---
date: 2021-04-12
tags:
  - web/quick-notes
created: 2021-10-27T19:35
updated: 2024-01-01T23:13
---


# 2021-04-12

## Chat with Phil

#projects/research 

- Some motivation for [K3 surfaces](K3%20surfaces.md) : Fermat hypersurfaces $\sum x_i ^k$ for some fixed $k$.   
  Look for $\QQ\dash$points, since by homogeneity the denominators can be scaled out to get $\ZZ\dash$points

- [Unsorted/Faltings theorem](Unsorted/Faltings%20theorem.md) : for a [curves](curves.md) $C$ with $g(C) \geq 2$, the number of [Unsorted/rational points](Unsorted/rational%20points.md) is finite, i.e. $\size C(\QQ) < \infty$.

  - 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](Calabi-Yau.md) when $k=n+1$ (maybe a bound instead..?), sharp change in behavior of finiteness of rational points at this threshold.

## 15:23: Topology Talk

#projects/notes/seminars 

- [Dehn surgery](Dehn%20surgery) : remove a tubular neighborhood of a knot, i.e. a solid torus, glue back in by some diffeomorphism of the boundary.

- [L Space conjecture](L%20Space%20conjecture) simplest [Heegard-Floer homology](Heegard-Floer%20homology.md), rank of $\HF$ equals cardinality of $H_\sing$.

- 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](taut%20foliation) : a geometric condition. Admits a decomposition into leaves where a simple closed curve intersects each transversally?

- [fibred](fibred) 3-manifolds: take $\Sigma \cross I$ for $\Sigma$ a surface, glue the top and bottom by some diffeomorphism $\phi: \Sigma: \selfmap$.

- Osvath-Szabo: admitting a taut foliation implies being a non-$L\dash$space.
  Is the converse true?

- Interesting [knot invariants](knot%20invariants) : $\tau, s, g_4(K), \sigma$.
  Also the Jones, Conway, Alexander polynomials, or even just a coefficient.
  Note that some of these polynomials can not admit cabling formulas.