Historically, geometry studies the shape and relative position of objects in the Euclidean space. In modern geometry, the concept of `space’ is much wider and means a set equipped with a particular special structure such as, for example, projective space, Riemannian space, or topological space. The study of the simplest very general structure, which allows us to speak of continuity, led to the separation of topology from geometry as a large independent part of mathematics.
Classical differential geometry studies smooth geometric objects (for instance, Riemannian manifolds). Its most interesting features are connections between `microscopic’ properties of spaces and their global structure. Metric geometry is a common term for several directions studying non-smooth spaces with more rough structures (such as metric spaces) whose geometry is in one way or another similar to that of classical spaces. Among those are Alexandrov spaces, hyperbolic groups, and other classes of spaces.
S.V. Ivanov is working in this field.
Configuration spaces and moduli spaces arise in various mathematical frameworks. The following ones are of a traditional interest: moduli spaces of (pointed) algebraic curves, Teichmueller spaces, configuration spaces of bar-and-joint mechanisms, configuration spaces of convex polytopes and point configurations, etc. There exists a number of open problems: universality of configuration spaces, computation of (co)homology groups, finding extremal configurations, cellular (or simplicial) models. This subject requires knowledge of algebraic topology, hyperbolic geometry, Morse theory, algebraic geometry.
The topology of low-dimensional manifolds occupies a special place in modern geometry and topology since most of the powerful methods of multidimensional topology do not work in dimensions 3 and 4. Classification type questions are crucial ones for geometric topology. In the 20th century, it was the desire to solve classification problems that motivated a considerable part of the research in the field of the three-dimensional topology, the problems which include, in particular, the Poincare conjecture and Thurston’s geometrization conjecture resolved by Perelman. However, the significant advancement in this area leaves widely open many classical problems, which remain relevant.
A knot is a smooth embedding of the one-dimensional sphere (the circle) into the three-dimensional one. Knot theory has a rich history and is used in many areas of mathematics, in cryptography, physics, chemistry, and biology. For example, using knots one can describe any three-dimensional manifold or study quantum field theory while the knottedness of a polymer molecule is related to its physical properties. The main task of knot theory is to understand the structure of knots and effectively classify and recognize them. This problem has not yet received a satisfactory solution. The search for such a solution has led to many outstanding discoveries (hyperbolic knots and Thurston’s classification, polynomial invariants, invariants of finite degree, etc.) but the main results, apparently, are still ahead.