Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence Tao
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) Let , and […]
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 equivalent to being locally invertible. (The implication of local […]
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 above five favorable […]
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 the […]
(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 year. Programs are selected on the basis of […]
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 testing and review has […]
With the advent of modern coding agents, many visualization projects that I had proposed in the past, but dropped due to the time and complexity of the coding portion of the task, have now become relatively feasible, in that a reasonable quality prototype (suitable for non-mission-critical tasks such as providing secondary visual aids, where it […]
I have been interested in machine-assisted ways to do and teach mathematics from as far back as 1999, when I started coding several applets in Java 1.0, both for my complex analysis and linear algebra courses, as well as to visualize various mathematical objects I was interested in (such as honeycombs or Besicovitch sets). This […]
Zachary Chase, Zach Hunter and I have uploaded to the arXiv our preprint Gilbreath’s conjecture: a Cramér random model and a deterministic analysis. This paper is motivated by a notorious conjecture of Gilbreath (also proposed eighty years prior by Proth), which one can state as follows: if one starts with the sequence of primes and […]