# Ideas for Posts or Notes ## Technical Topics - Fiber bundles - Method of killing homotopy - Brouwer's fixed point theorem - Hairy ball theorem - Why gimbal lock occurs - Results in SET and enumerative geometry - Einstein's equations - Interesting Results - Nash embedding - Aatiyah-Singer Index Theorem - Geometrization - Riemann-Roch - Descriptions of big problems - Fermat's Last Theorem - The abc conjecture - The Weil conjectures - Topology - Smooth 3d Poincare (Resolved) - Smooth 4d Poincare (Unresolved) - Homotopy groups of spheres - Smooth 4d Schoenflies - The Ravenel conjectureso - The Kervaire invariant problem - The Hopf invariant one problem - Hilbert's Problems - Continuum hypothesis - Peano is consistent - Millenium Prize Problems - Poincare - Riemann - P vs. NP - Navier Stokes - Yang-Mills and Mass Gap - Hodge - Birch-Swinnerton-Dyer - When you can swap limiting operations - List of mind blowing facts from undergrad - |Z| = |Q| < |R|, or even |Q| < |(0,1)| - There are holes in Q! - sum i = -1/12 - Calculators use series to compute functions (sin/cos, e/log, etc) and you can too! - Birthday paradox - Integrals depend on holes - Gabriel's horn - Euler's identity - The high dimensional sphere is spiky - The harmonic series diverges - 0.999.... = 1 - Zeno's paradox converges - There's no general formula for solving quintics - Only some regular polygons can be constructed with a ruler/compass - relates geometry and number theory - A conditionally convergent series can be rearranged to converge to anything - sqrt(2) is not rational - A normal vector to a surface is not always well-defined - Trig is just complex analysis - You can balance chemical equations using matrices - Factorials are really, super large - 52! is huge - chances are very low that any two people (ever) have held the name shuffled deck - Compare to total time since big bang in seconds, estimated size of the observable universe, etc - Monty Hall problem - Random walks in 2D almost never return to the origin (happens in 1d) - Fixed point theorems (brouwer) - The word problem - Banach Tarski - Godel completeness - There are continuous everywhere, differentiable nowhere functions - Every function can be expressed in terms of trig functions - There is no uniform distribution on N, Q, R, etc - We've only had algebraic symbols/formulas for ?00 years, and cartesian coordinates for ?00 years - Euclid's 5th postulate - Having boy/girl then boy problem - i^i is real - Green's Theorem and Stoke's Theorem are the same thing - Deriving formulas using differentiation/integration of power series - What is the Axiom of Choice? - Constructing large cardinals - Euclid's proof of the infinitude of primes: \$1 + \prod p_i\$ is not necessarily prime!! - The Kernel Trick - Eigenfaces - "What every math major should know" ## Personal - Undergrad degree in review - Review of research program - Review of Witt Vectors conference # From Lightning Talks - Epsilon delta proofs - Approaches to finding the limit of a function (single and/or multivariable) - Parametricity and the algebra of pure functional programs - Set theory (maybe the ZFC axioms, some history etc.) - Graph theory (e.g. basic theory, applications) - Turing machines - Non-standard analysis (hyper-real numbers etc.) - Construction of number systems (e.g. the natural numbers from the Peano axioms, or the real numbers as Cauchy sequences of rationals) - Something fun from probability theory (e.g. Markov processes and their connection to linear algebra) - Game theory - Problem solving techniques (maybe someone has read Polya, or happens to have exerience from contest math at some level?) - Solving recurrence relations (using the language of ODEs) - Applications of Linear Algebra - PageRank, Computer Graphics, and more - Mathematical formulation of Brownian motion and its role in mathematics - Stochastic Calculus