Topology studies properties that are invariant under homeomorphism and its standard variations (diffeomorphism, piecewise linear homeomorphism, homeomorphism of pairs, fiberwise homeomorphism, etc.) Geometric topology confines its attention to spaces that are accessible to geometric intuition (for example, subsets of Euclidean spaces $\mathbb{R}^n$) and proceeds from elementary questions that have intuitive geometric meaning. Yet methods used to solve these questions can be far from elementary and can dive deeply into other fields (algebraic topology, group theory, algebraic K-theory, general topology, metric geometry, representation theory, functional analysis, etc.)

As an illustration, here are five old unsolved problems of geometric topology with short statements.

• Schoenflies Problem: Can the 3-sphere be piecewise linearly (or smoothly) knotted in $\mathbb{R}^4$?
• Rolfsen's Problem: Is every knot (=embedding of a circle in $\mathbb{R}^3$) isotopic (=homotopic through knots) to the trivial knot?
• Fenn's Problem: Does every 2-dimensional contractible finite simplicial complex piecewise linearly embed in $\mathbb{R}^4$?
• Borsuk's Problem: Does every $n$-dimensional contractible, locally contractible compact metric space embed in $\mathbb{R}^{2n}$?
• Hilbert–Smith Problem: Does there exist a free action of the group of $p$-adic integers on a manifold?

Financial support. The seminar is supported by the Ministry of Science and Higher Education of the Russian Federation (the grant to the Steklov International Mathematical Center, Agreement no. 075-15-2019-1614).

Seminar organizers
Melikhov Sergey Aleksandrovich
Shchepin Evgeny Vital'evich

Institutions
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
National Research University "Higher School of Economics", Moscow
Steklov International Mathematical Center