• The dependency graph for the first target of the Liquid Tensor Experiment now suggests that "half" of the challenge is done. Earlier this week I saw a talk by Johan Commelin explaining that the challenge was split into two parts (namely a proof of Lemma 9.4, and then a proof that Lemma 9.4 implies Theorem 9.1), and that they were very close to finishing the first one. And apparently on May 28 2021 they managed to do just this! Congratulations!

  • Giosuè Muratore, Csaba Schneider: Effective computations of the Atiyah-Bott formula computes integrals on the moduli space $\overline{\mathrm{M}}_{0,n}(\mathbb{P}^n,d)$, giving genus-zero Gromov-Witten invariants. The agreement of these invariants with other numbers were an important test for mirror symmetry. On page 7 the computation of quartic curves on a quintic 3-fold was done by Kontsevich using a computer, taking 5 minutes, and the current implementation takes less than a second to do this. Is this speed up what one should expect from Moore's law? Anyway, the code is at GitHub.

    I somehow consider the code for some of these early computer experiments of the same nature as the original notes of some influential mathematicians, and of equal historical interest. It would be interesting to (a) preserve the code, (b) preserve the infrastructure (hardware and software) to run them. Not just Kontsevich's computations, there have been equally important computations in the 1970s, 1980s and 1990s (maybe 1960s too? I guess that's where evidence for the BSD conjecture was gathered?) It would be a very niche project to work on though...

  • On the automorphisms of Mukai varieties describes the automorphism groups of (some families of) Fano varieties of Picard number one and coindex 3. Mukai used homogeneous varieties to describe the "interesting" cases which are of a somewhat exceptional origin. Understanding their geometry is an important question, and now their automorphism groups have been described. They are trivial whenever one expects them to be (for a general member of the family) except for some surprising cases outlined in Theorem 5. Cool stuff!

    On Fanography these cases are 1-10 (genus 12), 1-9 (genus 10), 1-8 (genus 9), 1-7 (genus 8) and 1-6 (genus 7). The cases 1-1 to 1-5 exist in arbitrary dimension, unlike the first 5. They are constructed using $\operatorname{Gr}(3,7)$, $\operatorname{G_2Gr}(2,7)$, $\operatorname{LGr}(3,6)$, $\operatorname{Gr}(2,6)$ and $\operatorname{OGr}_+(5,10)$.