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Lecture# Local Homeomorphisms and Coverings

Description

This lecture covers the concepts of local homeomorphisms and coverings in the context of manifolds, emphasizing the conditions under which a map between manifolds is considered a local homeomorphism or a covering. The lecture also introduces the notion of connected components and the mapping of these components under local homeomorphisms. The Riemann-Hurwitz formula is presented as a tool to analyze the behavior of non-constant holomorphic maps between compact connected Riemann surfaces. The lecture concludes with exercises on triangulations, branch points, and the construction of compact Riemann surfaces.

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In course

Instructor

MATH-410: Riemann surfaces

This course is an introduction to the theory of Riemann surfaces. Riemann surfaces naturally appear is mathematics in many different ways: as a result of analytic continuation, as quotients of complex

Related concepts (302)

Triangle

A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted . In Euclidean geometry, any three points, when non-collinear, determine a unique triangle and simultaneously, a unique plane (i.e. a two-dimensional Euclidean space). In other words, there is only one plane that contains that triangle, and every triangle is contained in some plane.

Equilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. It is also a regular polygon, so it is also referred to as a regular triangle.

Medial triangle

In Euclidean geometry, the medial triangle or midpoint triangle of a triangle △ABC is the triangle with vertices at the midpoints of the triangle's sides AB, AC, BC. It is the n = 3 case of the midpoint polygon of a polygon with n sides. The medial triangle is not the same thing as the median triangle, which is the triangle whose sides have the same lengths as the medians of △ABC. Each side of the medial triangle is called a midsegment (or midline). In general, a midsegment of a triangle is a line segment which joins the midpoints of two sides of the triangle.

Integer triangle

An integer triangle or integral triangle is a triangle all of whose side lengths are integers. A rational triangle is one whose side lengths are rational numbers; any rational triangle can be rescaled by the lowest common denominator of the sides to obtain a similar integer triangle, so there is a close relationship between integer triangles and rational triangles. Sometimes other definitions of the term rational triangle are used: Carmichael (1914) and Dickson (1920) use the term to mean a Heronian triangle (a triangle with integral or rational side lengths and area);cite book |last=Carmichael |first=R.

Special right triangle

A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45°–45°–90°. This is called an "angle-based" right triangle. A "side-based" right triangle is one in which the lengths of the sides form ratios of whole numbers, such as 3 : 4 : 5, or of other special numbers such as the golden ratio.

Related lectures (1,000)

Open Mapping TheoremMATH-410: Riemann surfaces

Explains the Open Mapping Theorem for holomorphic maps between Riemann surfaces.

Fundamental GroupsMATH-410: Riemann surfaces

Explores fundamental groups, homotopy classes, and coverings in connected manifolds.

Differential Forms IntegrationMATH-410: Riemann surfaces

Covers the integration of differential forms on smooth manifolds, including the concepts of closed and exact forms.

Topology of Riemann SurfacesMATH-410: Riemann surfaces

Covers the topology of Riemann surfaces, focusing on orientation and orientability.

Meromorphic Functions & DifferentialsMATH-410: Riemann surfaces

Explores meromorphic functions, poles, residues, orders, divisors, and the Riemann-Roch theorem.