32 problems

Farber–Oprea conjecture. The formal power series $f_X(t)$ represents a rational function of the form

Let $X$ be an irreducible smooth complex variety. For a partition $\lambda$ of $k$, let $W_\lambda(X)$ denote the symmetric complement associated with the stratum indexed by…

Ample-cone conjecture. If $C$ is a very general curve of genus $g\ge 4$, then

Let $({\mathbb C}^2)^{[k]}$ be the Hilbert scheme of $k$ points on $\mathbb C^2$, let $({\mathbb C}^2)^k$ be its $k$-fold Cartesian power with the permutation action of the symmetr…

Let $SP^n(X):=X^n/S_n$ be the $n$th symmetric product of a space $X$. A topological, commutative $(m+k,m)$-group is a commutative $(m+k,m)$-group $(M,\mu)$ whose multiplication map…

Chen–Ruan's conjecture. The ring $H_{CR}^*(X)$ coincides with the cohomology ring $H^*(S^{[n]})$ of the Hilbert scheme of $n$ points on $S$.

Generalized Vafa–Witten conjecture. The analysis relating Yang–Mills theory on $K3$ to generating functions of Euler numbers should generalize so that the generating function of th…

Let $C$ be a smooth projective curve of genus $g$, let $J_C^d$ be its degree-$d$ Jacobian, and let $x\in J_C^d$ with $1\leq d\leq g$. Let ${\cal L}_x$ be the line bundle on $S^dC$…

Let $\mathrm{SP}^d(\mathbb{P}^n)$ be the $d$-th symmetric product of complex projective space, and let $cl$ denote the cycle class map from Lawson homology to singular homology wit…

Grojnowski's conjecture. The Grojnowski–Nakajima result should hold for $X$ of any dimension if one replaces $X^{[n]}$ by the symmetric quotient orbifold $X^n/S_n$ and replaces rat…

PFH–symmetric-product Floer homology conjecture. The periodic Floer homology is isomorphic to the symplectic Floer homology of

Let $X$ be a smooth projective curve of genus $g\geq 3$, let $L$ be a fixed line bundle of odd degree on $X$, and let $M$ be the moduli space of rank-$2$ stable vector bundles on…

Let $X$ be a smooth projective curve of genus $g\geq 2$, let $L$ be a fixed line bundle of odd degree on $X$, and let $M$ be the moduli space of rank-$2$ stable vector bundles on…

Let $X$ be a smooth projective curve, let $d$ be an integer, and let … be the morphism sending a line bundle $L$ to its tautological bundle $L^{[2]}$. In the cases where $X$ has ge…

Let $C$ be a projective smooth curve of genus $g\geq 2$, and let $C_d$ denote its $d$-th symmetric product. Let $D(C_d)$ be its derived category, and call a semi-orthogonal decompo…

Fix $n\ge 2$, $g,g'\ge 0$, and $k,k'\ge 1$ such that … and $g\ne g'$. Let $\overline{M}^2_{g,k}$ and $\overline{M}^2_{g',k'}$ be the corresponding compact surfaces with boundary, a…

Let $M^2$ and $N^2$ be Riemann surfaces, each either closed or obtained from a closed surface by removing finitely many points. Suppose that $M^2$ and $N^2$ are homeomorphic, equiv…

Wang's conjecture. One has

Let $X$ be a connected finite CW-complex, including the possibility that $X$ is 1-connected. Consider the integral cohomology ring $H^*(X^2;\mathbb{Z})$ associated with the square…

Let $C$ be a smooth complex curve of genus $g>1$, let $C^{(2)}$ be its second symmetric product, let $\Delta$ be the image of the diagonal in $C^{(2)}$, and let $K$ be a canonical…

Let $X$ be an irreducible smooth complex variety, let $^j\lambda$ denote the partition obtained by adjoining $j$ parts equal to $1$ to a partition $\lambda$, and let…

Let $i$ be a nonnegative integer, let $bb$ be a partition, and let $X$ be an irreducible smooth complex variety. For each $j$, let $\bar{w}^c_{1^j\lambda}(X)$ denote the complement…

Let $S$ be the sphere spectrum, let $p$ be a prime, and let $SP^{p^k}(S)$ denote the $p^k$-fold symmetric product of $S$. Whitehead's conjecture. For every $k$, the map … is zero f…

Let $\mathfrak{N}_m$ be the degree-$m$ piece of the nilCoxeter sequential 2-algebra, and let $\mathrm{Sym}^m(\mathbb{R}^2)$ denote the $m$-fold symmetric product of the plane. A ni…

Let $X$ be a topological space, let $A=C(X)$ be its algebra of functions, and let $p,q$ be nonnegative integers. Write $\operatorname{Sym}^{p|q}(X)$ for the corresponding supersymm…