• Deck Transformations

    Deck transformations are important because we want to study the fundamental group of the base space (X) by looking at subgroups induced by the image of the covering map (q: E —> X).(I don’t like the way I phrased the first sentence so I will rewrite this lecture with pictures and come back to it…

  • Intractable Integrals (AMATH 112)

    Required reading: https://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/chapter25/contents.html [learn_press_profile]

  • Introduction to Julia Programming (MATH 173)

    Course Textbook: https://services.publishing.umich.edu/wp-content/themes/mpub-services/library/pdf/PDSdownload102.pdfPrerequisites: Calculus 1-3, Linear Algebra, Mathematical Foundations [learn_press_profile]

  • Koch Curve, Fractal Geometry, and Measures (MATH 334)

    I ended up back here coincidentally, and I think my intuition about this was on the right track. I just need to review it a bit. This is entirely a note to self. Required Reading: https://math.stackexchange.com/questions/3005409/koch-curve-from-cantor-sets-paradoxhttps://math.stackexchange.com/questions/12906/the-staircase-paradox-or-why-pi-ne4https://www.wikiwand.com/en/Fractal_dimension [learn_press_profile]

  • Introduction to Algebraic Cycles (MATH 568)

    Required Reading: Chapter 1 of Hartshorne, Algebraic GeometryGrothendieck, A. (1969), “Standard Conjectures on Algebraic Cycles”, Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968) (PDF), Oxford University Press, pp. 193–199, MR 0268189. Grothendieck, A. (1958), “Sur une note de Mattuck-Tate”, J. Reine Angew. Math., 1958 (200): 208–215, doi:10.1515/crll.1958.200.208, MR 0136607, S2CID 115548848 Background In mathematics, an algebraic cycle on an algebraic variety V is a formal linear combination of subvarieties of V. These are the part of…

  • Polar Coordinates and Coxeter Groups (MATH 127)

    Course Textbook: Reflection Groups and Coxeter Groups, James E. Humphreys,Preparing for this Section (Review)Rectangular CoordinatesDefinition of the Trigonometric FunctionsThe Distance FormulaInverse Tangent FunctionCompleting the SquareAngles; Degree Measure; Radian Measure [learn_press_profile]

  • Persistent Homology (MATH 528)

    Required Reading (Lectures): https://mrzv.org/Course Website: http://graphics.stanford.edu/courses/cs468-09-fall/Persistent homology is a method for computing topological features of a space at different spatial resolutions. More persistent features are detected over a wide range of spatial scales and are deemed more likely to represent true features of the underlying space rather than artifacts of sampling, noise, or particular choice of…

  • An Automorphism of de Moivre’s Theorem (Math 347)

    Introduction to de Moivre’s Theorem: https://brilliant.org/wiki/de-moivres-theorem/Practice Problem: Books: PreCalculus text, Dummit and FootePreliminary Reading: https://www.wikiwand.com/en/Abraham_de_MoivreAbraham de Moivre (French pronunciation: ​[abʁaam də mwavʁ]; 26 May 1667 – 27 November 1754) was a French mathematician known for de Moivre’s formula, a formula that links complex numbers and trigonometry, and for his work on the normal distribution and probability theory.In mathematics, de Moivre’s formula (also known as de Moivre’s theorem and de Moivre’s identity)…

  • Newton’s minimal resistance problem (Phys 540)

    Newton’s Minimal Resistance Problem is a problem of finding a solid of revolution which experiences a minimum resistance when it moves through a homogeneous fluid with constant velocity in the direction of the axis of revolution, named after Isaac Newton, who studied the problem in 1685 and published it in 1687 in his Principia Mathematica. This is the first example…

  • Hilbert’s twenty-third problem (Math 540)

    Hilbert’s twenty-third problem is the last of Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. In contrast with Hilbert’s other 22 problems, his 23rd is not so much a specific “problem” as an encouragement towards further development of the calculus of variations. His statement of the problem is a summary of the state-of-the-art (in…