Mathematics - An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1-Tori and Elliptic Curves (English)

1    Wednesday, April 02, 2025   

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Lecture 1: The Idea of a Riemann Surface

Lecture 2: Simple Examples of Riemann Surfaces

Lecture 3: Maximal Atlases and Holomorphic Maps of Riemann Surfaces

Lecture 4: A Riemann Surface Structure on a Cylinder

Lecture 5: A Riemann Surface Structure on a Torus

Lecture 6: Riemann Surface Structures on Cylinders and Tori via Covering Spaces

Lecture 7: Moebius Transformations Make up Fundamental Groups of Riemann Surfaces

Lecture 8: Homotopy and the First Fundamental Group

Lecture 9: A First Classification of Riemann Surfaces

Lecture 10: The Importance of the Path-lifting Property

Lecture 11: Fundamental groups as Fibres of the Universal covering Space

Lecture 12: The Monodromy Action

Lecture 13: The Universal covering as a Hausdorff Topological Space

Lecture 14: The Construction of the Universal Covering Map

Lecture 15: Completion of the Construction of the Universal Covering: Universality of the Universal Covering

Lecture 16: ompletion of the Construction of the Universal Covering: The Fundamental Group

Lecture 17: The Riemann Surface Structure on the Topological Covering of a Riemann Surface

Lecture 18: Riemann Surfaces with Universal Covering the Plane or the Sphere

Lecture 19: Mod-04 Lec-18 Classifying Complex Cylinders Riemann Surfaces

Lecture 20: Characterizing Moebius Transformations with a Single Fixed Point

Lecture 21: Characterizing Moebius Transformations with Two Fixed Points

Lecture 22: Torsion-freeness of the Fundamental Group of a Riemann Surface

Lecture 23: Characterizing Riemann Surface Structures on Quotients of the Upper Half-Plane with Abelian

Lecture 24: Classifying Annuli up to Holomorphic Isomorphism

Lecture 25: Orbits of the Integral Unimodular Group in the Upper Half-Plane

Lecture 26: Galois Coverings are precisely Quotients by Properly Discontinuous Free Actions

Lecture 27: Mod-05 Lec-26 Local Actions at the Region of Discontinuity of a Kleinian Subgroup

Lecture 28: Quotients by Kleinian Subgroups give rise to Riemann Surfaces

Lecture 29: The Unimodular Group is Kleinian

Lecture 30: The Necessity of Elliptic Functions for the Classification of Complex Tori

Lecture 31: The Uniqueness Property of the Weierstrass Phe-function associated to a Lattice in the Plane

Lecture 32: The First Order Degree Two Cubic Ordinary Differential Equation satisfied by the Weierstras

Lecture 33: The Values of the Weierstrass Phe-function at the Zeros of its Derivative are nonvanishing

Lecture 34: The Construction of a Modular Form of Weight Two on the Upper Half-Plane

Lecture 35: The Fundamental Functional Equations satisfied by the Modular Form of Weight Two on the Upper Half

Lecture 36: The Weight Two Modular Form assumes Real Values on the Imaginary Axis in the Upper Half-plane

Lecture 37: The Weight Two Modular Form Vanishes at Infinity

Lecture 38: The Weight Two Modular Form Decays Exponentially in a Neighbourhood of Infinity

Lecture 39: A Suitable Restriction of the Weight Two Modular Form is a Holomorphic Conformal Isomorphism

Lecture 40: The J-Invariant of a Complex Torus (or) of an Algebraic Elliptic Curve

Lecture 41: A Fundamental Region in the Upper Half-Plane for the Elliptic Modular J-Invariant

Lecture 42: The Fundamental Region in the Upper Half-Plane for the Unimodular Group

Lecture 43: A Region in the Upper Half-Plane Meeting Each Unimodular Orbit Exactly Once

Lecture 44: Moduli of Elliptic Curves

Lecture 45: Punctured Complex Tori are Elliptic Algebraic Affine Plane Cubic Curves in Complex 2-Space

Lecture 46: The Natural Riemann Surface Structure on an Algebraic Affine Nonsingular Plane Curve

Lecture 47: Complex Projective 2-Space as a Compact Complex Manifold of Dimension Two

Lecture 48: Complex Tori are the same as Elliptic Algebraic Projective Curves