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People have for a long time been interested in the properties of geometric shapes. In geometry one is usually interested in terms like distance, angle, area and volume. Topologists study the qualitative properties of geometric space. As the math has evolved, geometry and topology have grown to an active research area with links to physics and many other parts of mathematics.
The Faculty of Mathematics and Natural Sciences has selected the research group in Geometry and Topology as an emphasized research area, or more specifically as an "emerging top-tier research group". A large part of the group's research concentrates on algebraic topology and algebraic K-theory, with applications to geometric topology. The group is also involved in relating homotopy theory at large to other subjects. Motivic homotopy theory is an in vogue example of a homotopy theory that arises in algebraic geometry. An emerging example is a new homotopy theory of C*-algebras. Members of the group are also doing research related to Floer homology, manifolds of dimension 4 and symplectic geometry.
Projects
Equations in Motivic Homotopy, 2021-2025: Funded by RCN, this project aims at studying motivic homotopy theory, a relatively new subject that allows us to utilize algebraic topology methods to understand the objects of interest in algebraic geometry.
Motivic Hopf Equation, 2016 -2020: This is a project funded of RCN. The research aims at formulating and solving ground-breaking problems in motivic homotopy theory. As a relatively new field of research this subject has quickly turned into a well-established area of mathematics drawing inspiration from both algebra and topology.
Cooperation
Academic programmes and courses
Bachelor program:
Mathematics, Natural Sciences, Technology
Master program:
Mathematics
Courses: MAT3500/4500 - Topology MAT4510 - Geometric structures MAT 4520/9520 Manifolds MAT 4530/9530 Algebraic topology I
MAT 4540/9540 Algebraic topology II
MAT 4551/9551 Symplectic geometry MAT 9560 Lie groups MAT 9570 Algebraic K-theory MAT 9580 Algebraic topology III
MAT 4590/9590 Differential geometry
MAT 4595/9595 Geometry and analysis
Published Nov. 8, 2010 10:46 PM - Last modified Sep. 1, 2021 1:32 PM
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For the mathematical journal, see Geometry & Topology. In mathematics, geometry and topology is an umbrella term for the historically distinct disciplines of geometry and topology, as general frameworks allow both disciplines to be manipulated uniformly, most visibly in local to global theorems in Riemannian geometry, and results like the Gauss–Bonnet theorem and Chern–Weil theory.
Sharp distinctions between geometry and topology can be drawn, however, as discussed below.[clarification needed] It is also the title of a journal Geometry & Topology that covers these topics.
It is distinct from "geometric topology", which more narrowly involves applications of topology to geometry.
It includes:
It does not include such parts of algebraic topology as homotopy theory, but some areas of geometry and topology [such as surgery theory, particularly algebraic surgery theory] are heavily algebraic.
Geometry has local structure [or infinitesimal], while topology only has global structure. Alternatively, geometry has continuous moduli, while topology has discrete moduli.
By examples, an example of geometry is Riemannian geometry, while an example of topology is homotopy theory. The study of metric spaces is geometry, the study of topological spaces is topology.
The terms are not used completely consistently: symplectic manifolds are a boundary case, and coarse geometry is global, not local.
Local versus global structure
By definition, differentiable manifolds of a fixed dimension are all locally diffeomorphic to Euclidean space, so aside from dimension, there are no local invariants. Thus, differentiable structures on a manifold are topological in nature.
By contrast, the curvature of a Riemannian manifold is a local [indeed, infinitesimal] invariant[clarification needed] [and is the only local invariant under isometry].
Moduli
If a structure has a discrete moduli [if it has no deformations, or if a deformation of a structure is isomorphic to the original structure], the structure is said to be rigid, and its study [if it is a geometric or topological structure] is topology. If it has non-trivial deformations, the structure is said to be flexible, and its study is geometry.
The space of homotopy classes of maps is discrete,[a] so studying maps up to homotopy is topology. Similarly, differentiable structures on a manifold is usually a discrete space, and hence an example of topology, but exotic R4s have continuous moduli of differentiable structures.
Algebraic varieties have continuous moduli spaces, hence their study is algebraic geometry. These are finite-dimensional moduli spaces.
The space of Riemannian metrics on a given differentiable manifold is an infinite-dimensional space.
Symplectic manifolds
Symplectic manifolds are a boundary case, and parts of their study are called symplectic topology and symplectic geometry.
By Darboux's theorem, a symplectic manifold has no local structure, which suggests that their study be called topology.
By contrast, the space of symplectic structures on a manifold form a continuous moduli, which suggests that their study be called geometry.
However, up to isotopy, the space of symplectic structures is discrete [any family of symplectic structures are isotopic].[1]
- ^ Given point-set conditions, which are satisfied for manifolds; more generally homotopy classes form a totally disconnected but not necessarily discrete space; for example, the fundamental group of the Hawaiian earring.[citation needed]
- ^ Introduction to Lie Groups and Symplectic Geometry, by Robert Bryant, p. 103–104
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