15 problems

Let $r$-graphs be finite uniform hypergraphs, let $\lambda(m,r)$ denote the maximum Lagrangian of an $r$-graph with $m$ edges, and let $\binom{x}{r}$ be the generalized binomial co…

Higher-dimensional generalization conjecture. Generalizing these concepts to higher dimensions may offer nontrivial implications.

Let $N$ be a closed manifold, let $DT^*N$ denote the unit cotangent disc bundle, and let ${\mathfrak L}(T^*N)$ be the class of Lagrangians used in the source. Viterbo's boundedness…

Let $(M,\omega)$ be a symplectic manifold and let $L$ be an exact Lagrangian. For any Hamiltonianly isotopic exact Lagrangian $L_1$ such that $\gamma(L,L_1)$ is sufficiently small,…

A correspondence is established between hyperaffine $(p_q,q)$ hyperforms and Lagrangian densities with projectable Euler–Lagrange equations. In local coordinates, the associated La…

Let a variational problem have independent variables and dependent variables, and let a Lagrangian have Euler–Lagrange equations of order strictly less than twice the Lagrangian's…

Let a Lagrangian depend on derivatives up to order $k$. Call its Euler–Lagrange equations reduced-order when their order is strictly less than $2k$. The paper constructs a family o…

Let $G$ be a $t$-intersecting $r$-graph, meaning that any two edges of $G$ have intersection of size at least $t$. For integers $t\geq 1$, $i\geq 0$, and $r\geq t+i$, let…

Let $r\geq 4$ and let $H_{2r}^{M_2^r}$ be the extension of the $r$-uniform matching $M_2^r$ consisting of two pairwise disjoint edges. Let $G$ be an $r$-graph on $[n]$, and let…

Hefetz–Keevash conjecture. The maximum Lagrangian of an $r$-uniform intersecting family is achieved by a feasible weight vector on this star. The conjecture concerns the extremal L…

Pikhurko–Zhao conjecture. The Lagrangian of $G$ satisfies

Let $t$, $m$, and $r\ge 3$ be positive integers satisfying … Let $G$ be an $r$-graph with $m$ edges that contains a clique of order $t-1$. Peng–Zhao's Motzkin–Straus-type conjectur…

Star optimality conjecture. Stars are optimal for $r\ge 4$, and their blow-ups are extremal.

Fundamental-entity conjecture. The objects $S$, $\gamma_i$, $\gamma_{ij}$, and $\Gamma_{ij}$ are fundamental entities to be used as algebraic building blocks of the Lagrangian…

Conjecture. A homogeneous Lagrangian $L$ is null if and only if