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.

15:23: Topology Talk

• 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.

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