12 problems

Farrell–Jones conjecture in $L$-theory. The assembly map is an isomorphism for all $n$. This conjecture predicts that the algebraic $L$-theory of the group ring can be computed fro…

Borel conjecture. Every homotopy equivalence

Let $M\to E\to B$ be a smooth fiber bundle whose fiber admits a negatively curved Riemannian metric. Assume that $B$ is a paracompact, Hausdorff space with the homotopy type of a s…

Let $D$ be the diagonal group of $\operatorname{SL}_n(\mathbb{R})$ where $n\ge 3$. A $D$-orbit in $\operatorname{SL}_n(\mathbb{R})/\operatorname{SL}_n(\mathbb{Z})$ is relatively co…

Let $X$ be a compact homogeneous manifold, let $f_0\in\operatorname{Aff}(X)$ be a $K$-system, and suppose that $\mathcal{Z}(f_0)$ contains a hyperbolic affine diffeomorphism. If…

Let $M$ be a Scott complex or quaternionic manifold with its natural stratification. Let $N$ be a stratified space and let $f:N\to M$ be a stratified homotopy equivalence. Stratifi…

Let $M^n$ be a manifold with a “nice” group action or a space with a “nice” stratification, and let $N^n$ be another manifold or space that is equivariantly or stratified homotopy…

For a Hamiltonian diffeomorphism or homeomorphism $\psi$, $C^0$ rigidity means that there is a sequence of natural numbers $k_i\to\infty$ such that $\psi^{k_i}\to\operatorname{Id}$…

A non-compact topological manifold without boundary is properly rigid if, whenever $N$ is another boundaryless topological manifold of the same dimension and $h\colon N\to M$ is a…

Let ${\cal F}$ be a family of maps, and let $f\in{\cal F}$ be non-hyperbolic. The map $f$ is topologically rigid when it is not topologically conjugate to any other map in a suffic…

Let $M$ be a closed aspherical manifold, and suppose it is equipped with a suitable action by a finite or compact group. Equivariant Borel conjecture. Such closed aspherical manifo…

Let $X^n$ be a Hadamard manifold of dimension $n>3$, and let $\Gamma$ be a discrete cocompact group of isometries of $X$. Assume that the fixed set $X^H$ has codimension greater th…