Sur les blogs — les derniers billets des blogs de mathématiques

Ben Krause, Hamed Mousavi, Joni Teräiväinen, and I have just uploaded to the arXiv our paper Quantitative bounds for sets lacking polynomial progressions with shifted prime difference. The purpose of this paper is to obtain quantitative versions of this theorem of Wooley and Ziegler: Theorem 1 Let be polynomials of...

In recent months there has been a proliferation of AI-generated proofs of various old and new results, some of which have been formalized in the proof assistant language Lean. However, checking that a given Lean repository actually proves the claimed statement is somewhat non-trivial, especially for an audience which...

This post concerns the following conjecture of Sendov, as well as its strengthening by Phelps–Rodriguez: Conjecture 1 (Sendov's conjecture) Let , and let be a degree polynomial with all zeroes in the unit disk. Then for every zero of , there exists a critical point of with . Conjecture 2 (Phelps–Rodriguez conjecture)...

A function of one variable can be translated in space by a spatial shift to obtain a new function and also modulated in frequency by a frequency shift to obtain a new function One can compose these two operations to obtain a time-frequency shift: As per the time-frequency uncertainty principle, the two shifts do not...

The notorious Jacobian conjecture can be formulated concretely over the complex numbers as follows. Conjecture 1 (Jacobian Conjecture) Let be a polynomial map in complex variables, whose Jacobian is a non-zero constant. Then is invertible (with polynomial inverse). The condition that the Jacobian is non-zero is...

I believe that the creation of visualization apps to illustrate mathematical or scientific concepts is a particularly favorable use case for modern coding agents, as many of the downside risks attached to other LLM use cases are limited: I would however caution against unrestricted LLM use when one or more of the...

One byproduct of learning how to use coding agents to create visualization apps is that it now becomes straightforward to convert any figure in one's papers that had already been generated by code (e.g., in Python) into a more interactive, animated applet. I can illustrate this with Figure 1 from my recent paper on...

(I am writing here in my capacity as Director of Special Projects at IPAM.) IPAM seeks program proposals from the mathematical, statistical, and scientific communities for long programs, workshops, and summer schools.  Most program proposals are reviewed at IPAM's Science Advisory Board meeting, held in November each...

I am finding the newly revealed capability to code old applet ideas into reality to be very tempting to sink more time into, though I am certainly encountering the common "vibe coding" experience that the process can produce something that superficially resembles a finished product well before a satisfactory level of...