51 problems

Strong positivity conjecture. One has $c_{v,w}>0$ for every $v\le w$. The weaker non-negativity assertion is known, but strict positivity remains the stronger conjectural statement…

Equivariant positivity conjecture. The coefficient $c_{v;w}(\alpha)$ is a polynomial in the positive roots $\alpha$ with non-negative coefficients. Numerical evidence supports this…

Sign conjecture. The coefficient of $[w]$ in the expansion of $\mathcal{C}[p]$ is a Laurent polynomial in $\tau_1,\ldots,\tau_n,y$ whose terms have sign $(-1)^{\ell(p)-\ell(w)}$.

Epsilon-class conjecture. If $\nabla$ is an integrable, admissible, absolute connection, then

Mumford's conjecture. This homomorphism should be an isomorphism in the stable range of Harer's stability theorem; in the formulation stated earlier, it is an isomorphism in degree…

Let $M$ be a smooth projective variety, let $\pi$ be a group, and let $f:M\to B\pi$ be a continuous map. For $\alpha\in H^*(B\pi;\mathbb{Q})$, let $\operatorname{Td}(M)$ denote the…

Let $M$ be a Weil–Petersson variety, and let $X$ be a Weil–Petersson subvariety of dimension $q$ in $M$. Let $c_k(\omega_{WP})$ denote the $k$-th elementary polynomial of the curva…

Let $\phi\colon\widehat V\to V$ be a blowing down between nonsingular complex projective varieties, with $\widehat V$ obtained by blowing up $V$ along a nonsingular irreducible cen…

Let $V$ be a nonsingular complex projective variety of complex dimension $n$, viewed as a complex manifold, with fundamental class $[V]\in H_{2n}(V,\mathbb Z)$ and total Todd class…

Let $C_{2r+1}$ be an odd cycle, let $K_n$ be the complete graph on $n$ vertices, and let $\varpi_1^{n-2}(\text{\tt Hom}(C_{2r+1},K_n))$ denote the indicated Stiefel-Whitney charact…

Let $B$ be the base of a surface bundle, let $p$ be a prime, and let $\kappa_i$ denote its Morita–Mumford classes. Vanishing conjecture. For $s\geq 1$ and $v\geq 0$, one has … The…

Characteristic-class conjecture. The characteristic classes of codimension-$p$ foliations on $X$ coincide with the elements of

Let $X$ be a complex manifold of dimension $d$, let ${\cal M}^\bullet$ be an object of $\operatorname{D}^b_{good}({\cal D}^{op}_X)$, and let $\Lambda$ be a conic subvariety of…

Beilinson–Cheeger–Chern class conjecture. For all $p \geq 1$ and $n \in {\mathbb N} \cup\{\infty\}$,

Pontryagin-class non-vanishing conjecture. For the orientable flat manifolds $\widehat{\mathsf{LS}}_n$, the mod $2$ reduction of $p_i(\widehat{\mathsf{LS}}_n)$ is non-zero whenever…

Cappell–Shaneson conjecture.

Gamma-class overconvergence conjecture. For the quantum connection on any monotone symplectic manifold, the Frobenius structure with constant term above, where $b=\Gamma_p(TM)$, is…

Mod-8 signature conjecture. The evaluation

Let $l$ be an integer, and let $\overline{A_2}(l+1)$ and $\overline{\Sigma}^2(l)$ denote the corresponding singularity loci. Their Segre–Stiefel–Whitney classes are defined in the…

Let $M$ be a manifold with a tangential structure, and let $F$ be an embedded submanifold that is Poincaré dual to a characteristic class of $M$. A characteristic structure is the…

Let $n$ be a positive integer, let $\mathbb{P}^n$ be complex projective space, and let $L^{2j}(\mathbb{P}^n)\in H^{4j}(\mathbb{P}^n,\mathbb{Q})$ denote its degree-$4j$ cohomology…

Let $X$ be a simplicial manifold. A fixing cycle is a $(3n-2)$-cycle $\phi\in Z_{3n-2}(Y,\pi^*({\mathcal D}))$ satisfying … where $n$ is the dimension of $X$, ${\mathcal D}$ is its…

Let $(E,h)\to X$ be a Griffiths semipositive Hermitian holomorphic vector bundle. Let $P$ be a non-negative combination of Schur polynomials, and let $P\big(c_{\bullet}(E,h)\big)$…

Let $M$ be a smooth manifold and let $E\subset TM$ be an integrable subbundle defining a foliation $\mathcal{F}$ whose leaves are rationally contractible, meaning that for every le…

Let $k$ be a perfect field, let $X$ be a closed subscheme of a smooth scheme over $k$, and let $\Lambda$ be the finite local coefficient ring in the setup. Let $\mathcal F$ be a co…