20 problems

Strong Barratt exponent conjecture. The double loop map

Let $\operatorname{Fl}_{n+1}$ be the flag variety, let $\beta$ be a strictly monotonic class, and write $\Omega^{2}_{\beta}(\operatorname{Fl}_{n+1})$ for the corresponding algebrai…

Cotangent-fiber Hodge microsheaf conjecture. Hain's Hodge structure is induced by a Hodge microsheaf whose underlying microsheaf corresponds to a cotangent fiber.

Let $\mathcal{K}$ be a simplicial complex and let $\mathbf{k}$ be a ring such that $H_*(\Omega\mathrm{DJ}(\mathcal{K});\mathbf{k})$ is a free $\mathbf{k}$-module. Recall that a Hop…

Let $n>3$ and let $q otin Z(\operatorname{Quat}{n+1})$. Consider the space ${cal L}n(1;q)$ of locally convex curves with endpoint data $(1,q)$. If it contains convex curves, let…

Let $\mathbb{B}_k$, for $0 \leq k \leq 8$, and $\mathbb{C}\mathbb{P}^1 \times \mathbb{C}\mathbb{P}^1$ be the indicated series of del Pezzo surfaces. Let $\mathcal{L}^{k}_c S^4$, fo…

Charming-map conjecture. The following statements are true:

Noncentral component conjecture. If

Let $p$ be an odd prime, let $q$ be a power of $p$, and let $\mathbb{F}_q$ be the finite field with $q$ elements. For $n\geq 3$, let $\mathrm{Spin}(n)$ be the spin group, let…

Let $p$ be a prime, let $n$ be a positive integer, and let $r$ be a positive integer. Set $T_{2n}=S^{2n+1}\{p^r\}$, and let $BW_n$ be the space occurring in the EHP sequence for th…

Higher-loop-space conjecture. (a) The space $Maps((T^\infty,\dot{T}^\infty),(X,\{x_n\}))$, together with all its quotients $Maps((T^n,\dot{T}^n),(X,x_1,\dots,x_n))$, captures the h…

Let $X$ be a pointed space, let $n\geq 1$, and let $\widetilde{\theta}_{\iota}$ be the operation on homotopy groups induced by the distinguished class $\iota$. For $i,j\geq 1$, ide…

Let $X$ be the space in the preceding construction, let $n\geq 2$, and let $\phi_n$ and $\psi_n$ be the two maps defined there from the relevant smash-product construction into…

Let $Z$ be an affine smooth scheme of finite type over the ground field, viewed as a $0$-coconnective DG scheme, and let $Z\qqart$ denote its loop DG scheme. 0-coconnectivity conje…

Let $G$ be a compact Lie group and let $P$ be a principal $G$-bundle over a Riemann surface $\Sigma$. Write $\mathcal A(P)$ for the space of connections on $P$, $\mathcal G_0(P)$ f…

Let $p$ be a prime, let $n,r$ be positive integers, and let $P^{2n}(p^r)\xrightarrow{i}T_{2n-1}(p^r)$ be the canonical map. A space $Z$ is strongly $(2n,r)$-subexponential if … for…

Cohen's Gerstenhaber structure conjecture. There is an isomorphism of Gerstenhaber algebras

Let $M^n$ be a simply connected positively curved manifold, and let $\Lambda M$ denote its loop space. The Betti numbers of $\Lambda M$ are the homology ranks with coefficients in…

Functorial trace conjecture. There exists a morphism

Natural lift conjecture. The complex $Tr(u)$ has a natural lift