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A weak homotopy equivalence between connected CW complexes is a homotopy equivalence. The analogous statement holds for Kan complexes.

If a path-connected space is $(n-1)$-connected for $n\ge 2$, then its first potentially nonzero homotopy group maps isomorphically to homology:

The rational homotopy group is obtained by tensoring a homotopy group with the rational numbers. It retains the free part and discards torsion.

For a based space $X$, adjunction gives

If $A\subseteq X$ is contractible and $(X,A)$ has the homotopy extension property, the quotient mapis a homotopy equivalence. Extend a contraction of $A$ to $X$; its endpoint is constant on $A$ and therefore factors through $q$ to supply a homotopy inverse.

The Sidorenko conjecture states that every bipartite graph $H$ satisfiesfor every bipartite host graph $G$. Thus a random map is at least as likely to preserve every edge as the heuristic that treats the edge constraints independently predicts.

If $g\in S{U}_2$ has eigenvalues $z,z^{-1}$, then

At nonzero charge density, a translationally invariant holographic state has infinite D.C. conductivity because an electric field continuously injects conserved momentum. Momentum relaxation broadens the zero-frequency delta function and makes the D.C. value finite.

For a bulk gauge field, the leading boundary value of its time component is the dual chemical potential, while the normalizable radial coefficient determines the charge density.

Fefferman--Graham coordinates put an asymptotically anti-de Sitter metric in the form $d{s}^2=L^2[d{Z}^2/Z^2+g_{ij}(Z,x)d{x}^{i}d{x}^j/Z^2]$.

The Baumslag-Solitar group is

Suppose an urn starts with $m$ green balls and $m+2$ red balls. Drawing green removes one green ball, while drawing red adds one green and one red ball. The difference remains two, and $h(g)=1/(g+1)$ is harmonic for the green-ball chain. Stopping $h$ on hitting $0$ or $N$ and then letting $N$ tend to infinity gives termination probability

A bounded operator $T$ on a separable Hilbert space is trace class when $|T{|}^{1/2}$ is Hilbert--Schmidt. Its trace norm is

The Higgs boson is the neutral radial excitation of the Higgs field about its vacuum expectation value.

In the Higgs mechanism, gauge fields associated with broken generators acquire longitudinal polarizations and masses by absorbing Goldstone modes. Gauge fields of the unbroken subgroup remain massless.

If a well-order is the union of suborders of types $\xi$ and $\eta$, its type is at most $\xi \#\eta$.

For a basis $(v_i)$, the matrix $A$ of a Hermitian form has entries $A_{ij}=H(v_i,v_j)$. With the convention that $H$ is linear in its first argument, coordinate columns $x,y$ satisfy $H(x,y)=x^{T}A\overline{y}$.

A function $d\in {\omega }^{\omega }$ dominates $h\in {\omega }^{\omega }$ when $d(n)>h(n)$ for all but finitely many $n$. A dominating real over a ground model dominates every such function belonging to that model.

With transverse and axial moments $I,J$, the Hamiltonian is

For $e_k=x^{\ast }-x_k$,If $A$ is diagonal with entries ${\lambda }_i$, a coordinate permutation turns this matrix into blocks