Topology Research Group
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In topology, we are interested in symmetry and geometry. The objects we study are called topological spaces and include everything that has something geometrical about it. The main tools for the study of topological spaces are algebra, geometry and combinatorics. A class of topological spaces that are related to physical theories, such as relativity and sting theory, is called Riemannian manifolds. These are topological spaces equipped with geometrical structure, which gives meaning to concepts such as distance, angle and volume. Broadly speaking, one can say that there are spaces it is possible to live in in the same manner that we live in a three-dimensional space.
A part of our research focuses on the application of algebraic tools to describe the so-called homotopy invariants of topological spaces. Broadly speaking, a homotopy is a deformation that is possible to create using an elastic space made of rubber. We are interested in both new descriptions of homotopy invariants and in computing the classical homotopy invariants, such as the homotopy groups of specific spaces.
Conversely, we also use topological spaces to study algebra. For example, we work with algebraic K-theory. This is a space that describes, among other things, how much of an error the algorithm for Gaussian elimination makes when we work with linear algebra over the integers instead of the real numbers.