# 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