9 problems

Let $G$ be a connected graph, let $X_G$ be its graph hypersurface, and write $[X_G]_q=|X_G(\mathbb F_q)|$ for its point-counting function over finite fields. Inspired by the appear…

Kontsevich's conjecture. Every graph hypersurface $\hat X_\Gamma$ should be a mixed Tate motive.

Aluffi's conjecture. The Euler characteristic of $Y_G$ satisfies

Aluffi–Marcolli–Qaisar conjecture. The sequence $a_0,a_1,\ldots,a_n$ is log-concave, meaning that

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…

Mixed Tate factorisation conjecture. The periods of Feynman integrals of primitive graphs in $\phi^4$ theory factorise through a category of mixed Tate motives.

The $c_2$ residue conjecture. If $I_{G_1}=I_{G_2}$ for two primitive log-divergent graphs $G_1$ and $G_2$, then

Let $X_\Gamma=\{\Psi_\Gamma=0\}$ be the graph hypersurface associated with a graph $\Gamma$, embedded in its ambient projective space, and let $H$ denote the hyperplane class. Writ…