Curves of all stripes, 2026
Information about the course.
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Thursday 15th of January 2026 Lecture 1 Examples of curves and historical motivations
Notes
Challenge problem: Show that there is no map from an open subset of C to the elliptic curve {y^2 = x^2 + x + 5} which is rational (ratio of two polynomials) in both coordinates.
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Thursday 22th of January 2026 Lecture 2 Topological surfaces
Notes
Challenge problem: Show that the complex projective plane CP^2 does not admit any orientation-reversing homeomorphism.
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Thursday 29th of January 2026 Lecture 3 Hyperbolic surfaces and Fuchsian groups
Notes
Challenge problem 1: Show that any complete, simply-connected Riemannian surface of constant curvature 0 (resp. -1) is isometric to the Euclidean plane (resp. the hyperbolic plane).
Challenge problem 2: Show that the set of biholomorphisms of the complex plane is the set of affine maps {z |---> az + b, a different from 0}.
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Thursday 5th of February 2026 Lecture 4 (Compact) Riemann surfaces are algebraic
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Thursday 12th of February 2026 Lecture 5 Harmonic functions and Dirichlet principle
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Thursday 26th of February 2026 Lecture 6 Existence of meromorphic functions and the Riemann-Roch theorem
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Thursday 5th of March 2026 Lecture 7 Counting primes and other things, and the Riemann hypothesis
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Thursday 12th of March 2025 Lecture 8 Pretending curves over F_p are Riemann surfaces
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Thursday 19th of March 2025 Lecture 9 The geometry of curves over finite fields
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