neverendingbooks

Learners’ logic

In the Learners and Poly-post we’ve seen that learners from $A$ to $B$ correspond to set-valued representations of a directed graph $G$ and therefore form a presheaf topos. Any topos comes with its Mitchell-Benabou language, allowing us to speak of formulas, propositions and their truth values. Two objects play a special role in this: the […]

February 22, 2022
Yet more topos news

Every topos has its own internal language, the so called Mitchell-Bénabou language, allowing us to speak about formulas and their truth values. Sadly, Jean Bénabou died last week. Here’s a nice interview with Bénabou (in French) on category theory, Grothendieck, logic, and a rant on plagiarism among topos theorists (starting at 1:00:16). Yesterday, France Culture’s…

February 18, 2022
The hype cycle of an idea

These three ideas (re)surfaced over the last two decades, claiming to have potential applications to major open problems: (2000) $\mathbb{F}_1$-geometry tries to view $\mathbf{Spec}(\mathbb{Z})$ as a curve over the field with one element, and mimic Weil’s proof of RH for curves over finite fields to prove the Riemann hypothesis. (2012) IUTT, for Inter Universal Teichmuller…

February 13, 2022
Chevalley’s circle of friends

Last week, Danielle Couty ArXiVed her paper Friendly views on Claude Chevalley (in French). From the abstract: “We propose to follow the itinerary of Claude Chevalley during the last twenty years of his life, through the words of Jacques Roubaud, Denis Guedj and Alexander Grothendieck. Our perspective is that of their testimonies filled with friendship.”…

February 8, 2022
Christine Bessenrodt (1958-2022)

We were pretty close once. It is a shock to read about her passing on Twitter. EWM mourns Christine Bessenrodt, who passed away on January 24 after a serious illness. Christine held the Chair of Algebra and Number Theory at The University of Hannover and she worked in many ways to promote equality for women…

February 1, 2022
Grothendieck stuff

January 13th, Gallimard published Grothendieck’s text Recoltes et Semailles in a fancy box containing two books. Here’s a G-translation of Gallimard’s blurb: “Considered the mathematical genius of the second half of the 20th century, Alexandre Grothendieck is the author of Récoltes et semailles, a kind of “monster” of more than a thousand pages, according to…

January 30, 2022
Learners and Poly

Brendan Fong, David Spivak and Remy Tuyeras cooked up a vast generalisation of neural networks in their paper Backprop as Functor: A compositional perspective on supervised learning. Here’s a nice introduction to neural networks for category theorists by Bruno Gavranovic. At 1.49m he tries to explain supervised learning with neural networks in one slide. Learners…

January 29, 2022
Poly

Following up on the deep learning and toposes-post, I was planning to do something on the logic of neural networks. Prepping for this I saw David Spivak’s paper Learner’s Languages doing exactly that, but in the more general setting of ‘learners’ (see also the deep learning post). And then … I fell under the spell…

January 26, 2022
Smirnov on $\mathbb{F}_1$ and the RH

Wednesday, Alexander Smirnov (Steklov Institute) gave the first talk in the $\mathbb{F}_1$ world seminar. Here’s his title and abstract: Title: The 10th Discriminant and Tensor Powers of $\mathbb{Z}$ “We plan to discuss very shortly certain achievements and disappointments of the $\mathbb{F}_1$-approach. In addition, we will consider a possibility to apply noncommutative tensor powers of $\mathbb{Z}$…

January 21, 2022
Deep learning and toposes

Judging from this and that paper, deep learning is the string theory of the 2020s for geometers and representation theorists. String theory is the 90s answer to the tears of algebraic geometers worldwide trying to write the "Applications" part of their grant proposals. https://t.co/AboZ5WkPtc — algebraic geometer BLM (@BarbaraFantechi) December 17, 2021 If you want…

January 16, 2022