2021-04-12

Chat with Phil

#projects/research

  • Some motivation for Unsorted/K3 surfaces : Fermat hypersurfaces \(\sum x_i ^k\) for some fixed \(k\).
    Look for \({\mathbb{Q}}{\hbox{-}}\)points, since by homogeneity the denominators can be scaled out to get \({\mathbb{Z}}{\hbox{-}}\)points

  • Unsorted/Faltings theorem : for a curves \(C\) with \(g(C) \geq 2\), the number of Unsorted/rational points is finite, i.e. \({\sharp}C({\mathbb{Q}}) < \infty\).

    • 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

#projects/notes/seminars

  • 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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