\operatorname {li} (x^{\rho })-\ln 2+\int _{x}^{\infty }{\frac {dt}{t(t^{2}-1)\ln t}}.$	$\ln \zeta (s)=s\int _{0}^{\infty }\Pi _{0}(x)x^{-s-1}\,\mathrm {d} x.$
$\pi _{0}(x)=\operatorname {R} (x)-\sum _{\rho }\operatorname {R} (x^{\rho })-{\frac {1}{\ln x}}+{\frac {1}{\pi }}\arctan {\frac {\pi }{\ln x}}$	$\ln \zeta (s)=s\int _{0}^{\infty }{\frac {\pi (x)}{x(x^{s}-1)}}\,\mathrm {d} x$	$\pi (x)=\operatorname {li} (x)+O\left({\sqrt {x}}\log x\right)$
$\sum _{n\leq x}\Lambda (n)=x-\sum _{\rho }{\frac {x^{\rho }}{\rho }}-\ln 2\pi -{\tfrac {1}{2}}\ln(1-x^{-2})=x-\sum _{\rho }{\frac {x^{\rho }}{\rho }}-{\frac {\zeta '(0)}{\zeta (0)}}-\sum _{k=1}^{\infty }{\frac {x^{-2k}}{-2k}}$	$\ln \zeta (s)=\sum _{n=2}^{\infty }{\frac {\Lambda (n)}{\log(n)}}\,{\frac {1}{n^{s}}},\qquad {\text{Re}}(s)>1.$

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$\zeta (s)=s\int _{0}^{\infty }S(x)\,x^{-s-1}dx=\int _{0}^{\infty }S'(x)\,x^{-s}dx$	$\zeta '(s)=-s\int _{0}^{\infty }T(x)\,x^{-s-1}dx=-\int _{0}^{\infty }T'(x)\,x^{-s}dx$
$\ln \zeta (s)=s\int _{0}^{\infty }\Pi _{0}(x)\,x^{-s-1}dx=\int _{0}^{\infty }\Pi _{0}'(x)\,x^{-s}dx$	${\frac {\zeta '(s)}{\zeta (s)}}=-s\int _{0}^{\infty }\psi (x)\,x^{-s-1}dx=-\int _{0}^{\infty }\psi '(x)\,x^{-s}dx$

${\frac {1}{\zeta (s)}}=s\int _{1}^{\infty }{\frac {M(x)}{x^{s+1}}}\,dx$ + $\psi _{0}(x)={\dfrac {1}{2\pi i}}\int _{\sigma -i\infty }^{\sigma +i\infty }\left(-{\dfrac {\zeta '(s)}{\zeta (s)}}\right){\dfrac {x^{s}}{s}}ds\quad$ ${\frac {\zeta ^{\prime }(s)}{\zeta (s)}}=-s\int _{1}^{\infty }{\frac {\psi (x)}{x^{s+1}}}\,dx$ +

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$\int _{0}^{\infty }x^{s}\ln(1-e^{-x})dx=-\int _{0}^{\infty }x^{s-1}{\frac {e^{-x}}{1-e^{-x}}}dx$
$\int _{0}^{1}|{\mathsf {Li}}_{s}(e^{2\pi ix})|^{2}dx=\sum _{k\geq 1}\left|{\frac {e^{2\pi ikx}}{k^{s}}}\right|^{2}=\sum _{k\geq 1}{\frac {1}{k^{2s}}}=\zeta (2s)$
+

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¿? $\sum _{p}{\frac {\log p}{p^{s}}}=\int _{1}^{\infty }{\frac {d\vartheta (x)}{x^{s}}}=s\int _{1}^{\infty }{\frac {\vartheta (x)}{x^{s+1}}}dx$
$\sum _{p}{\frac {\log p}{p^{s}-1}}=\int _{1}^{\infty }{\frac {d\vartheta (x)}{x^{s}}}=s\int _{1}^{\infty }{\frac {\vartheta (x)}{x^{s+1}}}dx$

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$\quad \psi '(x)=\ln(x)\,\Pi _{0}'(x)$ $\quad T'(x)=\ln(x)\,S'(x)$

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$\quad S[x]=\sum _{n=1}^{\lfloor x\rfloor }1=\lfloor x\rfloor$ $\quad T[x]=\sum _{n=1}^{\lfloor x\rfloor }\log n$

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$\zeta (s)={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }(e^{x}-1)^{-1}x^{s-1}\,\mathrm {d} x={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }x^{s-1}\sum _{n>0}e^{-nx}\mathrm {d} x\quad$ $\Gamma (s)=\int _{0}^{\infty }e^{-x}\,x^{s-1}\,\mathrm {d} x$ +
$\zeta (s)={\frac {1}{2\pi ^{-{\frac {s}{2}}}\Gamma \left({\frac {s}{2}}\right)}}\int _{0}^{\infty }{\bigl (}\theta (ix)-1{\bigr )}x^{{\frac {s}{2}}-1}\,\mathrm {d} x,\quad$ $\theta (\tau )=\sum _{n=-\infty }^{\infty }e^{\pi in^{2}\tau }$
$\zeta (s)={\frac {1}{2\pi ^{-{\frac {s}{2}}}\Gamma \left({\frac {s}{2}}\right)}}({\frac {1}{s-1}}-{\frac {1}{s}}+{\frac {1}{2}}\int _{0}^{1}\left(\theta (ix)-x^{-{\frac {1}{2}}}\right)x^{{\frac {s}{2}}-1}\,\mathrm {d} x+{\frac {1}{2}}\int _{1}^{\infty }{\bigl (}\theta (ix)-1{\bigr )}x^{{\frac {s}{2}}-1}\,\mathrm {d} x)$ +

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$\sum _{p}{\frac {1}{p}}=\infty$+ $\sum _{p}{\frac {1}{p-1}}=1$+

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$\gamma =\lim _{n\to \infty }\left(\ln n-\sum _{p\leq n}{\frac {\ln p}{p-1}}\right)=\lim _{n\to \infty }\left(-\ln n+\sum _{k=1}^{n}{\frac {1}{k}}\right)=\int _{1}^{\infty }\left(-{\frac {1}{x}}+{\frac {1}{\lfloor x\rfloor }}\right)\,dx.$

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$M=\lim _{n\to \infty }(\sum _{p}^{n}{\frac {1}{p}}-\ln(\ln n))$

$\gamma =\lim _{n\to \infty }(\sum _{k=1}^{n}{\frac {1}{k}}-\ln n)$

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${\frac {\zeta ^{\prime }(s)}{\zeta (s)}}=-\sum _{n=1}^{\infty }{\frac {\Lambda (n)}{n^{s}}}=-\sum _{p\in {\mathcal {P}}}{\frac {\log(p)}{p^{s}-1}}=P_{p-1}'(s)$ +

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$L(s,\chi )=s\int _{1}^{\infty }{\frac {A(x)}{x^{s+1}}}\,dx\quad A(x)=\sum _{n\leq x}\chi (n)$
$\zeta (s)=s\int _{1}^{\infty }{\frac {\lfloor x\rfloor }{x^{s+1}}}\,dx$+
$\zeta \left({s}\right)={\frac {s}{s-1}}-s\int _{1}^{\infty }\left\{{x}\right\}x^{-s-1}dx$
+ +

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$\zeta '(s)=-\sum _{n\mathop {=} 2}^{\infty }{\frac {\ln \left({n}\right)}{n^{s}}}$ +
$\left({\frac {\zeta '(s)}{\zeta (s)}}\right)^{2}=\sum _{n=1}^{\infty }\sum _{d|n}{\frac {\Lambda (d)\Lambda (n/d)}{n^{s}}}$ +
${\frac {d}{ds}}\left({\frac {\zeta ^{(k)}(s)}{\zeta (s)}}\right)={\frac {\zeta ^{(k+1)}(s)}{\zeta (s)}}-{\frac {\zeta '(s)}{\zeta (s)}}{\frac {\zeta ^{(k)}(s)}{\zeta (s)}}$ +

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$L(s,\chi )=\sum _{n=1}{\frac {\chi (n)}{n^{s}}}$	$\Delta (z)=\sum _{n=1}\tau (n)q^{n}$
$L(s,\chi )=\prod _{p}\left(1-\chi (p)p^{-s}\right)^{-1}$	$L(s,\tau )=\prod _{p}\left(1-{\frac {\tau (p)}{p^{s}}}+{\frac {1}{p^{2s-11}}}\right)^{-1}$

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$\zeta (s)=\prod _{p{\text{ prime}}}(1-p^{-s})^{-1}=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}=(\sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{s}}})^{-1}$

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$f(q)=\prod _{n=1}^{\infty }(1-q^{n})=\sum _{n=0}^{\infty }a_{n}q^{n}=(\sum _{n=0}^{\infty }p(n)q^{n})^{-1}$

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$\eta (\tau )=q^{\frac {1}{24}}\prod _{n=1}^{\infty }(1-q^{n})=(\sum _{n>0}\tau (n)q^{n})^{-24}=(\sum _{n=0}^{\infty }p(n)q^{n-{\frac {1}{24}}})^{-1}$
+ +

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$\left({1 \over p}-{1 \over q}\right)\prod _{n,m=1}^{\infty }(1-p^{n}q^{m})^{c_{nm}}=j(p)-j(q)$ +

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$e^{x}=\prod _{n\geq 1}(1-x^{n})^{-\mu (n)/n}$ + + + + + +
$x=\sum _{n\geq 1}{\frac {\mu (n)}{n}}\ln((1-x^{n})^{-1})$

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${\frac {1}{1-x}}=\prod _{n\geq 0}(1+x^{2^{n}})$ + +
$\prod _{n\geq 0}{\frac {1}{1-x^{2n+1}}}=\prod _{n\geq 0}(1+x^{n})$+

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Rodrigues's formula Li's criterion

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Szegő_limit_theorems Jensen's formula five value theorem

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ELSV formula

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https://en.wikipedia.org/wiki/Fredholm's theorem Fredholm alternative Farkas_lemma Hyperplane separation theorem Hanh Banach separation theorem Positive-definite matrix Positive-definite kernel Positive definiteness

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homology-homotopy dictionary+
number field-function field dictionary
Kapranov-Reznikov-Mazur dictionary/arithmetic topology + + + + + +
Diophantine dictionary/Arithmetic dynamics
algebraic geometry dictionary +
wu-yang dictionary
elliptic-parabolic dictionary feynman-intersection number dictionary-like

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petterson-weil volume+ witten's volume orbifolds volume orbifold euler characteristic+

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varieties: Grassman Segre veronese

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global-local homology global-local homotopy +

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nowhere differentiable: everywhere continuos, nowhere continuos

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$\max(|A+A|,|A\cdot A|)\geq c\cdot |A|^{1+\varepsilon }$
$\max(|a|,|b|,|c|)\geq C\cdot rad(abc)^{1+\varepsilon }$+
$\max\{\deg(a),\deg(b),\deg(c)\}\leq \deg(\operatorname {rad} (abc))-1.$

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mle-entropy
euler-lagrange-gauss-principle

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(Elliptic function-Elliptic curve) (Modular form-Modular curve)

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Arithmetic geometry Fermat's squares theorem Minkowski's theorem + Gauss circle problem Dirichlet's divisor problem

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Class field theory Class number Class number formula
List of number fields with class number one Lists of discriminants of class number 1
Stark-Heegner_theorem Kronecker-Weber_theorem Kummer theory Fundamental discriminant
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Perron's formulaShimura correspondence

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$\zeta (s)={\frac {\pi ^{\frac {s}{2}}}{\Gamma \left({\frac {s}{2}}\right)}}{\int }_{0}^{\infty }{{v^{\frac {s}{2}}}\left({\frac {\theta (iv)-1}{2}}\right){\frac {{\rm {d}}v}{v}}}$+
$\zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}$
$\zeta (s)=\prod _{p{\text{ prime}}}{\frac {1}{1-p^{-s}}}$
$\lim _{s\to 1}(s-1)\zeta (s)=1$

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$L(s,\chi )=\sum _{n=1}^{\infty }{\frac {\chi (n)}{n^{s}}}$
${L_{D}}(s)\!\!:=\mathop {\prod } \limits _{p\,{\rm {prime}}}{\frac {1}{1-{\chi _{D}}(p){p^{-s}}}}$
${\frac {\sqrt {\left|D\right|}}{2\pi }}{L_{D}}(1)={\frac {h(D)}{2}}$
$Q(x,y)=Ax^{2}+Bxy+Cy^{2},D=B^{2}-4AC$
$Q(x,y)=Q(ax+by,cx+dy),ad-bc=1$

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${L_{E}}(s)={\frac {{(2\pi )}^{s}}{\Gamma (s)}}{\int }_{0}^{\infty }{{v^{s}}{f_{E}}(iv){\frac {{\rm {d}}v}{v}}}$
${L_{E}}(s)\!\!:=\mathop {\prod } \limits _{p\,{\rm {prime}}}{\frac {1}{1-{a_{p}}{p^{-s}}+\varepsilon (p){p^{1-2s}}}}$
${\rm {li}}{{\rm {m}}_{s\to 1}}{(s-1)^{-{r_{E}}}}{L_{E}}(s)<\infty$
$\mathop {\lim } \limits _{s\to 1}{\frac {1}{\Omega _{E}}}{\frac {{L_{E}}(s)}{{(s-1)}^{r_{E}}}}={c_{E}}\left|{{\text{Ш}}(E)}\right|$
${y^{2}}={x^{3}}+Ax+B,4A^{3}+27B^{2}\neq 0$
${(Ncz+d)^{-2}}{f_{E}}\left({\frac {az+b}{Ncz+d}}\right)={f_{E}}(z),ad-Nbc=1$

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$\zeta _{K}(s)=\sum _{I\subseteq {\mathcal {O}}_{K}}{\frac {1}{(N_{K/\mathbf {Q} }(I))^{s}}}$
$\zeta _{K}(s)=\prod _{P\subseteq {\mathcal {O}}_{K}}{\frac {1}{1-(N_{K/\mathbf {Q} }(P))^{-s}}},{\text{ for Re}}(s)>1.$
$\lim _{s\to 1}(s-1)\zeta _{K}(s)={\frac {2^{r_{1}}\cdot (2\pi )^{r_{2}}\cdot \operatorname {Reg} _{K}\cdot h_{K}}{w_{K}\cdot {\sqrt {|D_{K}|}}}}$

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Winding number-Eisenbud Levine Khimshiashvili signature formula

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Roth's_theorem-Duffin-Schaeffer conjecture

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Kutsenov trace formula-Gutzwiller trace formula

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Min-Max theoremMax-min_inequality

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Kolmogorov equation-Fokker Planck equation
Koopman operator-Perron-Frobenius operator

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not recursive function not computable function not ZFC-dependent function bound+

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Euler product + +

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Feller-Tornier constant + Pólya conjecture Chebyshev's bias + + Goldfeld conjecture + Parity_problem

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$\pi _{n,a}(n)\sim {\frac {\pi (n)}{\varphi (n)}}$ $\pi _{q^{2},1}(n)\approx {\frac {\pi (n)}{q\cdot (q-1)}}$ + Artin's conjecture +

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$\Pi ^{*}(x;q,a)=\sum _{p\leq x,p\equiv a{\pmod {q}}}^{*}1+\sum _{p^{2}\leq x,p^{2}\equiv a{\pmod {q}}}^{*}{\tfrac {1}{2}}+\sum _{p^{3}\leq x,p^{3}\equiv a{\pmod {q}}}^{*}{\tfrac {1}{3}}+\cdots =\pi ^{*}(x;q,a)+{\tfrac {1}{2}}\sum _{b{\pmod {q}},b^{2}\equiv a{\pmod {q}}}\pi ^{*}(x^{1/2};q,b)+{\tfrac {1}{3}}\sum _{c{\pmod {q}},c^{3}\equiv a{\pmod {q}}}\pi ^{*}(x^{1/3};q,c)+\cdots$ +

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$\pi (x)\sim {\frac {x}{\ln x}}\sim \int _{2}^{x}{dt \over \ln t}$	$C_{1}=1$
$\pi _{2}(x)\sim 2C_{2}{\frac {x}{(\ln x)^{2}}}\sim 2C_{2}\int _{2}^{x}{dt \over (\ln t)^{2}}$	$C_{2}=\prod _{\textstyle {p\;{\rm {prime}} \atop p\geq 3}}\left(1-{\frac {1}{(p-1)^{2}}}\right)\approx 0.660161$
$N(x)\sim C_{LR}{\dfrac {x}{\sqrt {\ln(x)}}}\sim {\text{¿}}C_{LR}\int _{2}^{x}{dt \over {\sqrt {\ln t}}}{\text{?}}$	$C_{LR}={\frac {\pi }{4}}\prod _{p\equiv 1{\pmod {4}}}\left(1-{\frac {1}{p^{2}}}\right)^{1/2}={\frac {1}{\sqrt {2}}}\prod _{p\equiv 3{\pmod {4}}}\left(1-{\frac {1}{p^{2}}}\right)^{-1/2}\approx 0.764223...$

Bateman-Horn conjecture square free distribution +

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${\frac {\partial C}{\partial t}}={\frac {\partial ^{2}C}{\partial s^{2}}}=\kappa n,$	$\int _{L}\kappa (n)ds+\sum \epsilon _{i}=2\pi n$
${\frac {\partial S}{\partial t}}=D{\frac {\left(1+\left({\frac {\partial S}{\partial x}}\right)^{2}\right){\frac {\partial ^{2}S}{\partial y^{2}}}-2{\frac {\partial S}{\partial x}}{\frac {\partial S}{\partial y}}{\frac {\partial ^{2}S}{\partial x\partial y}}+\left(1+\left({\frac {\partial S}{\partial y}}\right)^{2}\right){\frac {\partial ^{2}S}{\partial x^{2}}}}{\left(1+\left({\frac {\partial S}{\partial x}}\right)^{2}+\left({\frac {\partial S}{\partial y}}\right)^{2}\right)^{3/2}}}{\sqrt {1+\left({\frac {\partial S}{\partial x}}\right)^{2}+\left({\frac {\partial S}{\partial y}}\right)^{2}}}$	$\int _{M}K\;dA+\int _{\partial M}k_{g}\;ds=2\pi \chi (M),\,$	Minakshisundaram–Pleijel
$\partial _{t}g_{ij}=-2R_{ij}.$
	$\int _{M}\left(|{\text{Riem}}|^{2}-4|{\text{Ric}}|^{2}+R^{2}\right)d\mu =32\pi ^{2}\chi (M)$	Lovelock

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https://en.wikipedia.org/wiki/Homology_(mathematics)

Topological characteristics of closed 1- and 2-manifolds

Manifold		Euler No. χ	Orientability	Betti numbers			Torsion coefficient (1-dimensional)
Symbol	Name			b0	b1	b2
$S^{1}$	Circle (1-manifold)	0	Orientable	1	1	N/A	N/A
$S^{2}$	Sphere	2	Orientable	1	0	1	none
$T^{2}$	Torus	0	Orientable	1	2	1	none
$P^{2}$	Projective plane	1	Non-orientable	1	0	0	2
$K^{2}$	Klein bottle	0	Non-orientable	1	1	0	2
	2-holed torus	−2	Orientable	1	4	1	none
	g-holed torus (Genus = g)	2 − 2g	Orientable	1	2g	1	none
	Sphere with c cross-caps	2 − c	Non-orientable	1	c − 1	0	2
	2-Manifold with g holes and c cross-caps (c > 0)	2 − (2g + c)	Non-orientable	1	(2g + c) − 1	0	2

NOTES:

1. For a non-orientable surface, a hole is equivalent to two cross-caps.
2. Any 2-manifold is the connected sum of g tori and c projective planes. For the sphere $S^{2}$, g = c = 0.

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$\mathbb {S} ^{2},\mathbb {E} ^{2},\mathbb {H} ^{2},SO(3),ISO(\mathbb {R} ^{2})^{+},SL(2,\mathbb {R} )=SO(1,3)^{+},\pi _{1}(\mathbb {S} ^{2})=0,\pi _{1}(T\cong \mathbb {R} ^{2}/\mathbb {Z} ^{2})=0$

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RPⁿ CPⁿ HPⁿ

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Legendre form	hypergeometric differential equation	$j={\frac {256(1-\lambda +\lambda ^{2})^{3}}{\lambda ^{2}(1-\lambda )^{2}}}$
weierstrass form	Picard–Fuchs differential equation	$j={\frac {g_{2}^{3}}{g_{2}^{3}-27g_{3}^{2}}}$
Riemann form	Riemann's differential equation

???????????????

---

• https://en.wikipedia.org/wiki/Modular_elliptic_curve
• https://en.wikipedia.org/wiki/Weierstrass's_elliptic_functions#General_theory
  • https://en.wikipedia.org/wiki/Period_mapping

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• https://en.wikipedia.org/wiki/Poincaré–Bendixson_theorem
• https://en.wikipedia.org/wiki/Nielsen-Thurston_classification

---

• https://en.wikipedia.org/wiki/Wirtinger_derivatives
• https://en.wikipedia.org/wiki/Creation_and_annihilation_operators

---

Real n-Cauchy-Riemann:$Df^{T}Df=(\det(Df))^{2/n}I$

---

• https://en.wikipedia.org/wiki/Darboux's_theorem
• https://en.wikipedia.org/wiki/Period_mapping

---

Volume form + Connection_form torsion form curvature form + spin connection khäler form + Solder form

---

Poincaré series Igusa zeta function

---

Character theory Weyl character formula Kirillov_character_formula

---

Representation theory Representation of Lie algebra Representation of Lie group Representation of finite groups ℓ-adic representations

---

Hyperbolic geometryHyperbolic manifold++

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{5,5}-tilling-Poincaré Sphere+

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Minimal program model +

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Abel's theorem converse Abel's theorem

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Associahedron Pemutohedron

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https://en.wikipedia.org/wiki/Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain

• invariant factors + companion matrix yields Frobenius normal form (aka, rational canonical form)
• primary decomposition + companion matrix yields primary rational canonical form
• primary decomposition + Jordan blocks yields Jordan canonical form (this latter only holds over an algebraically closed field)

---

rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields

---

zeta table

	q	q	q
	$H(Q,P,t)=\sum _{s=0}^{\infty }{\frac {\phi (s)}{s!}}(-t)^{s}\!\quad ?$	$K(P,Q,s)={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }H(P,Q,t)t^{s-1}dt$	$K\propto \left(\int _{V}{\mathcal {D}}\phi \;e^{-\langle \phi ,K\phi \rangle }\right)^{-2}=\det(S)^{-2}=\sum _{n=0}^{\infty }{(-i)^{n} \over n!}\left(\prod _{k=1}^{n}\int _{t_{0}}^{t}dt_{k}\right){\mathcal {T}}\left\{\prod _{k=1}^{n}e^{iH_{0}t_{k}}Ve^{-iH_{0}t_{k}}\right\}.$
	$H(Q,P,t)=\sum _{n=1}^{\infty }f_{n}(P)f_{n}(Q)e^{-\lambda _{n}t}$	$K(P,Q,s)=\sum _{n=1}^{\infty }{\frac {f_{n}(P)f_{n}(Q)}{\lambda _{n}^{s}}}$
	$\operatorname {tr} H(Q,P,t)=\int _{M}H(P,P,t)dP=H(s)=\operatorname {tr} e^{-tH}=\sum _{i=1}^{\infty }e^{-\lambda _{i}t}$	$\operatorname {tr} K(Q,P,s)=\int _{M}K(P,P,s)dP=K(s)=\operatorname {tr} K^{-s}=\sum _{\lambda _{i}}\lambda _{i}^{-s}=\zeta _{K}(s)$
		$\det K=e^{-\operatorname {tr} K'(0)}=e^{-(\operatorname {tr} K^{-s})'|_{s=0}}=e^{-(\sum _{\lambda _{i}}\lambda _{i}^{-s})'|_{s=0}}=e^{-\sum _{\lambda _{i}}\ln(\lambda _{i})\lambda _{i}^{-s}|_{s=0}}=e^{-\zeta _{K}'(0)}$
	q	$\det(I-\lambda K)=\exp(\operatorname {tr} (\ln(I-\lambda K)))=\exp {\left(-\sum _{n=1}^{\infty }{\frac {\operatorname {Tr} (K^{n})}{n}}\lambda ^{n}\right)}=\sum _{n=0}^{\infty }(-\lambda )^{n}\operatorname {Tr} \Lambda ^{n}(K)={\frac {1}{\zeta _{K}(\lambda )}}$
		$K(t,x,y)={\frac {1}{(4\pi t)^{d/2}}}e^{-|x-y|^{2}/4t}\,$
	$H(t)\sim (4\pi t)^{-n/2}\sum _{m=0}^{\infty }a_{m}t^{m}$

+ + + + + + + +

---

$\zeta '(\Delta ,0)={\frac {1}{12}}\int _{M}KdA$ +
$S=\operatorname {tr} _{g}\operatorname {Ric}$ +
$R_{k{\overline {l}}}=\partial _{k}\partial _{\overline {l}}\ln(\operatorname {det} (g)),\ {\text{Ricci-Chern form}}$ + + +
${\frac {\log(T_{an}M_{i})}{{\textrm {volume}}(M_{i})}}\rightarrow -{\frac {1}{6\pi }};$$exp(-\zeta '(0))/Vol(S)$ + +

---

$n!=\exp(\ln(n!))=\exp(\sum _{n}\ln n)=\exp(\sum _{n}{\frac {\ln n}{n^{s}}}|_{s=0})=\exp(-\zeta '(0)))$
$p\#=\exp(\ln(p\#))=\exp(\sum _{p}\ln p)=\exp(\sum _{p}{\frac {\ln p}{p^{s}}}|_{s=0})=\exp(-P'(0)))$

---

$\zeta (s)={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }x^{s-1}\sum _{n>0}e^{-nx}dx$
$P(s)={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }x^{s-1}\sum _{p>0}e^{-px}dx$

---

$\zeta (s)=\exp(-\sum _{p}\ln(1-p^{-s}))=\exp(\sum _{p,n}{\frac {p^{-ns}}{n}})$ +

---

$\phi (q)=\exp(-\sum _{k}\ln(1-q^{k}))=\exp(\sum _{k,n}{\frac {q^{-nk}}{n}})=\exp(\sum _{n=1}^{\infty }{\frac {1}{n}}\,{\frac {q^{n}}{q^{n}-1}})$ +

---

$\phi (q)=\exp(-\sum _{k}\ln(1-q^{k}))=\exp(\sum _{k,n}{\frac {q^{-nk}}{n}})=\exp(\sum _{k|m,m}{\frac {kq^{-m}}{m}})=\exp(\sum _{m}{\frac {q^{-m}}{m}}\sum _{k|m}k)=\exp(\sum _{m}{\frac {q^{-m}}{m}}\sigma (m))$
$\log Z(X,T)=\sum _{x\in X}-\log \left(1-T^{\operatorname {deg} (x)}\right)=\sum _{x\in X}\sum _{n=1}^{\infty }{\frac {T^{\operatorname {deg} (x)\cdot n}}{n}}=\sum _{m=1}^{\infty }\left(\sum _{\operatorname {deg} (x)|m}\operatorname {deg} (x)\right){\frac {T^{m}}{m}}=\sum _{m=1}^{\infty }\left|X\left(\mathbb {F} _{q^{m}}\right)\right|{\frac {T^{m}}{m}}$ + +

---

$Tr_{V}q^{L_{0}}=\sum _{n\in \mathbf {Z} }\dim V_{n}q^{n}=\prod _{n\geq 1}(1-q^{n})^{-1}$ +

---

$\wp (z;\Lambda )=-\zeta '(z;\Lambda )=\ln ''(\sigma (z;\Lambda )),{\mbox{ for any }}z\in \mathbb {C}$ +
${\frac {1}{2\pi i}}{\frac {d}{dz}}\log \Delta (z)=1-24\sum _{n=1}^{\infty }{\frac {ne^{2\pi inz}}{1-e^{2\pi inz}}}=1-24\sum _{m=1}^{\infty }\sigma _{1}(m)e^{2\pi imz}=1-24\sum _{n>0}\sigma _{1}(n)q^{n}=E_{2}(z)$ + +

---

$\sum _{n\leq x}\Lambda (n)={\dfrac {1}{2\pi i}}\int _{\sigma -i\infty }^{\sigma +i\infty }\left(-{\dfrac {\zeta '(s)}{\zeta (s)}}\right){\dfrac {x^{s}}{s}}ds\quad =x-\sum _{\rho }{\frac {x^{\rho }}{\rho }}-\ln 2\pi -{\tfrac {1}{2}}\ln(1-x^{-2})=x-\sum _{\rho }{\frac {x^{\rho }}{\rho }}-{\frac {\zeta '(0)}{\zeta (0)}}-{\frac {1}{2}}\sum _{k=1}^{\infty }{\frac {x^{-2k}}{-2k}}$

---

$\ln {\mathcal {L}}(\mu ,\Sigma )=-{n \over 2}\ln \det(\Sigma )-{1 \over 2}\operatorname {tr} \left[\Sigma ^{-1}\sum _{i=1}^{n}(x_{i}-\mu )(x_{i}-\mu )^{\mathrm {T} }\right].$

---

$\sum _{\rho }F(\rho )=\operatorname {Tr} (F({\widehat {T}})).\!$ + ++++++

convergence stuff

as=>p=>d+

	as	p	d
LLN	T	T	T
LIL	F	T	T
CLT	F	F	T

+

	$\epsilon$	$x$	$f$
Pointwise continuity	x	x	x
Uniform continuity	x		x
equicontinuity	x	x
Uniform equicontinuity	x

---

function field analogy+ + +

PNT	Landau PNT
RH	Goss zeta function
Hilbert's reciprocity	Artin's recirpocity
abc conjecture	Mason–Stothers_theorem
falting's theorem	function field falting's theorem

	Riemann zeta	P-adic L-function
	Modular form	P-adic modular form
	Taniyama-Shimura conjecture	Iwasawa conjecture¿?
	Ideal class group+	Tate-Shafarevich group
	Chinese remainer theorem	Hasse principle
	Chowla-Selberg formula	Gross-Koblitz formula
	Multiplication theorem	Hasse-Davenport relation
	Gamma function	Gauss sump-adic Γ+
	Beta function	Jacobi sum
	Newton's method	Hensel's_lemma

---

Geodesics		PGT		Selberg zeta function	selberg trace formula
P-adic				P-adic L-function
Motives			Motivic zeta function	Motivic L-function
Varietes	Weil Conjectures++		Local zeta function	Hasse-Weil zeta function+
Schemes	Grand RH¿?			arithmetic zeta function
Ideals	ERH			Dedekind zeta function
Characters	GRH			Dirichlet L-function
Naturals	RH	PNT		Riemann zeta	Explicit_formulae

Selberg class Abstract analytic number theory

PNT	RH
$\zeta (1+it)\neq 0\iff {\text{PNT=true}}$	$\zeta (1/2+it)=0\iff {\text{RH=true?}}$

	Counting	LLN	CLT	Zeros
naturals	primes	PNT		RH
naturals	factors	Hardy ramanujan	Erdős-Kac_theorem
Geodesics	primes	PGT		++
Geodesics	factors		?
Elliptic curves	factors		?

+ + One can view Selberg’s theorem as a sort of Fourier-analytic variant of the Erdös-Kac theorem.
PNT scaled models++
mertens=/=PNT+ +

---

---

	g=0	g=1	g>1
homeomorphism	$\mathbb {S} ^{2}$	$\mathbb {T}$	$\#^{n}\mathbb {T}$
Universal cover	$\mathbb {S} ^{2}$	$\mathbb {R} ^{2}$	$\mathbb {H} ^{2}$
fundamental group	0	$\mathbb {Z} ^{2}$
automorphism	$PGL(2,\mathbf {C} )$		Hurwitz's bound
moduli	Example	j-invariat/cross ratio	3g-3
curvature	0	1	-1
Diophantine		Mordell-Weil theorem+	Faltings's theorem

$\Delta \cong \mathbf {H} \subset \mathbf {C} \subset {\widehat {\mathbf {C} }}$(complex disk,plane,sphere) +

automorphisms	${\frac {az+b}{cz+d}}$	$ad-bc$
$\Delta$	${\frac {az+b}{{\overline {a}}z+{\overline {b}}}}$	$|a|^{2}-|b|^{2}=1$
$\mathbf {H}$	${\frac {az+b}{cz+d}}$	$ad-bc=1$
$\mathbf {C}$	$az+b$	$a\neq 0$
${\widehat {\mathbf {C} }}$	${\frac {az+b}{cz+d}}$	$ad-bc\neq 0$

+

---

n>0:TOP=PL=DIFF
n>3:TOP=\=PL=DIFF
n>6:TOP=\=PL=\=DIFF

	True			False
	TOP	PL	DIFF	TOP	PL	DIFF
Manifold classification (MC)	1,2,3	1,2,3	1,2,3	n>3	n>3	n>3
Simple connected closed MC	1,2,3,5	1,2,3	1,2,3		n>3	n>3
Poincaré Conjecture	1,2,3,4,5,...	2,3,?,5,...	2,3,?,5,6			n>6

		TOP	PL	DIFF
n=2	Manifold classification (MC)	True Uniformization Poincare&koebe	True Uniformization Poincare&koebe	True Uniformization Poincare&koebe
n=2	Simple connected closed MC	True Uniformization Poincare&koebe	True Uniformization Poincare&koebe	True Uniformization Poincare&koebe
n=2	Poincaré conjecture	True Uniformization Poincare&koebe	True Uniformization Poincare&koebe	True Uniformization Poincare&koebe
n=3	Manifold classification (MC)	True Thurston's geometrization Perelman	True Thurston's geometrization Perelman	True Thurston's geometrization Perelman
n=3	Simple connected closed MC	True Thurston's geometrization Perelman	True Thurston's geometrization Perelman	True Thurston's geometrization Perelman
n=3	Poincaré conjecture	True Thurston's geometrization Perelman	True Thurston's geometrization Perelman	True Thurston's geometrization Perelman
n=4	Manifold classification (MC)	False Uncountable Markov	False Uncountable Markov	False Uncountable Markov
n=4	Simple connected closed MC		False untriangulable & uncountable Casson & Donaldson	False untriangulable & uncountable Casson & Donaldson
n=4	Poincaré conjecture	True Cobordism Freedman
n=5	Manifold classification (MC)	False Uncountable Markov	False Uncountable Markov	False Uncountable Markov
n=5	Simple connected closed MC	True Barden	False untriangulable Manolescu	False untriangulable Manolescu
n=5	Poincaré conjecture	True Engulfing Newman	True Cobordism Smale	True Cobordism Smale
n=6	Manifold classification (MC)	False Uncountable Markov	False Uncountable Markov	False Uncountable Markov
n=6	Simple connected closed MC		False untriangulable Manolescu	False untriangulable Manolescu
n=6	Poincaré conjecture	True Engulfing Newman	True Cobordism Smale	True Cobordism Smale
n>=7	Manifold classification (MC)	False Uncountable Markov	False Uncountable Markov	False Uncountable Markov
n>=7	Simple connected closed MC		False untriangulable Manolescu	False untriangulable Manolescu
n>=7	Poincaré conjecture	True Engulfing Newman	True Cobordism Smale	False Exotic sphere Milnor

---

---

---

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Analogies: Quine McCluskey algorithmBuchberger's algorithm

---

Correspondences:
algebraic sets & Ideals
Field subextension & Galois Subgroups
Space covering & Fundemantal group + + +
Modular forms & Elliptic curves
Automorphic forms & Algebraic curves
Elliptic modular forms & Group representations

Geometric langlands
Kobayashi–Hitchin correspondence
Simpson correspondence
Riemann-Hilbert correspondence
Robinson-Schensted correspondence
Shimura correspondence-Theta correspondence

---

Enumerative invariants:
Sympletic category: Donaldson invariants Seiberg–Witten invariants Gromov-Witten invariants
FJRW theory Algebraic category: Donaldson–Thomas +

---

characterization: Gamma function+ Determinant exponential function

---

Obstructions characteristic classes:

• homotopy=simple homotopy: Whitehead torsion+
• homeorphism=PL homeomorphism: Kirby-Siebenmann_class+
• homeomorphims=homotopic homeomorphism:Reidemeister torsion
• homeomosphism=homological homeomorphism?:Euler class
• vector bundle morphism=vector bundle isomorphism: Stiefel-Whitney_class
• complex vector bundle morphism=complex vector bundle isomorphism:Chern-class (unitary group)

+ Pontryagin_class (orthogonal group)+++

• Todd classreciprocal characteristic class
• Segre class-inverse chern class
• Hasse invariant
• Manin obstruction

---

---

---

---

Generalizations: (Hodge conjecture-Arithmetic Hodge Conjecture) (Milnor conjecture-Thom conjecture) (Witten conjecture-Virasoro conjecture) (K theory-L theory) (Pontryagin_duality-Tannaka-Krein duality) (Maximun principle-Hopf's Maximum principle) (Padé series-Laurent series-Puiseux series) (Weierstrass factorization-Mittag-Leffer's factorization) (Stone-Weierstrass theorem-Arakelyan's_theorem) (Cantor's_paradox-Ordinal Cantor's paradox(Introduction to Mathematical Logic Mendhelson page 2))(Stone's theorem-Stone-von Neumann theorem) (Morse theory-Picard-Lefschetz theory) (Invariant theory-Geometric invariant theory) (https://en.wikipedia.org/wiki/Differential_Galois_theory)

---

(Farkas's lemma-Fredholm alternative) (sdf matrix-sdf kernel)

---

• CLT-frequentist CLT-Bayesian CLT-Arithmetic CLT-Ensemble CLT-Flow CLT

Martingale CLT Functional integralPath integral

• Gibbs entropy-LK entropy-arithmetic entropy?-maximun entropy-openprob entropy
• AEP+LLN+CLT=LAN
• $S=k_{\text{B}}\ln \Omega _{\rm {mic}}=k_{\text{B}}(\ln Z_{\rm {can}}+\beta {\bar {E}})=k_{\text{B}}(\ln {\mathcal {Z}}_{\rm {gr}}+\beta ({\bar {E}}-\mu {\bar {N}}))$
  

Hammersley–Clifford theorem

---

https://en.wikipedia.org/wiki/Generalized_extreme_value_distribution

---

Vector bundles on algebraic curves Birkhoff–Grothendieck theoremAtiyah-Birkhoff–Grothendieck theorem

---

Bott periodicity&Hurwitz's_theorem Bott_periodicity_theorem Hurwitz theorem

---

Vector bundles on algebraic curves Birkhoff–Grothendieck theoremAtiyah-Birkhoff–Grothendieck theorem

---

Hesisenberg group transform Special linear group transform +?

---

Special_unitary_group (spin group) + +2 +3 +4 +5

---

String group + +2 anomaly conformal invariants + +2

---

• soliton (localised in space)(fixed points of the flow?)
• breather (periodic soliton), oscilon (standing periodic soliton), kink(steady state periodic soliton)
• ADHM instanton
• Instanton(localised in time soliton)(1d:tunneling,2d:vortices,3d:higgs field-TP monopole, 4d: gauge bundle)
• Renormalon(Borel istanton)
• Cotangent localization

---

Helmholtz decomposition FLRW decomposition Ricci tensor decomposition Riemann tensor decomposition Electromagnetic tensor decomposition+

---

cauchy-pompieu decomposition poloidal decomposition clebsch decomposition

---

Sokhotski Plemelj theorem +

---

Covariance matrix Covariance process +

---

Var(x)=2Dt + + +

---

Cayley-Hamilton theorem Fredholm determinant +

variational quotients

Dirichlet eigenvalue Rayleigh quotient Variational method Fisher's linear discriminant Poincaré constant Cheeger constant

---

Poincaré inequality Bauer Fike theorem

---

Min-max theorem Minimax theorem Max-min inequality Duality gap Lagrange multiplier

---

Geometric_phase Fubini-Study_metric Fisher_information_metric

---

Coarea formula Smooth coarea formula coarea+ coarea+

---

Distance preserving Angle preserving Area preserving Volume preserving + Sympletic preserving + +

---

Knot group Link group Braid group

---

Knot theory Link theory Braid theory Ribbon theory +

---

Homology group Homotopy group Holonomy group

---

Cohomology group Cohomotopy group

---

Homological algebra Homotopical algebra

---

stable unitary group stable orthogonal group J-homomorphism

---

Riemann zeta function Hurwitz zeta function Polylogarithm

---

Eta invariant Dirichlet_eta_function [2] 1+2+3+4+... Riemann+Euler-Maclaurin Darboux's formula

---

Analytic torsion Heat kernel signature

---

Signature operator Atiyah Singer index theorem+

---

Witten index + +2 +3 +4 +5 +6

---

Supersymmetric atiyah Singer index theorem + +2 +3 +?

---

Equivalences:Abel–Plana_formula<=>Euler–Maclaurin_formula<=>Poisson summation formula

---

Singular Kernel Regularization

---

Riemman, Ricci, scalar curvature +

---

isothermal conformal uniformization isothermal/conformal? Ricci flow

---

Inclusion-exclusion principle Isomorphism theorems

---

• atiyah-singer proof techniques: pseudodifferential operator, cobordism, k theory, heat operator

Manifold decomposition

• https://en.wikipedia.org/wiki/Borel_summation (best summation)
• https://en.wikipedia.org/wiki/Padé_approximant (best rational approximant)

• https://en.wikipedia.org/wiki/Category:Mathematics-related_lists
• https://en.wikipedia.org/wiki/List_of_cohomology_theories +
• https://en.wikipedia.org/wiki/Vanishing_theorem
• https://en.wikipedia.org/wiki/Fixed-point_theorem#List_of_fixed-point_theorems
• https://en.wikipedia.org/wiki/List_of_zeta_functions ++
• https://en.wikipedia.org/wiki/Trace_formula
• https://en.wikipedia.org/wiki/List_of_dualities
• https://en.wikipedia.org/wiki/Category:Duality_theories
• https://en.wikipedia.org/wiki/List_of_complex_and_algebraic_surfaces
• https://en.wikipedia.org/wiki/List_of_small_groups
• https://en.wikipedia.org/wiki/List_of_prime_knots (https://en.wikipedia.org/wiki/Knot_tabulation)
• https://en.wikipedia.org/wiki/List_of_mathematical_knots_and_links
• https://en.wikipedia.org/wiki/List_of_manifolds

---

https://en.wikipedia.org/wiki/Uniqueness_theorem

---

• https://en.wikipedia.org/wiki/Classification_theorem
• https://en.wikipedia.org/wiki/Complete_set_of_invariants
• https://en.wikipedia.org/wiki/Invariant_(mathematics)
• (https://en.wikipedia.org/wiki/Invariant_theory)
• https://en.wikipedia.org/wiki/Enriques-Kodaira_classification#Invariants_of_surfaces
• https://en.wikipedia.org/wiki/Decomposition_(disambiguation)
• $\Delta \subset \mathbf {C} \subset {\widehat {\mathbf {C} }}$(complex disk,plane,sphere)
• algebraic classification 2-manifolds: sum connected toris,sum connected projective plane, sphere, geometric classification, topological classification, combinatoric classification
• https://en.wikipedia.org/wiki/Holonomy#The_Berger_classification
• https://en.wikipedia.org/wiki/Wigner's_classification
• https://en.wikipedia.org/wiki/Bianchi_classification
• https://en.wikipedia.org/wiki/Petrov_classification
• https://en.wikipedia.org/wiki/Classification_of_Clifford_algebras
• https://en.wikipedia.org/wiki/Enriques–Kodaira_classification
• https://en.wikipedia.org/wiki/Bernstein–Zelevinsky_classification
• https://en.wikipedia.org/wiki/Langlands_classification
• https://en.wikipedia.org/wiki/Segre_classification
• https://en.wikipedia.org/wiki/Crystal_system#Classification_of_lattices
• https://en.wikipedia.org/wiki/Painlevé_transcendents
• https://en.wikipedia.org/wiki/List_of_finite_simple_groups

+ + +

---

• complexity classes lists: list complexity classes+complexity zoo

$\det(\exp(A))=\exp(\mathrm {tr} (A))$

$\ln(\det(A))=\mathrm {tr} (\ln(A))$

[https://en.wikipedia.org/wiki/Schrödinger_equation#Time_indep endent $\mathrm {\hat {H}} \Psi =E\Psi$]

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Twelvefold way +

---

Plane partition number partition Multiplicative_partition (unordered factorization)

---

[3]

---

Numerical methods: Garlekin Homotopy analysis Finite difference Finite element Finite volume ≈

Physics

---

https://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics)

---

https://physics.stackexchange.com/questions/295714/whats-the-relation-between-path-integral-and-dyson-series http://bolvan.ph.utexas.edu/~vadim/Classes/2011f/dyson.pdf

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$\Delta H\equiv \Delta H_{\text{L}}+\Delta H_{\text{T}}$ ${\mathcal {H}}={\mathcal {H}}_{\rm {Coulomb}}+{\mathcal {H}}_{\mathrm {kinetic} }+{\mathcal {H}}_{\mathrm {SO} }+H_{\mathrm {Darwin} }\!$ + + +

---

https://en.wikipedia.org/wiki/Quantum_invariant https://en.wikipedia.org/wiki/Periodic_table_of_topological_invariants

---

$e^{-{\frac {F}{kT}}}=\operatorname {Tr} \exp {\big (}-{\tfrac {1}{kT}}{\hat {H}}{\big )}$
$-{\frac {F}{kT}}=\ln \operatorname {Tr} \exp {\big (}-{\tfrac {1}{kT}}{\hat {H}}{\big )}$

---

https://en.wikipedia.org/wiki/Bose_gas https://en.wikipedia.org/wiki/Gas_in_a_box https://en.wikipedia.org/wiki/Gas_in_a_harmonic_trap

---

thermal fluctuations + +

---

Dilaton Perelman's renormalization Perleman's entropy Perelman's fluctuation Quantropy Jacobson's entropic field Verlinde's entropic force [4]

---

Jacobson	$T={\frac {\kappa }{2\pi }}$	$S={\frac {k_{\mathrm {B} }A}{4l_{P}^{2}}}$	$\delta Q=TdS$	$R_{\mu \nu }-{\tfrac {1}{2}}R\,g_{\mu \nu }+\Lambda g_{\mu \nu }={\frac {8\pi G}{c^{4}}}T_{\mu \nu }$
Verlinde	$T={\frac {\hbar GM}{2\pi ck_{\mathrm {B} }r^{2}}}$	$S={\frac {2\pi k_{\mathrm {B} }mc}{\hbar }}$	$F=T\nabla S$	$F=G\,{\frac {Mm}{r^{2}}}$

---

velocity limit: light velocity entropy limit: bekenstein bound

---

Unruh radiation Hawking radiation +

---

+ Locality breaker: EPR paradox Unitarity breaker: BHI paradox causality breaker: relativity

---

Newton	----	HJ	Koopman–von Neumann
Ehrenfest theorem	Schrodinger	Bhom	Moyal
GR	ADM	EHJ	chern-simmonns?

---

Spacetime symmetries

---

Sagnac effect

---

Crystallographic periodic table crystal structure

Bravais lattice	Wigner-Seitz_cell
Recriprocal lattice	Brillouin zone

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YB equation BMW algebra +

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ΛCDM common observations: SNIa CMB H(z) BAO LSS
Planck spacecraft + standard candle standard ruler

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Hamilton Jacobi_equation Circulation Biot-Savart

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Kutta Joukowski theorem Magnus effect Vorticity equation

---

Kelvin's circulation theorem tao's post Euler fluid equations Hamiltonian fluid mechanics Madelung equations

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https://en.wikipedia.org/wiki/Category:Scattering

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https://en.wikipedia.org/wiki/Moiré_pattern https://en.wikipedia.org/wiki/Bragg's_law https://en.wikipedia.org/wiki/Compton_scattering

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https://en.wikipedia.org/wiki/List_of_quasiparticles https://en.wikipedia.org/wiki/List_of_particles

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mass oscilation	charge oscillation	spin oscilations
PhononRotonMaxon	polarons	magnons

---

Molecular partition function + + +

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volumetric entropy surface entropy gas entropy + + +

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+ thermodynamics +

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Deconfinement

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Bose-Einstein statistics	Fermi-Dirac statistics	anyon statistics	braid statistics
boson	fermion	anyon	plekton

https://en.wikipedia.org/wiki/Scale-free_ideal_gas

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• electromagnetism
• relativistic charge distribution
• relativistic charge point
• +

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https://en.wikipedia.org/wiki/Bragg's_law

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wave: string, springs, bars

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• https://en.wikipedia.org/wiki/Chapman-Enskog_theory(https://en.wikipedia.org/wiki/Navier-Stokes_equations https://en.wikipedia.org/wiki/Boltzmann_equation)

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	Non-rotating (J = 0)	Rotating (J ≠ 0)
Uncharged (Q = 0)	Schwarzschild	Kerr
Charged (Q ≠ 0)	Reissner–Nordström	Kerr–Newman

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Minkowski	$\Lambda =0$	$ISO(3,1)$
De Sitter	$\Lambda >0$	$SO(4,1)$
Anti De sitter	$\Lambda <0$	$SO(3,2)=SO(3,1)xSO(3,1)$?

++

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Group contractionQuantization commutes with reduction

---

D'Alembert->KdV Hierarchy/Dirac operator->BBGKY

---

Hamiltonian fluid mechanics Madelung equation De_Broglie–Bohm_theory Convection diffusion equation Langevin dynamics

---

Lagenvin motion + +

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Dyson Series + Functional determinant

---

Electromagnetism Gravitoelectromagnetism + Kaluza-klein theory Yang-Mills theory

---

Thomas precession	Gravitational Thomas precession
Larmor precession++?	de Sitter precession (Lense-Thirring precession)
Total angular momentum quantum operator	Total angular momentum tensor
Einstein–de Haas effect Barnett effect

Precession +

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Bohr-Sommerfeld EBK-Bohr-sommerfeld cyclotron Bohr-Sommerfeld

---

Aharonov-Bohm effect Landau quantization

---

Wilson loop t'hooft loop

---

T*C->Poison bracket
T*M_g->ADM bracket
+

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• https://en.wikipedia.org/wiki/Supersymmetry_nonrenormalization_theorems
• https://en.wikipedia.org/wiki/Loop_representation_in_gauge_theories_and_quantum_gravity

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Black hole: Black hole radiation dark matter

---

Cosmic Backgrounds:
Relics: Photon(CMB) Neutrino Gravitation
DEBRA: Infrared X-ray Extragalactic light Radio Gamma-ray CMB

---

Angles: Weinberg Peccei–Quinn GIM CKM PMKS +

---

Planck temperature Hagedorn temperature +

---

Noether's Theorem extensions Wald entropy formula++

---

stringification=categorization

---

Operator equations

$i\hbar {\frac {\partial {\hat {\rho }}}{\partial t}}+[{\hat {\rho }},{\hat {H}}]_{-}=0$
${\hat {\rho }}{\frac {\partial {\hat {S}}}{\partial t}}+{\frac {1}{2}}[{\hat {\rho }},{\hat {H}}]_{+}=0$

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Types of radioactive decay

Mode of decay	Daughter nucleus
Decays with emission of nucleons:
Alpha decay	(A − 4, Z − 2)
Proton emission	(A − 1, Z − 1)
Neutron emission	(A − 1, Z)
Double proton emission	(A − 2, Z − 2)
Spontaneous fission	—
Cluster decay	(A − A1, Z − Z1) + (A1, Z1)
Different modes of beta decay:
β− decay	(A, Z + 1)
Positron emission (β+ decay)	(A, Z − 1)
Electron capture	(A, Z − 1)
Bound state beta decay	(A, Z + 1)
Double beta decay	(A, Z + 2)
Double electron capture	(A, Z − 2)
Electron capture with positron emission	(A, Z − 2)
Double positron decay	(A, Z − 2)
Transitions between states of the same nucleus:
Isomeric transition	(A, Z)
Internal conversion	(A, Z)

Nuclide radioactive decay map
Nucleosynthesis periodic table
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