15 problems

Segal's conjecture. Analogues of Segal's theorem should hold for holomorphic maps from the Riemann sphere $S^2$, or more generally from any compact Riemann surface, to various comp…

Naive-function-field conjecture. For arbitrary $Z$, one has

Let $G/T$ be a flag manifold, let $F(n)$ be the complete flag manifold, and let $Map(G/T,F(n))$ denote the space of continuous maps $G/T\to F(n)$. For $f\in Map(G/T,F(n))$, let…

Segal's stability conjecture. The stability theorem for spaces of rational curves should extend to the spaces of maps from ${\bf CP}^{m}$ to ${\bf CP}^{n}$ for $m>1$.

Let $\mathcal{C}$ be an $(\infty,n)$-category, and let $\mathrm{O}_{n-1}$ and $\Theta_n$ be the indexing shapes used to define the iterated constructions…

The stabilisation conjecture. The pairing

Let $G$ be a non-commutative compact connected Lie group, let $B_nG$ denote the $n$-th stage of the classifying-space construction, and let $\iota_n\colon B_nG\to BG$ be the canoni…

Consider the space of maps $\mathcal{M}(S^2,S^2)$ and an energy functional on it, where the energy is chosen reasonably, whether topologically defined or otherwise. No-energy-barri…

Optimality conjecture. The main result, Theorem, is optimal.

Let $A_d^{\mathbb{C}}(m,n)$ be the space of normalized $(n+1)$-tuples of homogeneous complex polynomials of degree $d$ defining algebraic maps from $\mathbb{R}\operatorname{P}^{m}$…

Let $m,n$ and $d$ be integers, and let $\operatorname{Alg}^{*}_{d}(\mathbb{R}\operatorname{P}^{m},\mathbb{C}\operatorname{P}^{n})$ denote the based space of degree-$d$ algebraic ma…

Let $A_d(m,n)$ be the polynomial-representative space and let $\Psi_d:A_d(m,n)\to\operatorname{Alg}_d^*(\mathbb{R}\operatorname{P}^m,\mathbb{R}\operatorname{P}^n)$ be the natural p…

Let $A_d(m,n)$ and $\widetilde{A}_d(m,n)$ be the spaces of polynomial data representing based and free algebraic maps from $\mathbb{R}\operatorname{P}^m$ to…

Let $m,n,d$ be integers with $2\leq m<n$ and $d\geq1$. Write $\operatorname{Alg}_d(\mathbb{R}\operatorname{P}^m,\mathbb{R}\operatorname{P}^n)$ and…

For real algebraic varieties, consider the inclusion of algebraic maps into the space of continuous maps, and in particular the inclusions…