Open Math Notes

Introductory course: Includes elementary solution via separation of variables and change of variables, general theory (Peano existence theorem, uniqueness of solutions, maximal solutions, continuity in initial conditions), linear ODE (existence, uniqueness, vector space of solutions, fundamental matrix solution, variation of constants, higher-order, Wronskian, constant coefficients), qualitative theory (stability at fixed points, phase portraits, Lyapunov's theorem, linearization, limit sets).

Peter Philip · LMU Munich · Date posted: August 9, 2023

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Part I is an intermediate level probability course given at the Independent University of Moscow. The unusual feature is an emphasis on discrete-parameter martingales. The development using examples based on gambling schemes makes no mention of conditional expectation.

Part II is a more systematic review of basic theory, leading up to the common definition of martingales in terms of conditional expectation.

William Faris · University of Arizona · Date posted: August 8, 2023

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These notes cover a large range of topics, including general topology, geometric topology, algebraic topology etc.

These notes are still work in progress.

Stefan Friedl · University of Regensburg · Date posted: July 24, 2023

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This is a collection of high-school algebra problems, with elements of calculus. Some of the problems go somewhat beyond the average high-school level and may require an in-depth analysis and good imagination. For each problem I provide a solution. Prerequisites include high-school mathematics and curiosity.

Oleg Saenko · University of Victoria, BC, Canada · Date posted: June 19, 2023

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A 1-semester introduction to modular forms

Kimball Martin · University of Oklahoma · Date posted: June 12, 2023

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