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New project: problemsilike.com, a website collecting open problems that I, personally, like, with comments on their context, difficulty, and interest.
The goal is to track progress on mathematical questions that I think are important, and to measure human understanding of these questions, as well as the usefulness of AI tools in helping to resolve them.
It is also a step for me towards thinking a
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New paper just dropped, joint with Thomas Krämer and Marco Maculan. It's about a (somewhat mysterious, to me) connection between cubic threefolds and the exceptional Lie group E_6. arxiv.org/abs/2604.20970
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One of the crown jewels of 19th century mathematics is the classification (by Killing and Cartan) of compact Lie groups--groups that are also compact manifolds. Dynkin later interpreted this classification in terms of the finite graphs below.
Loosely speaking, every compact Lie group is built out of simple pieces: the circle group, the (infinite families of) special unitary groups, special orthog
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I'll finish with a little mystery. The starting point for this work was the classical connection between E_6 and cubic surfaces. There are also surfaces famously connected to E_7 and E_8: del Pezzo surfaces of degree 2 and 1.
So it is natural to look for threefolds generalizing the story here, i.e. where one can find E_7 or E_8 local systems/motives in the cohomology of some natural auxiliary con
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🔥 Top post: New project: problemsilike.com, a website collecting open proble · 54 likes + reposts
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New project: problemsilike.com, a website collecting open problems that I, personally, like, with comments on their context, difficulty, and interest.
The goal is to track progress on mathematical questions that I think are important, and to measure human understanding of these …
Just added a 15th problem to https://www.problemsilike.com! This one is a "non-abelian" analogue of the variational Hodge or Tate conjectures.
Link here: https://www.problemsilike.com
I think this last is especially important: I’m committing in advance to a position on the interest of these problems, to prevent goalpost-moving. And I’m trying to say something about their difficulty, to help non-experts understand what i…
I'll finish with a little mystery. The starting point for this work was the classical connection between E_6 and cubic surfaces. There are also surfaces famously connected to E_7 and E_8: del Pezzo surfaces of degree 2 and 1.
So it is nat…
Assuming the Hodge or Tate conjecture, the geometry of any variety is controlled by some group: its motivic Galois group. Serre's question is to, given a group, find a variety with that as its Galois group. It's a fancy version of the inve…
One pleasant consequence is that it gives a new (and in my view, particularly beautiful) answer to a question of Serre, originally answered by Boxer, Calegari, Emerton, Levin, Madapusi Pera, and Patrikis.
Namely, Serre asked if there exis…
The main result of this paper is that the geometry of covers of this surface is controlled by the 27-dimensional representation of the group E_6. The proof involves a lot of the beautiful, classical geometry of cubic threefolds.
For the e…
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The lines on a cubic threefold are parametrized by a (complex) surface, whose real points are pictured below in the case where the cubic threefold is cut out by the equation
x_0^3+x_1^3+x_2^3+x_3^3+x_4^3=0.
You can play with it yourself h…
It has long been known that there is some connection between cubic *surfaces* and E_6: loosely speaking, the combinatorics of the 27 lines on a cubic surface are controlled by the Dynkin diagram for E_6.
The point of this paper is to expla…
One of the crown jewels of 19th century mathematics is the classification (by Killing and Cartan) of compact Lie groups--groups that are also compact manifolds. Dynkin later interpreted this classification in terms of the finite graphs bel…
New paper just dropped, joint with Thomas Krämer and Marco Maculan. It's about a (somewhat mysterious, to me) connection between cubic threefolds and the exceptional Lie group E_6. arxiv.org/abs/2604.20970
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