User:Nomalias

polynomials
Chebotarev theorem on roots of_unity generating question 1gq 2gq3gq4

fourier vs contour
+

terms
https://en.wikipedia.org/wiki/Arithmetic_geometry https://en.wikipedia.org/wiki/Arithmetic_dynamics https://en.wikipedia.org/wiki/Arithmetic_topology https://en.wikipedia.org/wiki/Topological_graph_theory https://en.wikipedia.org/wiki/Shape_theory_(mathematics)

weird
Replica trick++++

snippets
Hilbert 10th problem Pell's conics elliptic curves parallel

+ +

https://en.wikipedia.org/wiki/Template:Commutative_local_ring_classes https://en.wikipedia.org/wiki/Template:Commutative_ring_classes https://en.wikipedia.org/wiki/Template:Algebraic_structures https://en.wikipedia.org/wiki/Template:Analogous_fixed-point_theorems

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

stack stuff
$$\sum_{k=1}^{n} k^{m}=\sum_{b=1}^{m+1} \binom{n}b\sum_{i=0}^{b-1} (-1)^{i}(b-i)^{m}\binom{b-1}i$$+ $$\sum_{n=1}^\infty \zeta(2n)x^{2n} = -\frac{\pi x}{2}\cot(\pi x)$$ + toric=>rational+ $$\sum_{t \in \mathbb{F}_{p^n}} \zeta ^{Tr (t)} = 0$$+

stack stuff 2
$$d^3\neq a^3+b^3+c^3\Leftrightarrow_{*} d^3\equiv 4,5\pmod9 \Rightarrow d^3\equiv 1,2\pmod3\Leftrightarrow (a+b+c)\equiv 1,2\pmod3$$

stuff
$$g_n^{-1}(x) = \sup_{i \leq n} \{f_i^{-1}(x)\}=\bigcup_{i=1}^n f_i^{-1}((a,\infty))$$+

$$k \mid n\Leftrightarrow p^{k} - 1 \mid p^{n} - 1 \Leftrightarrow x^{p^{k}} - x \mid x^{p^{n}} - x.$$ + $$\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$$ + $$\gcd(m,n)=\prod p_i^{\min(m_i,n_i)}$$ $$x^2 \equiv a \mbox{ } \mbox{mod} \mbox{ } p\Leftrightarrow a^\frac{p-1}{2} \equiv 1\mbox{ } \mbox{mod} \mbox{ } p$$ + $$(x+yi)^p \equiv x^p+y^pi^p \equiv x + (-1)^{\frac{p-1}{2}}yi \pmod{p}$$ +

$$\left|\vec a\ \vec b\ \vec c\right|=\vec a \cdot (\vec b \times \vec c)$$ + $$d\Omega = 4 \pi \left(\frac{d\Sigma}{A}\right) \, (\hat{r} \cdot \hat{n})= 4 \pi \left(\frac{d\Sigma}{4\pi r^2}\right) \, (\hat{r} \cdot \hat{n})$$ + $$\frac{\mathrm{d}q}{\mathrm{d}t} =\iint_S \mathbf{j}\cdot\mathbf{\hat{n}}\,{\rm d}A\ = \underset{\mathbf{\hat{n}}}{\operatorname{arg\,max}}\, \mathbf{\hat{n}}_{\mathbf p} \frac{\mathrm{d}q}{\mathrm{d}t}(A,\mathbf{p}, \mathbf{\hat{n}})$$ + $$\frac{c+di}{c-di}=\frac{c^2-d^2}{c^2+d^2} + \frac{2cd}{c^2+d^2} i$$

https://en.wikipedia.org/wiki/Zsigmondy's_theorem

$$\text{Cotangent Bundles}\leftrightarrow \text{Pull-backs}\leftrightarrow \text{Differentials}$$ $$\text{Tangent Bundles}\leftrightarrow \text{Push-forward}\leftrightarrow \text{Tangent Vectors}$$ +

Hausdorff moment problem Stieljes moment problem Hamburger moment problem

https://en.wikipedia.org/wiki/Highly_composite_number https://en.wikipedia.org/wiki/Van_Eck's_sequence

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

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

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

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

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

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

combinatorics
$$\dbinom{n}{k}=\frac{1}{2\pi i}\oint_{\left|z\right|=1}\frac{\left(1+z\right)^{n}}{z^{k+1}}dz$$ + $$B_n = {1 \over e}\sum_{k=0}^\infty \frac{k^n}{k!}.$$+ $${j\brace k}=\frac{1}{k!}\sum_{i=0}^{k}\binom{k}{i}\left(-1\right)^{i}\left(k-i\right)^{j}$$ +

+++++++++++
$$f^{(n)}(a) = \frac{n!}{2\pi i} \oint_\gamma \frac{f(z)}{\left(z-a\right)^{n+1}}\, dz.$$+ $$f^{-(n+1)}(x)= \frac{1}{n!} \int_a^x \left(x-t\right)^n f(t)\,\mathrm{d}t$$+ $$\oint_{\gamma}(z-c)^{n-k-1}dz=2\pi i\delta_{nk}$$+

Universality stuff
https://en.wikipedia.org/wiki/Feigenbaum_constants https://en.wikipedia.org/wiki/Misiurewicz_point https://en.wikipedia.org/wiki/Albert_J._Libchaber#Research https://en.wikipedia.org/wiki/Sharkovskii's_theorem https://en.wikipedia.org/wiki/Kondo_effect https://en.wikipedia.org/wiki/Beta_function_(physics) https://en.wikipedia.org/wiki/Critical_exponent https://en.wikipedia.org/wiki/Universality_class https://en.wikipedia.org/wiki/Kibble-Zurek_mechanism

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

Correspondences&dictionaries
https://ncatlab.org/nlab/search?query=correspondence [https://mathoverflow.net/questions/14574/your-favorite-surprising-connections-in-mathematics? +] Schreiber's correspondences+

Correspondences: algebraic sets & Ideals Field subextension & Galois Subgroups Galois group & Fundemantal group + + + algebra/geometry & galois/fundamental group Modular forms & Elliptic curves Automorphic forms & Algebraic curves Elliptic modular forms & Group representations Geometric langlands Kobayashi–Hitchin correspondence Nonabelian Hodge correspondence Simpson correspondence Riemann-Hilbert correspondence Robinson-Schensted correspondence Shimura correspondence-Theta correspondence

Haussdorf-C* algebra duality + + + Lawrence theory Morita's equivalence

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

votja's conjecture+

Baez-Stay dictionary

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

https://en.wikipedia.org/wiki/Lie_group-Lie_algebra_correspondence#Proof_of_the_homomorphisms_theorem

Generalizations
(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(+)) (Russell's_paradox-Girard's_paradox) (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) (Borel–Weil–Bott theorem-GAGA) (Kloosterman sum-Ramanujan sum) (Weierstrass preparation theoremMalgrange preparation theorem)

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

+ + +
 * 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/Classification_of_finite_simple_groups
 * https://en.wikipedia.org/wiki/List_of_finite_simple_groups
 * https://en.wikipedia.org/wiki/Compact_group#Classification
 * https://en.wikipedia.org/wiki/Lie_algebra_representation#Classifying_finite-dimensional_representations_of_Lie_algebras

Invariant miscelanea
(Invariant theory-Geometric invariant theory)
 * 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
 * Eta invariant
 * https://en.wikipedia.org/wiki/J-invariant
 * https://en.wikipedia.org/wiki/Hasse_invariant

+ +
 * https://en.wikipedia.org/wiki/Curvature_invariant
 * conformal invariants+

https://en.wikipedia.org/wiki/Invariant_differential_operator https://en.wikipedia.org/wiki/Differential_invariant https://en.wikipedia.org/wiki/Differential_operator

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

Enumerative invariants
Enumerative invariants: Sympletic category: Donaldson invariants+ Seiberg–Witten invariants Gromov-Witten invariants FJRW theory Gopakumar-Vafa invariant

duality miscelanea
Reciprocities: https://en.wikipedia.org/wiki/Quadratic_reciprocity http://en.wikipedia.org/wiki/Weil_reciprocity_for_algebraic_curves http://en.wikipedia.org/wiki/Stanley's_reciprocity_theorem +

+

https://en.wikipedia.org/wiki/Topology_(electrical_circuits)#Duality

algebraic geo stuff
https://en.wikipedia.org/wiki/Transfer_principle https://en.wikipedia.org/wiki/Algebraic_geometry_and_analytic_geometry

$$\mu(\mathcal E)=\frac{deg \mathcal E}{rk \mathcal E}\quad$$ $$\mu(E)=\frac{dim H^0(X,E)-dim H^1(X,E)}{rank E}+g_X-1$$ + + +

category theory
five lemmasix operations

https://en.wikipedia.org/wiki/Universal_property Universal bundle? + https://en.wikipedia.org/wiki/Product_(category_theory) https://en.wikipedia.org/wiki/Exponential_object https://en.wikipedia.org/wiki/Exponential_sheaf_sequence +

$$(Z, \ker(f, g)) ≃ \ker(\text{Hom} (Z, X) ⇉ \text{Hom} (Z, Y ))$$ +

discrete stuff
discrete taylor series discrete taylor series table discrete integration by parts discrete laplacian discrete spectral theory

stackexchange
The Selberg trace formula is making $$PSL(2,\Bbb{R})$$ act on $$C^\infty(\Gamma \setminus \Bbb{H})$$, the Frobenius formula is making G act on $$\Bbb{C}[G/H]$$ +

generating functions
Generating functions+$$\sum_{n=0}^\infty \frac{s(n)}{n!}x^n = \left(1 - x^2\right)^{-\frac{1}{4}}\exp\left(\frac{x^2}{4}\right).$$+connected graph generating function

diferential representations
$$\sum_{n=0}^{\infty} \sum_{k=0}^{n} B(k,n) = \sum_{n=0}^{\infty} \sum_{k=0}^{\infty} B(k,n+k)$$+https://en.wikipedia.org/wiki/Shift_theorem https://en.wikipedia.org/wiki/Shift_operator https://en.wikipedia.org/wiki/Hasse-Schmidt_derivation


 * $$e^{D^2} f(x) = \sum_{k=0}^\infty \frac{D^{2k}}{k!}f(x).$$
 * $${e^D - 1 \over D}f(x)= \sum_{n=0}^\infty {D^n \over (n+1)!}f(x)$$
 * $$e^{D^2}f(x)=\frac{1}{\sqrt{4\pi}} \int_{-\infty}^\infty f(x-y) e^{-y^2/4}\;dy$$
 * $${e^D - 1 \over D}f(x) = \int_x^{x+1} f(u)\,du$$
 * $$e^{-D^2}\sum_{n=0}^\infty a_n x^n=\sum_{n=0}^\infty a_n H_n(x/2)$$
 * $${D \over e^D -1}\sum_{n=0}^\infty a_n x^n=\sum_{n=0}^\infty a_n B_n(x)$$

+ + +
 * $$\frac{D e^D}{e^D-1}=\sum_{n=0}^\infty \frac{B_n}{n!} D^n=\sum_{n=0}^{\infty} (-1)^{n-1} \frac{\zeta(1-n)}{(n-1)!}D^{n}=-1+\frac{D}{2}-\frac{D^2}{12}+...$$

++ +

$$e^{t D}f(x)= f(x+t)$$ ++ +

$$(X + D)^n = \sum_{j = 0}^{n }{n\choose j} H_{n-j}(X)D^j$$+

$$(1-D)^2X_t = X_t -2X_{t-1} + X_{t-2}\quad$$ $$(1 - B)^d= \sum_{k=0}^{\infty} \; {d \choose k} \; (-B)^k$$ +

Sequences
https://en.wikipedia.org/wiki/Category:Spectral_sequences

Surgery exact sequence Serre spectral sequence Mapping_class_group

https://en.wikipedia.org/wiki/Poisson_summation_formula#Derivation

congmatics
https://en.wikipedia.org/wiki/Möbius_transformation

$$Isom^+(\mathbb{H}^3)=PGL(2, \mathbb{C})=PSL(2, \mathbb{C})$$+, $$Isom(\mathbb{S}^2)=O(3)$$+

$$\mathbb{S}^2,\mathbb{E}^2,\mathbb{H}^2,SO(3),ISO(\mathbb{R}^2)^+,SL(2,\mathbb{R})=SO(1,3)^+,\pi_1(\mathbb{S}^2)=0,\pi_1(T \cong \R^2/\Z^2)=\Z^2$$

RPn CPn HPn

+ + +

+



+ +

Diagramatics
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 +]

matrices stuff
Khatri rao product

+

https://en.wikipedia.org/wiki/Matrix_determinant_lemma#See_also

Gaps
https://en.wikipedia.org/wiki/Gap_theorem_(disambiguation) https://en.wikipedia.org/wiki/Spectral_gap_(physics) https://en.wikipedia.org/wiki/Duality_gap https://en.wikipedia.org/wiki/Prime_gap https://en.wikipedia.org/wiki/Montgomery's_pair_correlation_conjecture https://en.wikipedia.org/wiki/Yang-Mills_existence_and_mass_gap + 

spectral stuff
https://en.wikipedia.org/wiki/Singular_integral https://en.wikipedia.org/wiki/Singular_trace

linear algebra
https://en.wikipedia.org/wiki/Matrix_determinant_lemma#See_also + https://en.wikipedia.org/wiki/Matrix_splitting#Matrix_iterative_methods https://en.wikipedia.org/wiki/Matrix_decomposition

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

https://en.wikipedia.org/wiki/Rouché-Capelli_theorem

stuff with det,tr
Poisson summation formulaFrobenius reciprocity Selberg trace formula+++++Poisson=Fourier on circle

$$\frac{d}{dt}e^{X(t)} = \int_0^1 e^{\alpha X(t)} \frac{dX(t)}{dt} e^{(1-\alpha) X(t)}\,d\alpha ~=e^{X}\frac{1 - e^{-\mathrm{ad}_{X}}}{\mathrm{ad}_{X}}\frac{dX(t)}{dt}.$$ + + +

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

https://en.wikipedia.org/wiki/Nahm_equations#Nahm-Hitchin_description_of_monopoles https://en.wikipedia.org/wiki/Particle_in_a_one-dimensional_lattice#Kronig-Penney_model

http://mathworld.wolfram.com/Convergent.html

simplex determinant Cayley-Menger_determinant

$$\mathbf{a}\cdot\mathbf{b}=\|\mathbf{a}\|\ \|\mathbf{b}\|\cos\theta ,$$ $$\zeta(n+m)=\sqrt{\zeta(2n)\zeta(2m)}\cos\theta ,$$ +

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

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

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

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

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

$$E_n= 2 n \pi k T=\hbar \omega n $$ + + +


 * $$\langle \hat{A}\rangle={1\over Z_0}\operatorname{Tr}\,[\hat{\rho_0}\hat{A}]={1\over Z_0}\sum_n \langle n | \hat{A} |n \rangle e^{-\beta E_n}$$
 * $$\hat{\rho_0}=e^{-\beta \hat{H}_0}=\sum_n |n \rangle\langle n |e^{-\beta E_n}$$

+ + + + + + +

$$ \langle \Gamma_i\Gamma_j \rangle=\operatorname{tr}\{\Gamma_i\Gamma_jR_0\}$$ $$ \langle \Gamma_i(t)\Gamma_j(t') \rangle \propto \delta(t-t')\quad\text{ideally} $$ +


 * $$|\psi(t)\rangle = \exp\left(-\frac{i}{\hbar} \hat H t\right) |q_0\rangle \equiv \exp\left(-\frac{i}{\hbar} \hat H t\right) |0\rangle

$$
 * $$ \left \langle F \bigg| \exp\left( {- {i \over \hbar } \hat H T} \right) \bigg |0 \right \rangle = \int Dq(t) \exp\left[ {i\over \hbar} S \right]$$

+

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

$$\Theta \mapsto \operatorname{E}\left[ \, \exp\left( \,i \operatorname{tr}\left(\,\mathbf{X}{\mathbf\Theta}\,\right)\,\right)\, \right] = \left|\, {\mathbf I} - 2i\, {\mathbf\Theta}\,{\mathbf V}\, \right|^{-n/2} $$+

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

$$\operatorname{Index}(D) = \dim\operatorname{Ker}(D)− \dim\operatorname{Ker}(D*)=\operatorname{tr}(\exp(D*D))-\operatorname{tr}(\exp(DD*))$$ + $$ \frac{1}{Z_{\text{GUE}(n)}} e^{- \frac{n}{2} \mathrm{tr} H^2} $$ +

$$\zeta(1-n,a)=-\frac{B_n(a)}{n} \!$$ for $$ n\geq1 \!$$+ $$\zeta(2n) = \frac{(-1)^{n+1}B_{2n}(2\pi)^{2n}}{2(2n)!}$$ $$\zeta(-n)=(-1)^n \frac{B_{n+1}}{n+1}$$

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

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

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

$$\varphi_{Z_n}(t)=(\varphi_{Y_1}(\frac{t}{\sqrt{n}}))^n=(e^{-\frac{1}{2}(\frac{t}{\sqrt{n}})^2})^n=e^{-\frac{t^2}{2}}$$

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

$$\psi_t(y) = \int \psi_0(x) K(x - y; t) \,dx = \int \psi_0(x) \int_{x(0) = x}^{x(t) = y} e^{iS} \,Dx,$$

$$K(x, y; T) = \int_{x(0) = x}^{x(T) = y} \prod_t \exp\left(-\tfrac12 \left(\frac{x(t + \varepsilon) - x(t)}{\varepsilon}\right)^2 \varepsilon \right) \,Dx,$$

$$\chi$$
$$V(G)-E(G)+F(G) = k+1$$+ //+ + + + //+ + +

"Index theory"
https://mathoverflow.net/questions/233144/atiyah-singer-theorem-a-big-picture https://mathoverflow.net/questions/1162/atiyah-singer-index-theorem http://www.concinnitasproject.org/portfolio/gallery.php?id=Atiyah_Michael

List of fixed-point theorems +

$$\sum_i(-1)^i\text{Tr}(\text{Frob},H^i_c(X,\bf{Q}_\ell))=|X({\bf F}_q)|$$ $$\sum_i(-1)^i\mathrm{Tr}(f_*|H_k(X,\mathbb{Q}))=\sum_{x\in\mathrm{Fix}(f)}\mathrm{index} _x f$$ $$\sum_i(-1)^i\dim H_k(X,\mathbb{Q})=\sum_{x\in\mathrm{Sing}(v)}\mathrm{index}_xv$$ + + + + +

https://en.wikipedia.org/wiki/Atiyah-Bott_fixed-point_theorem https://en.wikipedia.org/wiki/Supersymmetric_theory_of_stochastic_dynamics#Parisi-Sourlas_approach_to_Langevin_SDEs

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

$${\mathcal W} = \operatorname{Tr} (-1)^{\hat n} \langle M_{t't}^* \rangle_\text{noise} = \langle \operatorname{Tr} (-1)^{\hat n} M_{t't}^* \rangle_\text{noise} = I_{L}$$. + $$I_L=\operatorname{Tr} (-1)^{\hat n} M_{t't}^*= \sum_{x\in \operatorname{fix} M_{tt'}} \operatorname{sign} \operatorname{det} (\delta_j^i -\partial M_{tt'}^i(x) /\partial x^j)$$ $$\deg(f,\Omega,p):=\sum_{y\in f^{-1}(p)} \operatorname{sgn} \det(1- Df(y))$$ + $$\deg(f) = \sum_{x \in f^{-1}(y)} \operatorname{sgn} (\det(df_x))$$+

$$\sigma(n)=\sum_{i\in\mathbb{Z}} (-1)^{i+1}\left(\sigma(n{-}\frac12(3i^2{-}i))+\delta(n,\frac12(3i^2{-}i))\,n\right)=\sigma(n{-}1)+\sigma(n{-}2)-\sigma(n{-}5)-\sigma(n{-}7)+\sigma(n{-}12)+\sigma(n{-}15)+ \cdots$$ +

Maximum modulus principle +Fundamental theorem of algebra Hairy ball theorem real root

https://en.wikipedia.org/wiki/Dividing_a_circle_into_areas#Combinatorics_and_topology_method

Riemann zeta function Hurwitz zeta function Polylogarithm

delete me? 1+2+3+4+... Riemann+Euler-Maclaurin Darboux's formula

Analytic torsion Heat kernel signature

Atiyah Singer index theorem+ Signature operator Equivariant_index theorem

Bott residue formula

+ + +

Supersymmetric atiyah Singer index theorem + +?

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

Singular Kernel Regularization

$$[f]=[P_0+\frac{1}{P_1+\frac{1}{P_2+\ddots}}]=\sum_{i \geq 0} (-1)^i[P_i]$$ + +

Algebraic Geometry
https://en.wikipedia.org/wiki/Enriques-Kodaira_classification#Invariants_of_surfaces

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

Hilbert's basis theorem Hilbert's Nullstellensatz Hilbert's syzygy theorem

https://en.wikipedia.org/wiki/Genus-degree_formula https://en.wikipedia.org/wiki/Plücker_formula https://en.wikipedia.org/wiki/Riemann-Hurwitz_formula

Dual theorems As the real projective plane, $PG(2, R)$, is self-dual there are a number of pairs of well known results that are duals of each other. Some of these are:

+
 * Desargues' theorem ⇔ Converse of Desargues' theorem
 * Pascal's theorem ⇔ Brianchon's theorem
 * Menelaus' theorem ⇔ Ceva's theorem

https://mathoverflow.net/a/17139/142708

Hodge duality+->Poincaré duality->Grothendieck local duality->Serre duality poincare-serre connection

https://en.wikipedia.org/wiki/De_Rham_cohomology https://en.wikipedia.org/wiki/Dolbeault_cohomology https://en.wikipedia.org/wiki/Differential_form https://en.wikipedia.org/wiki/Complex_differential_form

kunneth theorem + + +

Riemann-Roch Stuff
$$\ell (\mathcal K_X - D) = \dim H^0 (X, \omega_X \otimes \mathcal L(D)^\vee),H^0 (X, \omega_X \otimes \mathcal L(D)^\vee)$$ Line bundle-Riemann surface Vector Bundle-Complex manifold Quotient stack sheaf-Orbifold Chain-complex sheaf-Scheme Arithmetic+ $$\ell(D)-\ell(K-D) = \deg(D) - g + 1=$$dimension − correction = degree − genus + 1. + $$ \chi(D) = \chi(0) + \frac{1}{2}(D.D - D.K) $$ +

six operationsImages of sheaves

$$e_i f_i= \mathrm{deg}(X/\, X/G)= |G|$$+

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

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

Hodge stuff
$$\dim H^0(X, \mathbb{C}) - \dim H^1(X, \mathbb{C}) + \dim H^2(X, \mathbb{C}) = 2 \left( \dim H^0(X, \mathcal{O}) - \dim H^1(X, \mathcal{O}) \right)$$ +

Homotopy stuff
https://math.stackexchange.com/a/3088/683216

algebraic topology
$$ \chi(M \# N) = \chi(M) + \chi(N) - \chi(S^n).$$ + $$\pi_1(X\times Y)=\pi_1(T)\ast_{\pi_i(S^1)}\pi_1(T)$$ +

$$\langle a,b,c,d\ |\ abcd=1\rangle$$ $$\langle a,b,c\ |\ [a,b]c=1\rangle$$ +

https://en.wikipedia.org/wiki/Homology_(mathematics)
 * NOTES:
 * For a non-orientable surface, a hole is equivalent to two cross-caps.
 * Any 2-manifold is the connected sum of g tori and c projective planes. For the sphere $$S^2$$, g = c = 0.

geometric algebra
https://en.wikipedia.org/wiki/Geometric_algebra https://en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra https://en.wikipedia.org/wiki/Rotation_formalisms_in_three_dimensions

https://en.wikipedia.org/wiki/Hodge_star_operator#Derivatives_in_three_dimensions https://en.wikipedia.org/wiki/Exterior_derivative#Exterior_derivative_in_vector_calculus https://en.wikipedia.org/wiki/Closed_and_exact_differential_forms https://en.wikipedia.org/wiki/Exterior_calculus_identities $$A \times B = \tfrac{1}{2}(AB - BA) $$+ $$a \times b = \star (a \wedge b) \,.$$+ $$\mathbf{a} \times \mathbf{b} = [\mathbf{a}]_{\times} \mathbf{b}$$++

sympletic geometry
$$\int_M e^{-tH} \omega^n/n! = \sum_p {e^{-tH(p)} \over t^n \prod \alpha_j(p)}.$$ + + + !

Differential geo
Maxwell's equations-Alternative formulations Mathematical descriptions of the electromagnetic field P-form electrodynamics

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

https://en.wikipedia.org/wiki/Laplace_operators_in_differential_geometry $$ \underbrace{\operatorname{R}_{ab}}_{\text{Ricci}}\equiv \underbrace{\operatorname{R}^c{}_{acb}}_{\text{Riemann}}= g^{cd} \underbrace{\operatorname{R}_{cadb}}_{\text{Riemann}}$$ ++ https://en.wikipedia.org/wiki/Weitzenböck_identity https://en.wikipedia.org/wiki/Bochner's_formula https://en.wikipedia.org/wiki/Bochner_identity

$$\int_M K\;dA+\int_{\partial M}k_g\;ds=2\pi\chi(M), \, $$ + $$\zeta'(\Delta, 0) = \frac{1}{12}\int_M K dA$$ + + $$ \iint_R |N_u \times N_v| \ du\, dv = \iint_R K|X_u \times X_v| \ du\, dv = \iint_S K \ dA$$ + $$\operatorname{R}_{ab} = Kg_{ab}. \, $$ + $$S=\operatorname{tr}_g \operatorname{Ric}$$ + $$R_{k \overline{l}}=\partial_{k} \partial_{\overline{l}} \ln (\operatorname{det}(g)),\ \text{Ricci-Chern form}$$ + + + $$\frac{\log(T_{an}M_{i})}{\textrm{volume}(M_{i})}\rightarrow -\frac{1}{6\pi};$$$$exp( - \zeta'(0)) / Vol(S)$$ + +


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

Real n-Cauchy-Riemann: $$Df^TDf = (\det(Df))^{2/n}I$$ + $$\mathcal{D}\psi\mathcal{D}\overline{\psi} = \prod\limits_i da^i db^i = \prod\limits_i da^{\prime i}db^{\prime i}{\det}^{-2}(C^i_j),$$ +


 * 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 Conformal connection $$\int_X \omega \wedge \alpha = \int_X dg \wedge \omega = \int_{\gamma\times (0,\varepsilon)} dg \wedge \omega = \int_{\gamma\times (0,\varepsilon)}d(g\omega) = \int_\gamma \omega.$$+

$$\mathrm{d}p\, \mathrm{d}q = \frac{\partial(p, q)}{\partial(\theta, \varphi)} \mathrm{d}\theta\, \mathrm{d}\varphi = \left(\frac{\partial p}{\partial \theta} \frac{\partial q}{\partial \varphi} - \frac{\partial p}{\partial \varphi} \frac{\partial q}{\partial \theta}\right) \mathrm{d}\theta\, \mathrm{d}\varphi = n^2 \cos \theta \sin \theta\, \mathrm{d}\theta\, \mathrm{d}\varphi = n^2 \cos \theta\, \mathrm{d}\Omega,$$ +

$$h = \frac{1}{4} \mathrm{E}\left[(d\log p)^2\right] + \mathrm{E}\left[(d\alpha)^2\right]- \left(\mathrm{E}\left[d\alpha\right]\right)^2- \frac{i}{2}\mathrm{E}\left[d\log p\wedge d\alpha\right] =\frac{1}{4} \mathrm{E}\left[\frac{\partial\log p}{\partial\theta_j}\frac{\partial\log p}{\partial\theta_k}\right]+ \mathrm{E}\left[\frac{\partial\alpha}{\partial\theta_j}\frac{\partial\alpha}{\partial\theta_k}\right]- \mathrm{E}\left[ \frac{\partial\alpha}{\partial\theta_j} \right]\mathrm{E}\left[\frac{\partial\alpha}{\partial\theta_k} \right]- \frac{i}{2}\mathrm{E}\left[\frac{\partial\log p}{\partial\theta_j}\frac{\partial\alpha}{\partial\theta_k}-\frac{\partial\alpha}{\partial\theta_j}\frac{\partial\log p}{\partial\theta_k}\right]$$ +

$$(\nabla \times \mathbf{F})(p)\cdot \mathbf{\hat{n}} \ \overset{\underset{\mathrm{def}}{}}{=} \lim_{A \to 0}\left( \frac{1}{|A|}\oint_C \mathbf{F} \cdot d\mathbf{r}\right)$$ + $${\frac{\partial g}{\partial \bar{z}}(z_0)}\overset{\mathrm{def}}{=}\lim_{r \to 0}\frac{1}{2\pi i r^2} \oint_{\Gamma(z_0,r)} g(z)\mathrm{d}z,$$ +

Degenerancy theory
covering degenerancy manifold degenerancy

Poincaré–Hopf theoremHairy ball theorem

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

Hall's marrriage theorem equivalences

Representation stuff
Character theory https://en.wikipedia.org/wiki/Schur_orthogonality_relations https://en.wikipedia.org/wiki/Frobenius-Schur_indicator Weyl character formula Kirillov_character_formula https://en.wikipedia.org/wiki/Dirichlet_character

--SU(2)&SO(3)-- https://en.wikipedia.org/wiki/3D_rotation_group#Connection_between_SO(3)_and_SU(2) https://en.wikipedia.org/wiki/Representation_theory_of_SU(2) https://en.wikipedia.org/wiki/Representation_of_a_Lie_group#An_example:_The_rotation_group_SO(3) https://en.wikipedia.org/wiki/Spin_spherical_harmonics https://en.wikipedia.org/wiki/Clebsch-Gordan_coefficients

https://en.wikipedia.org/wiki/Representation_theory_of_the_symmetric_group https://en.wikipedia.org/wiki/Jucys-Murphy_element

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

https://en.wikipedia.org/wiki/Representation_theorem https://en.wikipedia.org/wiki/Multiplicity-one_theorem

conmutative tuff
prime ideals are maximal if nonzero, i.e.  dim D≤1->prime ideals are principal->maximal ideals are principal->gcd(a,b)=1⇒(a,b)=1, i.e.  coprime then comaximal->D is Bezout->D is a PID + $$l_R(R/I^{[p^e]})=p^{ed}l_R(R/I)$$+$$\operatorname{reg}(I^r)=rd$$+

group stuff
group theory: definitions basics factsnon basic facts

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

burside:+

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

$$\sum_{i=1}^{p-1} i^{p-1} \equiv -1 \pmod p,\prod_{i=1}^{p-1} i^{p-1} \equiv 1 \pmod p\Leftrightarrow p \text{ prime}$$+ $$(p-1)! \equiv -1 \pmod p,\Leftrightarrow p \text{ prime}$$ $$\sum_{k=0}^{n-1} e^{2 \pi i \frac{k}{n}} = 0 .$$+ $$\prod_{k=0}^{n-1} e^{2 \pi i \frac{k}{n}} = (-1)^{n-1} .$$+ $$\gcd\left(p, \sum_{i=1}^{p-1} i^{p-1}\right)=1\Leftrightarrow p \text{ prime or carmichael}$$+

+ https://en.wikipedia.org/wiki/Daniel_da_Silva_(mathematician) https://en.wikipedia.org/wiki/Chebotarev_theorem_on_roots_of_unity 

Cayley's theorem equivalencesWagner-preston theorem +++

https://en.wikipedia.org/wiki/Deligne-Lusztig_theory

https://en.wikipedia.org/wiki/Zappa–Szép_product $$H\subset G,K\subset G$$=\=>$$HK\subset G$$+ $$H_1\cong K_1,H_2\cong K_2,\ G$$=\=>$$\frac{H_1}{K_1}\cong\frac{H_1}{K_1}$$+ $$F_{p}=\langle x\rangle\text{=\=>}F_{p^2}=\langle x\rangle$$+ $$(\pm1)^2,(\pm12)^2\pmod{143}\equiv 1$$+ $$G=\langle x\rangle\Rightarrow G/Z(G)=\langle xZ(G)\rangle$$+$$C_G(C_G(g)) = Z(C_G(g))$$+

$$\operatorname{gcd}(|G|,n)||\{x\in G:x^n=1\}|$$+

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

Thompson order formula

$$|G| = |Z(G)| + \sum_{i=1}^r |G:C_G(g_i)|$$+ $$|S| = |S_0| + \sum_{i=1}^r |G|/|G_i|$$+

$$\sum_{x \in \pi_0(X)} \frac{1}{|\text{Stab}(x)|} = \frac{|S|}{|G|},\qquad \sum_{x \in \pi_0(X)} \frac{1}{|\text{Aut}(x)|}$$ + + $$ \sum_{\Lambda}{1\over|{\operatorname{Aut}(\Lambda)}|}= 2\pi^{-n(n+1)/4}\prod_{j=1}^n\Gamma(j/2)\prod_{p\text{ prime}}2m_p(f)$$ + $$|G|=|G/G_x||G_x|=|G_x \backslash G||G_x|$$+ $$|G|=|G/H||H|=|H \backslash G||H|$$+ $$G/Z(G) \cong Inn(G)$$ $$Aut(G)/Inn(G) \cong Out(G)$$ $$X^g = gXg^{-1}$$ + + +

Chebotarev's density theorem[ PNTHardy-Ramanujan theorem + + +

Centralizer-Normalizer Orbit stabilizer coset-index

https://en.wikipedia.org/wiki/Wedderburn's_little_theorem https://en.wikipedia.org/wiki/Zappa–Szép_product

Fundamental theorem of abelian groups Fundamental theorem of cyclic groups Fundamental theorem of free groups Jordan–Hölder theorem Finitely generated abelian group Structure theorem for finitely generated modules over a principal ideal domain

https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups https://en.wikipedia.org/wiki/Abelian_group#Classification

mobius table
+ + + + +

mobius+
$$\sum_{d\mid n}\varphi(d)=n$$, $$\varphi (n) = \sum\limits_{k=1}^n \gcd(k,n) e^{-2\pi i\frac{k}{n}}$$; $$\sum_{d \mid n} \mu(d)=\delta_{n1}$$, $$\mu(n) = \sum_{\stackrel{1\le k \le n }{ \gcd(k,\,n)=1}} e^{2\pi i \frac{k}{n}}$$

Lambert
$$\ln(\sum_{n=1}^\infty\frac{1}{n^s}))=\ln((\sum_{n=1}^\infty \frac{\mu(n)}{n^s})^{-1})=\ln(\zeta(s))=\ln(\prod_{p \text{ prime}} (1-p^{-s})^{-1})=\sum_{p,n}\frac{p^{-ns}}{n}=\sum_{n=2}^\infty \frac{\Lambda(n)}{\log(n)}\,\frac{n}{n^{s+1}}=...=...$$

$$\ln(\sum_{k=0}^\infty a_kq^k)=\ln((\sum_{n=0}^\infty p(k) q^k)^{-1})=\ln(f(q))=\ln(\prod_{k=1}^\infty (1-q^k))=-\sum_{k,n}\frac{q^{nk}}{n}=-\sum_n\frac{\sigma(n)}{n}q^{n}=\sum_{n=1}^\infty\frac{1}{n}\,\frac{q^n}{q^n-1}=,,,$$

$$...=...=\ln(g(q))=\ln(\prod_{k \geq 1}(1-q^k)^{\mu(k)/k})=-\sum_{k,n}\frac{\mu(k)}{k}\frac{q^{nk}}{n}=-\sum_n\frac{\delta_{n1}}{n}q^{n}=-\sum_{n=1}^\infty \frac{\mu(n)}{n}\,\frac{q^n}{q^n-1}$$ + + + + + + +

$$...=...=\ln(g(q))=\ln(\prod_{k \geq 1}(1-q^k)^{\frac{\varphi(k)}{k}})=-\sum_{k,n}\frac{\varphi(k)}{k}\frac{q^{nk}}{n}=-\sum_n \frac{n}{n}q^{n}=-\sum_{n=1}^\infty \frac{\varphi(n)}{n}\,\frac{q^n}{q^n-1}$$

$$...=...=\ln(g(q))=\ln(\prod_{k \geq 1}(1-q^k)^{\frac{\Lambda(k)}{k}})=-\sum_{k,n}\frac{\Lambda(k)}{k}\frac{q^{nk}}{n}=-\sum_n \frac{\ln(n)}{n}q^{n}=-\sum_{n=1}^\infty \frac{\Lambda(n)}{n}\,\frac{q^n}{q^n-1}$$

$$\sum_{n=1}^{\infty} \mathrm{spt}(n) q^n=\frac{1}{(q)_{\infty}}\sum_{n=1}^{\infty} \frac{q^n \prod_{m=1}^{n-1}(1-q^m)}{1-q^n}$$ ++

$$\eta(\tau)=q^{\frac{1}{24}} \prod_{n=1}^{\infty} (1-q^{n})=(\sum_{n>0}\tau(n)q^n)^{-24}=(\sum_{n=0}^{\infty}p(n)q^{n-\frac{1}{24}})^{-1}$$ $$\sum_{n\geq 1}\tau(n)q^n=q\prod_{n\geq 1}(1-q^n)^{24} = \eta(z)^{24}=\Delta(z),$$ + +

$$\sum_{g=0}^\infty~\sum_{k=1}^\infty~\sum_{\beta\in H_2(M,\mathbb{Z})}\text{BPS}(g,\beta)\frac{1}{k}\left(2\sin\left(\frac{k\lambda}{2}\right)\right)^{2g-2}q^{k\beta}=\sum_{g=0}^\infty~\sum_{\beta\in H_2(M,\mathbb{Z})} \text{GW}(g,\beta)q^{\beta}\lambda^{2g-2}$$ +

elliptic stuff
$$\int\limits_0^A \omega + \int\limits_0^B \omega = \int\limits_0^{A \oplus B} \omega$$ + $$|\mathbb{Z}/p\mathbb{Z}[\alpha]|=|\mathbb{Z}/p\mathbb{Z}|^d$$ + $$\int_{\gamma} f(\zeta)\,d\zeta + \int_{\tau^{-1}} f(\zeta) \, d\zeta =\oint_{\gamma \tau^{-1}} f(\zeta)\,d\zeta = 0.$$ +

arithmetic
$$[ \overline{\mathbb{F_{p}}} : \mathbb{F_{p}} ] = \infty$$ ++ $$\sigma(\zeta_n) = \zeta_n^{p \mod n}$$ + Finite unramified extension $$L/K \Leftrightarrow $$ Finite unramified ring extension$$\mathcal O_L / \mathcal O_K \Leftrightarrow $$ Finite extension $$l/k$$++ $$[ \overline{\mathbb{F_{p}}} : \mathbb{F_{p}} ] = \infty$$+

primes stuff
$$\int_2^Y\left(\sum_{2<p\le x} \log p -\sum_{2<n\le x}1\right)^2\,dx$$+$$\frac1{\tau_n} \sum_{d\mid n} d^2 - \left(\frac1{\tau_n} \sum_{d\mid n} d\right)^2$$+

Digamma functionWeierstrass zeta function

$$\beta(s) = \prod_{p \ge 3 \atop p \text{ prime}} \frac{1}{1 -\, \scriptstyle(-1)^{\frac{p-1}{2}} \textstyle p^{-s}}.$$ + $$G \cong G_2 \oplus \bigoplus_{p \, \equiv \, 1 \, (\text{mod } 4)} G_p.$$ +

$$\sum_{p\leq x}\frac{1}{p}=\int_2^x \frac{1}{t}\,d(\pi(t))$$ +

+

$$\zeta(s) = \dfrac{1}{2(s-1)}(\pi e^{\gamma})^{s/2}\prod_{n=1}^{\infty}(1+\dfrac{s}{2n})e^{-s/2n}\prod_{\rho}(1-\dfrac{s}{\rho})$$ +

++ $$\frac{1}{\zeta(s)} = s\int_1^\infty \frac{M(x)}{x^{s+1}}\,dx$$ + $$\psi_0(x) = \dfrac{1}{2\pi i}\int_{\sigma-i \infty}^{\sigma+i \infty}\left(-\dfrac{\zeta'(s)}{\zeta(s)}\right)\dfrac{x^s}{s}ds\quad$$ $$\frac{\zeta^\prime(s)}{\zeta(s)} = - s\int_1^\infty \frac{\psi(x)}{x^{s+1}}\,dx$$ +

$$\int_0^\infty x^{s}\ln(1-e^{-x})dx = - \int_0^\infty x^{s-1} \frac{e^{-x}}{1 -e^{-x}} dx$$ $$\int_{0}^{1} | \mathsf{Li}_{s}(e^{2 \pi i x})|^{2} dx = \sum_{k \geq 1} \left| \frac{e^{2 \pi i k x}}{k^{s}} \right|^{2} = \sum_{k \geq 1} \frac{1}{k^{2s}} = \zeta(2s)$$ +

$$\sum_{p} \frac{\log p}{p^s} = \int_{1}^{\infty} \frac{ d \vartheta(x)}{x^s} = s \int_{1}^{\infty} \frac{ \vartheta(x)}{x^{s+1}} dx$$¿? $$\sum_{p} \frac{\log p}{p^s-1} = \int_{1}^{\infty} \frac{ d \vartheta(x)}{x^s} = s \int_{1}^{\infty} \frac{ \vartheta(x)}{x^{s+1}} dx$$ https://en.wikipedia.org/wiki/Von_Mangoldt_function

$$\quad \psi'(x)=\ln(x)\,\Pi_0'(x)$$  $$\quad T'(x)=\ln(x)\,S'(x)$$

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

$$\zeta(s) = \frac{1}{\Gamma(s)} \int_0^\infty (e ^ x - 1)^{-1}x ^ {s-1}\, \mathrm{d}x=\frac{1}{\Gamma(s)} \int_0^\infty x ^ {s-1}\sum_{n>0}e^{-nx}\mathrm{d}x\quad$$ $$\Gamma(s) = \int_0^\infty e^{-x} \,x^{s-1}\, \mathrm{d}x $$ + $$\zeta(s) = \frac{1}{2\pi^{-\frac{s}{2}}\Gamma\left(\frac{s}{2}\right)}\int_0^\infty \bigl(\theta(ix)-1\bigr)x^{\frac{s}{2}-1}\,\mathrm{d}x,\quad$$ $$\theta(\tau)= \sum_{n=-\infty}^\infty e^{\pi i n^2\tau}$$ $$\zeta(s) = \frac{1}{2\pi^{-\frac{s}{2}}\Gamma\left(\frac{s}{2}\right)}(\frac{1}{s-1}-\frac{1}{s} +\frac{1}{2} \int_0^1 \left(\theta(ix)-x^{-\frac12}\right)x^{\frac{s}{2}-1}\,\mathrm{d}x + \frac{1}{2}\int_1^\infty \bigl(\theta(ix)-1\bigr)x^{\frac{s}{2}-1}\,\mathrm{d}x)$$ +

$$\gamma = \lim_{n\to\infty}\left(\ln n - \sum_{p\le n}\frac{\ln p}{p-1}\right)= \lim_{n\to\infty}\left(-\ln n + \sum_{k=1}^n \frac1{k}\right)=\int_1^\infty\left(-\frac1x+\frac1{\lfloor x\rfloor}\right)\,dx.$$

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

$$L(s,\chi)=s\int_1^\infty \frac{A(x)}{x^{s+1}}\,dx\quad A(x)=\sum_{n\le x} \chi(n)$$ $$\zeta(s)=s\int_1^\infty \frac{\lfloor x\rfloor}{x^{s+1}}\,dx$$+ $$\zeta \left({s}\right) = \frac s {s - 1} - s \int_1^\infty \left\{ {x}\right\} x^{-s - 1} dx$$ + +

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


 * $$\beta(s) = \prod_{p \ge 3 \atop p \text{ prime}} \frac{1}{1 -\, \scriptstyle(-1)^{\frac{p-1}{2}} \textstyle p^{-s}}.$$

+
 * $$\omega^p = (x+yi)^p \equiv x^p+y^pi^p \equiv x + (-1)^{\frac{p-1}{2}}yi \pmod{p},$$

+

$$\pi_k(x)\sim\frac{x(\log\log x)^{k-1}}{(k-1)!\log x}\qquad\qquad(1)$$ + $$\pi_2(x) \sim 1.32 \frac {x}{(\log x)^2}$$ +

$$p_n\approx n\log n + n(\log \log n - 1),$$ $$\sum_{n\le x} \sigma_0(n)\approx x\log x + x(2\gamma-1)$$ $$\lim_{n\to\infty}\frac{1}{\log n}\prod_{p\le n}\frac{p}{p-1}=e^\gamma,$$ $$\limsup_{n\rightarrow\infty}\frac{\sigma(n)}{n\,\log \log n}=e^\gamma$$ + + + $$\lim\inf\frac{\varphi(n)}{n}\log\log n = e^{-\gamma}.$$++

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

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

$$\frac{1}{1-x}=\prod_{n\geq 0} (1+x^{2^{n}})$$ + + https://en.m.wikipedia.org/wiki/Prouhet%E2%80%93Thue%E2%80%93Morse_constant $$\frac{1}{1-x}=\prod_{i=0}(1+x^{1\times b^i}+x^{2\times b^i}+x^{3\times b^i}+...)$$

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

$$\prod_{i,j} (1-x_i y_j)^{-1}=\sum_\lambda m_\lambda(x) h_{\lambda}(y)=\sum_\lambda s_\lambda(x) s_{\lambda}(y)$$ +

Rodrigues's formula Li's criterion

Szegő_limit_theorems Jensen's formula five value theorem Identity theorem

ELSV formula

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

varieties: Grassman Segre veronese

global-local homology global-local homotopy +

nowhere differentiable: everywhere continuos, nowhere continuos

Moduli stuff
petterson-weil volume+ witten's volume orbifolds volume$$\chi(\mathcal M_g) = \zeta(1-2g)/(2-2g)$$+ ++

height stuff
$\max( |A+A|, |A \cdot A| ) \geq c \cdot |A|^{1+\varepsilon} $ $\max(|a|,|b|,|c|) \geq C \cdot rad(abc)^{1+\varepsilon} $+ $\max\{\deg(a),\deg(b),\deg(c)\} \le \deg(\operatorname{rad}(abc))-1.$ $\max\{|x|,|y|,|z|\}<C|\xi-\alpha|^{\frac{-1}{3}}\max(1,\xi^2)$ +

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

https://en.wikipedia.org/wiki/Valuation_(algebra) https://mathoverflow.net/questions/310020/summing-bernoulli-numbers

+

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

Class field theory Class number Class number formula List of number fields with class number one Lists of discriminants of class number 1+ Minkowski's bound Stark-Heegner_theorem Heegner number Kronecker-Weber_theorem Kummer theory Fundamental discriminant + + $$\rm\ (x-\alpha)\:(x-\alpha')\ =\ x^2 + x + k\, \text{ Euler-Heegner polynomial } \iff\ \mathbb Z[\alpha]=\mathbb Z[\frac{{1} + \sqrt{1-4k}}{2}]\, \text{ PID}$$ + +

elliptic and quadratics
+ +

+ + + + +

+

+ +

Perron's formulaShimura correspondence

$$Q\in E(\Bbb Q)\Rightarrow_{Nagel-Lutz} Q\in E(\Bbb Z)$$++

$$\zeta(s)=\frac{\int}_0^\infty{{v^{\frac{s}{2}}}\left({\frac{2}}\right)\frac{v}}$$+ $$\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s}$$ $$\zeta(s)=\prod_{p \text{ prime}} \frac{1}{1-p^{-s}}$$ $$ \lim_{s\to 1}(s-1)\zeta(s)=1$$

$$L(s,\chi)=\sum_{n=1}^\infty \frac{\chi(n)}{n^s}$$ $${L_D}(s)\!\!:=\mathop {\prod}\limits_{p\,{\rm{prime}}} {\frac{1}} $$ $$\frac{L_D}(1)=\frac{2}$$ $$Q(x, y)=Ax^2+Bxy+Cy^2,D=B^{2}-4AC$$ $$Q(x, y)=Q(ax+by,cx+dy),ad-bc=1$$

$${L_E}(s) = \frac{\int}_0^\infty {{v^s}{f_E}(iv)\frac{v}}$$ $${L_E}(s)\!\!:=\mathop{\prod}\limits_{p\,{\rm{prime}}}{\frac{1}}$$ $${\rm{li}}{{\rm{m}}_{s \to 1}}{(s-1)^{-{r_E}}}{L_E}(s)<\infty$$ $$\mathop \limits_{s \to 1}\frac{1}\frac = {c_E}\left|{{\text{Ш}}(E)}\right|$$ $${y^2}={x^3}+Ax+B,4A^3+27B^2\neq0$$ $${(Ncz + d)^{ - 2}}{f_E}\left({\frac}\right)={f_E}(z),ad-Nbc=1$$

$$\zeta_K (s) = \sum_{I \subseteq \mathcal{O}_K} \frac{1}{(N_{K/\mathbf{Q}} (I))^{s}}$$ $$\zeta_K (s) = \prod_{P \subseteq \mathcal{O}_K} \frac{1}{1-(N_{K/\mathbf{Q}}(P))^{-s}},\text{ for Re}(s)>1.$$ $$\lim_{s \to 1} (s-1) \zeta_K(s) = \frac{2^{r_1} \cdot(2\pi)^{r_2} \cdot \operatorname{Reg}_K \cdot h_K}{w_K \cdot \sqrt{|D_K|}}$$

$$L(C,s)=\prod_{p\mid\Delta}(1-a_{p}p^{-s})^{-1}\cdot\prod_{p\nmid\Delta}(1-a_{p}p^{-s}+p^{1-2s})^{-1}=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ +

Winding number-Eisenbud Levine Khimshiashvili signature formula

Roth's_theorem-Duffin-Schaeffer conjecture

Kutsenov trace formula-Gutzwiller trace formula

Min-Max theoremMax-min_inequality

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

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

tuple primes
+

Euler product + +

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

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

$$ \Pi^*(x;q,a) = \sum^*_{p\le x, p\equiv a\pmod q} 1 + \sum^*_{p^2\le x, p^2\equiv a\pmod q} \tfrac12 + \sum^*_{p^3\le x, p^3\equiv a\pmod q} \tfrac13 + \cdots = \pi^*(x;q,a) + \tfrac12 \sum_{b\pmod q, b^2\equiv a\pmod q} \pi^*(x^{1/2};q,b) + \tfrac13 \sum_{c\pmod q, c^3\equiv a\pmod q} \pi^*(x^{1/3};q,c) + \cdots $$ +

k-tuple conjecture+ Bateman-Horn conjecture+

$$\sum_{0<\gamma,\gamma'<T, 0<\gamma-\gamma'< \frac{2\pi\alpha}{\ln T}}1=\frac{T\ln T}{2\pi}(1+o(1))\int_0^\alpha 1-\left(\frac{\sin(\pi u)}{\pi u}\right)^2 \mathrm du.$$+ https://en.wikipedia.org/wiki/Montgomery's_pair_correlation_conjecture https://en.wikipedia.org/wiki/Sato-Tate_conjecture + +

square free distribution +

hypergeometric stuff
Painlevé's conditionsMalmquist's conditions++

??????????????? Gauss-Manin connection

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

hipergeometric equation (3 singularities) Heun's equation (4 singularities) Heine-Stieltjes polynomials(n singularities) Schwarz's list


 * 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

Poincaré series Igusa zeta function

Hyperbolic geometryHyperbolic manifold++

{5,5}-tilling-Poincaré Sphere+

Minimal program model +

Abel's theorem converse Abel's theorem

Associahedron Pemutohedron

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)

eta stuff
Eta invariant Dirichlet_eta_function Spectral asymmetry Witten index $$\eta(M)=\frac{1}{\sqrt{\pi}}\int_0^{\infty}t^{1/2}\mathrm{Tr}[D\exp(-tD^2)]dt$$+ $$\frac{1}{4}\left(\eta_{L_{-}}(A)-\eta_{L_{-}}(0)\right)=\frac{c_{2}(G)}{2\pi}I[A]$$+

heat table
+ + + + + + + + + + +

$$\zeta_{\Delta}(s) = \frac{1}{\Gamma(s)}\int_{0}^{\infty}t^{s-1}\sum_{\lambda\in\mathrm{Sp}(\Delta)}e^{-\lambda{t}}dt =\frac{1}{\Gamma(2s)}\int_{0}^{\infty}t^{2s-1}\sum_{\lambda\in\mathrm{Sp}(\Delta)}e^{-\sqrt{\lambda}t}dt.$$ + $$\zeta(k)=\sum_{n=1}^{\infty}\frac{1}{n^{k}}=\sum_{n=1}^{\infty}\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}e^{-\pi n^{2}(x_{1}^{2}+\cdots+x_{k}^{2})}dx_{1}\cdots dx_{k}=\sum_{n=1}^{\infty}(\frac{1}{k!}\int_{-\infty}^{\infty}\cdots)^{-k}$$?++

$$\zeta'(\Delta, 0) = \frac{1}{12}\int_M K dA$$ + +

$$n!=\exp(\ln(n!))=\exp(\sum_n\ln n)=\exp(\sum_n\frac{\ln n}{n^s}|_{s=0})=\exp(-\zeta'(0)))$$ $$p\#=\exp(\ln(p\#))=\exp(\sum_p\ln p)=\exp(\sum_p\frac{\ln p}{p^s}|_{s=0})=\exp(-P'(0)))$$ $$p_{\infty}\#=(2\pi)^2$$+ $$\infty!=\sqrt(2\pi)$$+

$$\ln(n!)\approx n\ln(n)$$ + $$\ln(p_n\#)\approx p_n$$+

$$\zeta(s)=\frac{1}{\Gamma(s)} \int_0^\infty x ^ {s-1}\sum_{n>0}e^{-nx}dx$$ $$P(s)=\frac{1}{\Gamma(s)} \int_0^\infty x ^ {s-1}\sum_{p>0}e^{-px}dx$$

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

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

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

$$Tr_V q^{L_0} = \sum_{n \in \mathbf{Z}} \dim V_n q^n = \prod_{n \geq 1} (1-q^n)^{-1}$$ +

$$\wp(z;\Lambda)= -\zeta'(z;\Lambda)=\ln''(\sigma(z;\Lambda)), \mbox{ for any } z \in \Complex $$ + $$\frac{1}{2 \pi i} \frac{d}{d z} \log \Delta(z)=1-24 \sum_{n=1}^{\infty} \frac{n e^{2 \pi i n z}}{1-e^{2 \pi i n z}}=1-24 \sum_{m=1}^{\infty} \sigma_{1}(m) e^{2 \pi i m z}=1-24\sum_{n>0}\sigma_1(n)q^n=E_{2}(z)$$ + +

https://en.wikipedia.org/wiki/Hecke_operator https://en.wikipedia.org/wiki/Witt_vector

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

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

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

convergence stuff
+

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

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

Littlewood's three principles of real analysis Hahn-Banach theorem Open mapping theorem Uniform boundedness principle Closed graph theorem Bounded inverse theorem zabreikos lemma + https://en.wikipedia.org/wiki/Template:Functional_analysis https://en.wikipedia.org/wiki/Category:Theorems_in_functional_analysis - Riesz representation theorem Herglotz-Riesz representation Riesz–Markov–Kakutani representation theorem Spectral theorem

https://en.wikipedia.org/wiki/Monotone_convergence_theorem https://en.wikipedia.org/wiki/Dominated_convergence_theorem

Katětov–Tong insertion theorem

Riesz-Fischer theoremHopf–Rinow theorem

https://en.wikipedia.org/wiki/Category:Theorems_in_approximation_theory https://en.wikipedia.org/wiki/Stone-Weierstrass_theorem https://en.wikipedia.org/wiki/Fejér's_theorem + + https://en.wikipedia.org/wiki/Runge's_phenomenon https://en.wikipedia.org/wiki/Gibbs_phenomenon

https://en.wikipedia.org/wiki/Morera's_theorem https://en.wikipedia.org/wiki/Localization_theorem

as=>p=>d+ homeomorphism preserving properties

+

Continuously differentiable $$\subseteq$$Lipschitz continuous$$\subseteq$$α-Hölder continuous$$\subseteq$$uniformly continuous$$\subseteq$$Continuous function=continuous $$0 < α ≤ 1\Rightarrow$$ Lipschitz continuous$$\subseteq$$absolutely continuous$$\subseteq$$bounded variation$$\subseteq$$differentiable$$\subseteq$$almost everywhere +

https://en.wikipedia.org/wiki/Modes_of_convergence_(annotated_index) +

LLN,LIL,CLT
LIL random matrix

PNT,RH,EK
+

+

https://mathoverflow.net/questions/6889/what-is-the-difference-between-a-zeta-function-and-an-l-function Selberg class Abstract analytic number theory

+

$$\frac{1}{T}\mu(t\le T\,|\,\arg(\zeta(1/2+i t)/\sqrt{1/2\log\log t}<x)=\lim\limits_{T\to\infty}T^{-1}\int_0^T\mathbf 1_{\arg(\zeta(1/2+i t)/\sqrt{1/2\log\log t}\leqslant x}\,\mathrm dt=\frac1{\sqrt{2\pi}}\int_{-\infty}^x\mathrm e^{-z^2/2}\mathrm dz=\mathbb P(Z\leqslant x) $$+ + One can view Selberg’s theorem as a sort of Fourier-analytic variant of the Erdös-Kac theorem. PNT scaled models+ + + mertens=/=PNT+ +

CLT Martingale CLT Functional integralPath integral Hammersley–Clifford theorem
 * CLT-frequentist CLT-Bayesian CLT-Arithmetic CLT-Ensemble CLT-Flow CLT
 * 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)) $

https://en.wikipedia.org/wiki/Glivenko-Cantelli_theorem https://en.wikipedia.org/wiki/Donsker's_theorem

https://en.wikipedia.org/wiki/Markov_chain_central_limit_theorem https://en.wikipedia.org/wiki/Circular_law

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

numbers, fields, curves, p-adic
+ +

$$ \lim_{s \to 1} (s-1) \zeta_K(s) = \frac{2^{r_1} \cdot(2\pi)^{r_2} \cdot \operatorname{Reg}_K \cdot h_K}{w_K \cdot \sqrt{|D_K|}}$$ + $$ \lim_{s \to 1} (s-1) \zeta_K(s) =\frac{(q-1)^{r_1-1} \cdot \operatorname{Reg}_K \cdot h_K}{\ln(q) \cdot \sqrt{|D_K|}}$$ +

$$\zeta_K(s)=\zeta(s)\prod_{\rho \neq 1} L(s,\rho)=\color{red}{\zeta(s)\prod_{\pi \neq 1} L(s,\pi)}$$ + $$\Phi(G):=\det(X_{gh^{-1}})=\prod_{\chi\in\widehat{G}}\left[\sum_{g\in G}\chi(g)X_g\right]=\prod_{\rho~\rm irred}\det\left(\sum_{g\in G}X_g\rho(g)\right)^{\deg\rho}$$ + $$\det\left(\sum_{n=1}^\infty \Phi(n) n^{-s}\right)= \prod_{j=1}^{\phi(k)}L(s,\tilde{\chi}_j)=\zeta_{\mathbb{Q}(\zeta_k)}(s) = \sum_{I \subset \mathcal{O}_{\mathbb{Q}(\zeta_k)}} N(I)^{-s}$$+

$$\zeta_{[t=N(\mathfrak{p})^{-s}]}L(s)=\prod_{\mathfrak{p} \in K} \frac{1}{\det \left [I-N(\mathfrak{p})^{-s} \rho(\mathbf{Frob}_\mathfrak{p}){|V_{\mathfrak{p},\rho}} \right ]}=\prod_{n}\frac{1}{(1-N(\mathfrak{p})^{-sf})^{\frac{n}{f}}} $$ + $$\psi(x,\rho)=\sum_{\substack{{\frak p}\textrm{ unram} \\ N({\frak p})^m\le x}}{\rm tr}\,\rho(\tau({\frak p}))\log N({\frak p})=\sum_{\substack{{\frak p}\textrm{ unram} \\ N({\frak p})^m\le x}}χ(g)\log N({\frak p})$$ ++ $$L^{\large\rm{Artin}}_{E/K}(\sigma,s)=L^{\large\rm{Hecke}}_{K}(\chi,s)$$ +

$$Z(X,t)=\prod_{i=0}^{2\dim X}\det\big(1-t \mbox{Frob}_q |H^i_c(\overline{X},{\mathbb Q}_\ell)\big)^{(-1)^{i+1}}.$$ $$Z(X, t)=\prod\ (1-t^{\deg(x)})^{-1}.$$ +

$$\zeta_{\mathbf A^n(X)}(s)= \zeta_X(s-n)$$$$\zeta_{\mathbf P^n(X)}(s)= \prod_{i=0}^n \zeta_X(s-i)$$ + $$Z({\mathbb A}^n,t)=\frac{1}{1-{\mathbb L}^n t}$$$$Z({\mathbb P}^n,t)=\prod_{i=0}^n\frac{1}{1-{\mathbb L}^i t}$$ +

$$\zeta_{\mathbb{F}_p[p^{-s}]}(s)=\frac{1}{1-p^{1-s}}=\sum_{n}p^{n(1-s)}=\sum_{g\in A\text{ monic}}\vert g\vert^{-s}=\prod_{f\text{ monic irreducible}} (\sum_{k} |g|^{-sk})=\prod_{f\text{ monic irreducible}}(1-\vert f\vert ^{-s})^{-1}=\prod_{f}\frac{1}{1-|\mathbb{F}_p[x]/(f)|^{-s}}$$++ $$\sum q^{-\deg \ f(x)}=\sum f^{-1}=\infty$$+

$$\zeta_{\mathbb{F}_p[x]}(s)=\frac{1}{1-px}=\sum_{n}(px)^{n}=\sum_{d(g)} x^{d(g)}=\prod_{d(f)} (\sum_{k} x^{kd(f)})=\prod_{d(f)}(1-x^{d(f)})^{-1}=\prod_{d} (1-x^d)^{-\sigma(d)},\quad (\prod_{d(f)} \frac{1}{1-x^{d(f)}} =\prod_{d}\prod_{d(f)=d}\frac{1}{1-x^{d}} =\prod_{d}\frac{1}{(1-x^{d})^{\sum_{d(f)=d}1}} \prod_{d} \frac{1}{(1-x^d)^{\sigma(d)}})$$ + ++ +

$$\zeta_R(s)=\prod_{m\subset R}\frac{1}{1-|R/m|^{-s}}$$ [+]

$$\zeta_G(s)=\sum_{H\le_{\Large f} G}[G:H]^{-s}=\sum_{n\ge1}\sigma(G)\,n^{-s}$$+

$$\sum_f \sigma(f)|f|^{-s}=(1-q^{1-s})^{-1}(1-q^{2-s})^{-1}$$+ $$\sum_{n=1}^\infty \frac{\sigma_{a}(n)}{n^s} = \zeta(s) \zeta(s-a)$$+ $$Z(t) = \frac{P(t)}{(1 - t)(1 - qt)}\ $$+ $$Z(X,t)=\frac{P(t)}{(1-t)(1-{\mathbb L}t)}$$+

$$\prod_{n\geq 0}\frac{1}{1-x^{2n+1}}=\prod_{n\geq 0} (1+x^{n})$$++ $$\sum_{n=0}^\infty[({\mathbb A}^2)^{[n]}]t^n=\prod_{m=1}^\infty \frac{1}{1-{\mathbb L}^{m+1}t^m}$$ +

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

function field analogy+ + + +

https://en.wikipedia.org/wiki/Siegel's_theorem_on_integral_points https://en.wikipedia.org/wiki/Mordell-Weil_theorem

similtonics
(Farkas's lemma-Fredholm alternative) (sdf matrix-sdf kernel) (Perron–Frobenius theoremKrein–Rutman theorem)

https://mathoverflow.net/questions/322590/analogy-between-metric-space-completion-and-algebraic-closure

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

Uniformization
conformal uniformization isothermal [https://en.wikipedia.org/wiki/Ricci_flow#Relation_to_diffusion isothermal/conformal? Ricci flow] elliptic uniformization +

Manifolds
https://en.wikipedia.org/wiki/Manifold_decomposition https://en.wikipedia.org/wiki/Category:3-manifolds https://en.wikipedia.org/wiki/Category:Structures_on_manifolds injectivity+1=surjectivity

Poincaré-Thurston
n>0:TO P=PL=DIFF n>3:TOP=\=PL=DIFF n>6:TOP=\=PL=\=DIFF

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

Algebraic category: Donaldson–Thomas +

characterization: Gamma function+ Determinant exponential function

Analogies: Quine McCluskey algorithmBuchberger's algorithm

Cobordism
Thom Transversality theorem Thom space h-cobordism

Obstructions
Obstructions characteristic classes: + Pontryagin_class (orthogonal group) ++ +
 * homotopy=simple homotopy: Whitehead torsion+(homotopy groups equality=\=> homotopy equivalents)
 * 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)
 * Surgery obstruction+
 * thom spectrum lack
 * Todd classreciprocal characteristic class
 * Segre class-inverse chern class
 * Hasse invariant
 * Manin obstruction
 * Kronheimer-Mrowka basic class

Knots
https://en.wikipedia.org/wiki/Volume_conjecture https://en.wikipedia.org/wiki/Khovanov_homology#The_relation_to_link_(knot)_polynomials

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

Bott periodicity&Hurwitz's_theorem Bott_periodicity_theorem Hurwitz theorem Frobenius 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 + Lippmann-Schwinger_equation

Covariance matrix Covariance process +

Var(x)=2Dt + + +

Cayley-Hamilton theorem Fredholm determinant +

isoperimetric&weyl stuff
Eigenvalues and eigenvectors of the second derivative

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

https://www.encyclopediaofmath.org/index.php/Blaschke-Santaló_inequality $$T \cdot (KA) \geq L^2$$+

Poincaré constant Cheeger constant Dirichlet eigenvalue$$\lambda_{n+1}^{Neumann} <\lambda_n^{Dirichlet}$$+

$$N(\lambda) \leq \frac{\lambda a b}{4 \pi}$$+

$$E(\lambda_{max}(A)) \le \frac{1}{2}E(\lambda_{max}(A-\text{diag}(u))) + \frac{1}{2}E(\lambda_{max}(A-\text{diag}(-u)))$$++

Coarea formula Smooth coarea formula coarea+ coarea+

Gaussian isoperimetric inequality

variational quotients
https://en.wikipedia.org/wiki/Variational_inequality https://en.wikipedia.org/wiki/Complementarity_theory

Rayleigh quotient Variational method Fisher's linear discriminant Bauer Fike theorem

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

Geometric_phase Fubini-Study_metric Fisher_information_metric

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 - Inclusion-exclusion principle Isomorphism theorems

Manifold decomposition +
 * atiyah-singer proof techniques: pseudodifferential operator, cobordism, k theory, heat operator
 * 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


 * 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 $$\mathrm{\hat H}\Psi=E\Psi$$]

Twelvefold way +

Plane partition number partition Multiplicative_partition (unordered factorization)



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

PHYSICSPHYSICSPHYSICSPHYSICSPHYSICS
$$F = E + B \wedge dt$$ + Maxwell's equations-Alternative formulations Mathematical descriptions of the electromagnetic field P-form electrodynamics Covariant formulation of classical electromagnetism

statistical physics
$$-\frac{F}{k T} = \ln \operatorname{Tr} \exp\big(-\tfrac{1}{kT} \hat H\big)$$ $$ f(E;\beta)=\beta\frac{(\beta E)^{m-1}}{\Gamma(m)}e^{-\beta E}\quad$$ $$ P(N) = \frac{\langle N \rangle^N \exp(-\langle N\rangle)}{N!}=\frac{(\lambda t)^N \exp(-\lambda t)}{N!}.$$

+ + + + +

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

physics stack
$$\partial_t \psi = -\Gamma \dfrac{\delta \mathcal{F}}{\delta \psi}\quad$$ $$\frac{\partial{\psi}}{\partial{t}}=\nabla^2 \dfrac{\delta \mathcal{F}}{\delta \psi}.$$ +

$$U(T,0) = {\cal T}( e^{-i \int_0^T H(t') dt'})=e^{-i T H_{\mathrm eff}}$$+

Conservation stuff
+ ++

https://en.wikipedia.org/wiki/Curvature_form#Bianchi_identities

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

NC
https://en.wikipedia.org/wiki/Noncommutative_quantum_field_theory https://en.wikipedia.org/wiki/IR/UV_mixing https://en.wikipedia.org/wiki/Kramers-Wannier_duality

Ising stuff
Kramers-Wannier duality IR/UV mixing +

homotophysics
$$\pi_3\left(\frac{SU(N)_L\times SU(N)_R}{SU(N)_\text{diag}}\cong SU(N)\right)=\mathbf{Z}$$ + $$\pi_2\left(\frac{SU(4)\times SU(2)}{[SU(3)\times U(1)]/\mathbf{Z}_3}\right)=\mathbf{Z}$$ + $$\pi_4(SU(2)) = \mathbf{Z}/2\mathbf{Z}$$ +

Black hole metrics
+ + + + + + +

Precession
Precession +

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

Madelung equations Circulation

Aharonov-Bohm effect Landau quantization

Wilson loop t'hooft loop

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

$$S_q=S_q^{FD}+S_q^{BE}$$ + $$S_\text{B}= S_s+k \log \omega.$$ + $$\Delta H \equiv \Delta H_\text{L} + \Delta H_\text{T}$$ + $$\mathcal{H}=\mathcal{H}_{\rm Coulomb}+\mathcal{H}_{\mathrm{kinetic}}+\mathcal{H}_{\mathrm{SO}}+H_{\mathrm{Darwin}}\!$$ + + $$\hat{H} = \hat{H}_{\text{field}} +\hat{H}_{\text{atom}} +\hat{H}_{\text{int}}$$ + $$H=D_1+D_2+D_3$$ + $$ E_\mathrm{total} = E_\mathrm{electronic} + E_\mathrm{vibrational} + E_\mathrm{rotational} + E_\mathrm{nuclear\,spin}$$ + $$\epsilon=\epsilon_{elec}+\epsilon_{vib}+\epsilon_{rot}+\epsilon_{trans}$$ + $$\sigma = \sigma_\text{a} + \sigma_\text{s} + \sigma_\text{l}.$$ + $$\frac{1}{\tau} = \frac{1}{\tau_{\rm impurities}} + \frac{1}{\tau_{\rm lattice}} + \frac{1}{\tau_{\rm defects}} + \cdots$$. + $$\frac{1}{\lambda_\mathrm{MFP}} = \frac{1}{\lambda_\mathrm{el-el}} + \frac{1}{\lambda_\mathrm{ap}} + \frac{1}{\lambda_\mathrm{op,ems}} + \frac{1}{\lambda_\mathrm{op,abs}} + \frac{1}{\lambda_\mathrm{impurity}} + \frac{1}{\lambda_\mathrm{defect}} + \frac{1}{\lambda_\mathrm{boundary}}$$ + $$ωtotal = ωGeneral Relativity + ωquadrupole + ωtide + ωperturbations$$ +

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

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

velocity limit: light velocity entropy limit: bekenstein bound

Unruh radiation Hawking radiation +

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

Spacetime symmetries

Sagnac effect

YB equation BMW algebra +

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

Hamilton Jacobi_equation Circulation Biot-Savart

Kutta Joukowski theorem Magnus effect Vorticity equation

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

P: Conserved quantities
Laplace–Runge–Lenz vector hidrodinamical helicity Enstrophy

P: Electronic transport mechanisms
+

+Hot-carrier injection Ballistic conduction

Field electron emission

P: Optics
https://en.wikipedia.org/wiki/Brewster's_angle

P: diffraction, refraction, reflection, interference stuff
https://en.wikipedia.org/wiki/Kirchhoff's_diffraction_formula https://en.wikipedia.org/wiki/Fresnel_diffraction + https://en.wikipedia.org/wiki/Fraunhofer_diffraction

https://en.wikipedia.org/wiki/N-slit_interferometric_equation

$$\mathbf{\hat{d}}_\mathrm{s} = \mathbf{R} \; \mathbf{\hat{d}}_\mathrm{i}=(\mathbf{I} - 2 \mathbf{\hat{d}}_\mathrm{n} \mathbf{\hat{d}}_\mathrm{n}^\mathrm{T})\; \mathbf{\hat{d}}_\mathrm{i}$$ +

https://en.wikipedia.org/wiki/Dynamical_theory_of_diffraction https://en.wikipedia.org/wiki/Diffraction_formalism

P: scattering & cross section
https://en.wikipedia.org/wiki/Scattering https://en.wikipedia.org/wiki/Scattering_theory https://en.wikipedia.org/wiki/Category:Scattering +

Elastic: https://en.wikipedia.org/wiki/Rayleigh_scattering https://en.wikipedia.org/wiki/Mie_scattering https://en.wikipedia.org/wiki/Compton_scattering (https://en.wikipedia.org/wiki/Thomson_scattering) https://en.wikipedia.org/wiki/Rutherford_scattering Inelastic: https://en.wikipedia.org/wiki/Raman_scattering https://en.wikipedia.org/wiki/Deep_inelastic_scattering +: https://en.wikipedia.org/wiki/Two-photon_physics https://en.wikipedia.org/wiki/Delbrück_scattering https://en.wikipedia.org/wiki/Quantum_mechanical_scattering_of_photon_and_nucleus

https://en.wikipedia.org/wiki/Cross_section_(physics) https://en.wikipedia.org/wiki/Scattering_cross-section https://en.wikipedia.org/wiki/Nuclear_cross_section https://en.wikipedia.org/wiki/Neutron_cross_section http://hep.physics.wayne.edu/~harr/courses/5210/w15/lecture29.htm

https://en.wikipedia.org/wiki/Moiré_pattern https://en.wikipedia.org/wiki/Scattering_amplitude https://en.wikipedia.org/wiki/Fluctuation-dissipation_theorem https://en.wikipedia.org/wiki/Mean_free_path https://en.wikipedia.org/wiki/Scattering_length

Electron scattering: + https://en.wikipedia.org/wiki/Bhabha_scattering https://en.wikipedia.org/wiki/Møller_scattering https://en.wikipedia.org/wiki/Compton_scattering https://en.wikipedia.org/wiki/Mott_scattering https://en.wikipedia.org/wiki/Bremsstrahlung https://en.wikipedia.org/wiki/Deep_inelastic_scattering https://en.wikipedia.org/wiki/Synchrotron_emission

Light scattering: + https://en.wikipedia.org/wiki/Rayleigh_scattering https://en.wikipedia.org/wiki/Geometric_optics https://en.wikipedia.org/wiki/Mie_scattering https://en.wikipedia.org/wiki/Umklapp_scattering

https://en.wikipedia.org/wiki/Scattering_parameters https://en.wikipedia.org/wiki/Forward_scattering_alignment https://en.wikipedia.org/wiki/Back_scattering_alignment

https://en.wikipedia.org/wiki/Lippmann–Schwinger_equation

$$\frac{I_0}{I}=\frac{\, d\Omega\, dA\cos(\theta)}{d\Omega_0\,\, dA_0\cos(\theta)}=\frac{\, d\Omega\, dA}{d\Omega_0\, dA_0}, \,$$ $$F_{tot} = \int\limits_0^{2\pi}\,\int\limits_0^{\pi/2}\cos(\theta)I_{max}\,\sin(\theta)\,\operatorname{d}\theta\,\operatorname{d}\phi, \,$$ $$\Phi_\mathrm{e} = \oint_\Sigma \mathbf{S} \cdot \mathbf{\hat{n}}\, \mathrm{d}A = \oint_\Sigma |\mathbf{S}| \cos \alpha\, \mathrm{d}A, \,$$ $$L_{\mathrm{e},\Omega} = \frac{\partial^2 \Phi_\mathrm{e}}{\partial \Omega\, \partial A \cos \theta},$$ $$\mathbf{F}(\mathbf{x}, t;\nu) = \oint_\Omega\ I(\mathbf{x}, t;\mathbf{\hat{n}},\nu) \,\mathbf{\hat{n}} \,d\omega(\mathbf{\hat{n}})$$ + + + +

P: Crystallography stuff
Crystallographic periodic table crystal structure Crystall system some 3D point groups character tables Peierls stress

Bloch's theorem $H(r+R)=T_RH(R)T_R^{-1}$ Floquet theory

Crystal_momentum Umklapp_scattering

$$\mathbf{M}_{\rm orb} = \frac{1}{2V}\int_V d^3\mathbf{r} \, \mathbf{r}\times\mathbf{J}(\mathbf{r}),\quad $$ $$\mathbf{M}_{\rm orb} = \frac{-e}{2m_e}\sum_n\int_{\rm BZ}\frac{d^3k}{(2\pi)^3}\, \langle\psi_{n\mathbf{k}}\vert\mathbf{r}\times\mathbf{p}\vert \psi_{n\mathbf{k}}\rangle \,,$$ +

+ +

$$ \mathbf{\Delta k} \cdot \mathbf x= \mathbf{\Delta k}\cdot (p\,\mathbf a+q\,\mathbf b+r\,\mathbf c)= p\,2\pi h + q\,2\pi k + r\,2\pi l= 2\pi(hp+kq+lr)=2\pi n,$$ + $$2d\sin\theta=n\lambda$$ +

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

Quark lepton_complementarityEigthfold way

Phonon-polaron-magnon spinon-orbiton-chargon https://en.wikipedia.org/wiki/Bose-Einstein_condensation_of_quasiparticles

Molecular partition function + + +

volumetric entropy surface entropy gas entropy + + +

+ thermodynamics +

Deconfinement

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


 * electromagnetism
 * relativistic charge distribution
 * relativistic charge point
 * +

wave: string, springs, bars


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

--- ++

Group contractionQuantization commutes with reduction

D'Alembert->KdV Hierarchy/Dirac operator->ANKNS hierarchyBBGKY hierarchy

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

Langevin motion + +

Electromagnetism Gravitoelectromagnetism + Kaluza-klein theory Yang-Mills theory

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


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

Black hole: Black hole radiation dark matter - Cosmic Backgrounds: Relics: Photon(CMB) Neutrino Gravitation DEBRA: Infrared X-ray Extragalactic light Radio Gamma-ray CMB

Non-linear
Primakoff_effectSchwinger limitEuler-Heisenberg Lagrangian

angle mechanism
Angles: Weinberg Peccei–Quinn GIM CKM PMKS + vacuum angle

Planck temperature Hagedorn temperature +

Noether's Theorem extensions Wald entropy formula++

stringification=categorization

Types of radioactive decay



coding
atributes(class/intance), method (function atribute), paramenter, argument "{}, you'll always be the {}th planet to me.".format(planet, position)="Pluto, you'll always be the 9th planet to me." + type,dir,help + https://pyformat.info/ https://docs.python.org/3/

ctf
ctf0:ctrl+I->Network->crtl+f5->background.png