38 problems

Let $k$ be a perfect field of characteristic $p\geq 7$, and let $(X,\emptyset)$ be a log canonical pair over $k$ of dimension at most $p-2$. Logarithmic Extension Theorem conjectur…

Let $A_n$ be the free associative algebra and let $\varphi(d,D):A_n\to\Omega(d,D)^{even}_*$ be the algebra map sending each generator $x_i$ to $x_i$. For $m_1,m_2,m_3,m_4\in A_n$,…

Let $F/k$ be the field extension in the source, let $\mathfrak m$ be the object and $\mathcal C$ the category appearing in the source, let $\mathcal A$ be the category of admissibl…

Let $F/k$ be the field extension in the source, let $G$ be its field automorphism group, and let $\mathcal I_G$ be the category of homotopy-invariant smooth $G$-representations. Le…

Let $M$ be a closed manifold with zero Euler characteristic, and let $\alpha$ be a non-closed differential $1$-form on $M$. Non-vanishing perturbation conjecture. There exists a smo…

Let $A$ be a projective manifold, and let ${\mathcal L} \subset \bigwedge^r \Omega^j_A$ be a line bundle. A curve $C$ on $A$ is movable if its deformations fill up $A$. The positiv…

Let $\mathrm{Vect}(\mathbb R^n)$ be the Lie algebra of vector fields on $\mathbb R^n$, let $\Omega(\mathbb R^n)$ denote the space of differential forms, and let…

Rational connectedness criterion. $X$ is rationally connected if and only if

Rational integrability conjecture. The function $f$ is rationally integrable with respect to $x_1,\ldots,x_m$ if and only if $n\neq m$. This extends the discussion of rational inte…

A zigzag multiplicity is the multiplicity of a zigzag summand in the bicomplex of differential forms of a compact complex manifold, and a Chern number is a characteristic number ob…

Let $X$ be a smooth projective scheme defined over a field of characteristic zero. After spreading out $X$, for a prime $p$, write $X_{p=0}$ for its reduction modulo $p$, and let…

Uniqueness conjecture. The family $(\alpha_q)$ has a unique limit as $q\to1$. This asks whether the limiting cohomology class arising from the $q$-harmonic conjugates is distinguis…

Absolute maximum principle for tight forms. The restriction of $|F|$ to $\overline{U}$ attains its maximum on $\partial U$. This would extend the absolute-minimization theorem from…

Let $T$ be a simplex, let $\widetilde{T}$ denote its blow-up, and let $b\mathcal P_1^-\Lambda^k$ be the shadow-form or blow-up Whitney-form complex and $b\mathcal P_r\Lambda^0$ the…

Let $M$ be a PL manifold with a given triangulation, and let the blow-up Whitney forms on this triangulation be equipped with the continuity conditions described above. Blow-up Whi…

Let $d$ and $q$ be integers with $1\leq q\leq d-1$, and consider the exterior derivative acting on $(d-q)$-forms. The operator is $k$-balanceable when it satisfies the balanceabili…

Let $(M^n,g)$ be a complete noncompact Riemannian manifold with asymptotically nonnegative Ricci curvature. For $0\leq k\leq n$, write $\sigma_{\rm ess}(k,\Delta,M)$ for the essent…

Let $\Omega^{\bullet}_{\mathrm{can}}$ be the graded space of canonical differential forms, let $\mathbb{L}(\Omega^{\bullet}_{\mathrm{can}})$ be the free Lie algebra on it, and let…

Geometric interpretation conjecture. Under this identification, the operation $\Delta$ on $\mathsf{HGK}_0$ corresponds to pullback under $\pi$, and this pullback induces a well-def…

Let $Y$ be a space, let $H^*(Y;\mathbb{R})$ denote its real cohomology algebra, and let $\bigwedge^*\mathbb{R}^N$ be the exterior algebra on a finite-dimensional real vector space.…

Let $M$ be a compact manifold and let several closed $1$-forms on $M$ be given. Their common level is the intersection of the corresponding level sets at a generic choice of levels…

Polynomial numerator conjecture. There exists a polynomial $H(S,A,B)$ in the entries of $S,A,B$ such that

Let $\mathcal{T}$ be a degree-one $(n,k,m)$-triangulation. The universal amplituhedron form $\omega^{(\mathcal{T})}_{\mathcal{A}_{n,k,\geq m}}$ is the signed sum of the pushforward…

Amplituhedron form conjecture. The form $\omega^{(\mathcal{T})}_{\mathcal{A}_{n,k,m}(Z)}$ does not depend on the choice of triangulation $\mathcal{T}$. The form is intended to be a…

Frames conjecture. The sets $B^J_{k,p}$ and $\widetilde B_{k,p}$ are frames for $H^1_k$, for every $k\in\{1,\ldots,m\}$ and $p\geq 0$.