26 problems

Simple generalized permutahedron conjecture. There is a natural map

For each loop order $L$, consider the period Feynman integrals of $\phi^4$ theory and sample them with probability inversely proportional to the symmetry factor of their associated…

Flat-space partial-fraction decomposition conjecture. The flat-space wavefunction can be decomposed into partial fractions as

Generalized permutahedron conjecture. The Newton polytope $\mathbf{N}[\mathcal{F}]$ is a generalized permutahedron in the Euclidean regime for all kinematics, including exceptional…

Let $I(z_1,\ldots,z_n)$ be an $L$-loop integral with nonnegative indices, and let $D_1,\ldots,D_n$ be the propagator generators in the associated noncommutative rational double-shi…

Let $G$ be a graphical function for which the radial representation and Gegenbauer expansion are defined. Universal Gegenbauer expansion conjecture. The Gegenbauer expansion (gfr)…

Let $G$ be a constructible graph in an even dimension $D\geq4$, and let $P_G$ be its associated period. The preceding reduction argument concerns the weight of graphical functions…

Let $G$ be a weighted graph whose graphical function $f_G(z)$ exists in dimension $D=2\lambda+2\geq3$. Let $\mathcal{S}\mathcal{V}_{\{0,1,\infty\}}$ denote the space of functions o…

Let $G$ be a primitive log-divergent Feynman graph, and let $[H_G,\omega_G,\sigma]^{\rm m}$ denote the motivic realisation of its Feynman integral $I_G^{\rm m}$. Let $\Delta$ denot…

Brown's coaction conjecture. The space $\mathcal{P}_{g}^{\rm m}$ is stable under the Galois coaction:

Let $G$ be a graph that is p-logarithmic in $d=4$ dimensions and whose vertices all have degree at most $4$; call such a graph a $\phi^4$ graph. Let $\mathcal{P}(G)$ denote its per…

$K_{6}$ forbidden-minor conjecture. The complete graph $K_{6}$ is a forbidden minor for reducibility with respect to the first Symanzik polynomial.

A graph is reducible with respect to the first Symanzik polynomial when it has the corresponding reducibility property for that polynomial. The $K_6$ irreducibility conjecture. The…

Schnetz's cosmic Galois closure conjecture. The space $V$ is itself closed under the action of the cosmic Galois group $\mathcal{C}_{0,0}$.

Loop-number weight conjecture. The weight of $I_G^{\mathfrak{m}}$ is less than or equal to $4h_G$.

Motic-descendant degree conjecture. The amplitude of $G$ is a regularised period of motic descendants of $G$ of degree at most $dh$.

Weight bound conjecture. The amplitude integrand satisfies

Graph-minor generation conjecture. The motivic periods of $G$ of weight at most $k$ lie in the algebra generated by regularised motivic periods of graph minors of $G$ having at mos…

A master integral is one of the finitely many basis integrals to which the integrals of a topology reduce through integration-by-parts identities. A master integral is uniformly tr…

Consider an $L$-loop amplitude in dimensional regularisation with $D=4-2\epsilon$, and write its Laurent expansion in $\epsilon$. Maximum-transcendentality conjecture. The Laurent c…

Let $G$ be a graph with loop number $h_G$ and number of edges $n_G$, and suppose that $G$ is log-divergent, meaning … with $h_G,n_G\geq 3$. A graph is duality admissible when it sa…

Let $G_1$ and $G_2$ be primitive log-divergent graphs with graph periods $I_{G_1}$ and $I_{G_2}$. For each finite field size $q$, define the $c_2$-invariant by … where $X_G$ is the…

Let $G_1$ and $G_2$ be primitive-divergent graphs in $\phi^4$ theory, let $I_{G_i}$ denote their residues, and let $c_2(G_i)$ be their graph invariants. Period-determines-$c_2$ conj…

Let $G$ be a primitive-divergent graph in $\phi^4$ theory, and let $c_2(G)$ denote its graph invariant. The completion relation identifies graphs whose completions are isomorphic; w…

Unbounded Tate-defect conjecture. There exists a sequence of primitive-divergent graphs $G_i$ in $\phi^4$ such that