2021-04-12

Chat with Phil

Some motivation for Unsorted/K3 surfaces : Fermat hypersurfaces $\sum x_i ^k$ for some fixed $k$ .
Look for ${\mathbf{Q}}{\hbox{-}}$ points, since by homogeneity the denominators can be scaled out to get ${\mathbf{Z}}{\hbox{-}}$ points
Unsorted/Faltings theorem : for a curves $C$ with $g(C) \geq 2$ , the number of Unsorted/rational points is finite, i.e. ${\sharp}C({\mathbf{Q}}) < \infty$ .
- Interesting consequence: there are only finitely many counterexamples to Fermat for any fixed $k$ . In fact, there are zero, but still.
Diagonal hypersurfaces $x_0^k + \cdots + x_n^k = 0$ .
Calabi-Yau when $k=n+1$ (maybe a bound instead..?), sharp change in behavior of finiteness of rational points at this threshold.

Dehn surgery : remove a tubular neighborhood of a knot, i.e. a solid torus, glue back in by some diffeomorphism of the boundary.
L Space conjecture simplest Heegard-Floer homology, rank of $\operatorname{HF}$ equals cardinality of $H_{\mathrm{sing}}$ .
Left-orderability on groups: a total order compatible with the group operation. Torsion groups can’t be LO: $x>1 \implies 1 = x^n > \cdots > x > 1$ .
taut foliation : a geometric condition. Admits a decomposition into leaves where a simple closed curve intersects each transversally?
fibred 3-manifolds: take $\Sigma \times I$ for $\Sigma$ a surface, glue the top and bottom by some diffeomorphism $\phi: \Sigma: {\circlearrowleft}$ .
Osvath-Szabo: admitting a taut foliation implies being a non- $L{\hbox{-}}$ space. Is the converse true?
Interesting knot invariants : $\tau, s, g_4(K), \sigma$ . Also the Jones, Conway, Alexander polynomials, or even just a coefficient. Note that some of these polynomials can not admit cabling formulas.