Almost 20 years ago, I wrote a textbook in real analysis called “Analysis I“. It was intended to complement the many good available analysis textbooks out there by focusing more on foundational issues, such as the construction of the natural numbers, integers, rational numbers, and reals, as well as providing enough set theory and logic to allow students to develop proofs at high levels of rigor.| What's new
Thomas Bloom’s erdosproblems.com site hosts nearly a thousand questions that originated, or were communicated by, Paul Erdős, as well as the current status of these questions (about a third of which are currently solved). The site is now a couple years old, and has been steadily adding features, the most recent of which has been […]| What's new
The Simons-Laufer Mathematical Sciences institute, or SLMath (formerly the Mathematical Sciences Research Institute, or MSRI) has recently restructured its program formats, and is now announcing th…| What's new
First things first: due to an abrupt suspension of NSF funding to my home university of UCLA, the Institute of Pure and Applied Mathematics (which had been preliminarily approved for a five-year NSF grant to run the institute) is currently fundraising to ensure continuity of operations during the suspension, with a goal of raising $500,000. Donations can be made at this page. As incoming Director of Special Projects at IPAM, I am grateful for the support (both moral and financial) that we hav...| What's new
First things first: due to an abrupt suspension of NSF funding to my home university of UCLA, the Institute of Pure and Applied Mathematics (which had been preliminarily approved for a five-year NS…| What's new
The history of every major galactic civilization tends to pass through three distinct and recognizable phases, those of Survival, Inquiry and Sophistication, otherwise known as the How, Why, and Wh…| What's new
A remarkable phenomenon in probability theory is that of universality – that many seemingly unrelated probability distributions, which ostensibly involve large numbers of unknown parameters, …| What's new
The Salem prize was established in 1968 and named in honor of Raphaël Salem (1898-1963), a mathematician famous notably for his deep study of the links between Fourier series and number theory and …| What's new
This is a blog version of a talk I recently gave at the IPAM workshop on “The Kakeya Problem, Restriction Problem, and Sum-product Theory”. Note: the discussion here will be highly non-…| What's new
There has been some spectacular progress in geometric measure theory: Hong Wang and Joshua Zahl have just released a preprint that resolves the three-dimensional case of the infamous Kakeya set conjecture! This conjecture asserts that a Kakeya set – a subset of that contains a unit line segment in every direction, must have Minkowski and Hausdorff dimension equal to three. (There is also a stronger “maximal function” version of this conjecture that remains open at present, although the...| What's new
Ayla Gafni and I have just uploaded to the arXiv the paper “On the number of exceptional intervals to the prime number theorem in short intervals“. This paper makes explicit some relationships between zero density theorems and prime number theorems in short intervals which were somewhat implicit in the literature at present.| What's new
Boris Alexeev, Evan Conway, Matthieu Rosenfeld, Andrew Sutherland, Markus Uhr, Kevin Ventullo, and I have uploaded to the arXiv a second version of our paper “Decomposing a factorial into large factors“. This is a completely rewritten and expanded version of a previous paper of the same name. Thanks to many additional theoretical and numerical contributors […]| What's new
Ayla Gafni and I have just uploaded to the arXiv the paper “On the number of exceptional intervals to the prime number theorem in short intervals”. This paper makes explicit some relati…| What's new
Almost 20 years ago, I wrote a textbook in real analysis called “Analysis I”. It was intended to complement the many good available analysis textbooks out there by focusing more on foun…| What's new
Rachel Greenfeld and I have just uploaded to the arXiv our paper Some variants of the periodic tiling conjecture. This paper explores variants of the periodic tiling phenomenon that, in some cases,…| What's new
In a recent post, I talked about a proof of concept tool to verify estimates automatically. Since that post, I have overhauled the tool twice: first to turn it into a rudimentary proof assistant th…| What's new
Many problems in analysis (as well as adjacent fields such as combinatorics, theoretical computer science, and PDE) are interested in the order of growth (or decay) of some quantity $latex {X}&…| What's new
This post was inspired by some recent discussions with Bjoern Bringmann. Symbolic math software packages are highly developed for many mathematical tasks in areas such as algebra, calculus, and num…| What's new
[This is a (lightly edited) repost of an old blog post of mine, which had attracted over 400 comments, and as such was becoming difficult to load; I request that people wishing to comment on that …| What's new
There has been some spectacular progress in geometric measure theory: Hong Wang and Joshua Zahl have just released a preprint that resolves the three-dimensional case of the infamous Kakeya set con…| What's new
A fundamental and recurring problem in analytic number theory is to demonstrate the presence of cancellation in an oscillating sum, a typical example of which might be a correlation $latex \display…| What's new
In Notes 1, we approached multiplicative number theory (the study of multiplicative functions $latex {f: {\bf N} \rightarrow {\bf C}}&fg=000000$ and their relatives) via elementary methods, in …| What's new
In analytic number theory, an arithmetic function is simply a function $latex {f: {\bf N} \rightarrow {\bf C}}&fg=000000$ from the natural numbers $latex {{\bf N} = \{1,2,3,\dots\}}&fg=0000…| What's new
I’ve just uploaded to the arXiv my paper “Planar point sets with forbidden -point patterns and few distinct distance“. This (very) short paper was a byproduct of my recent explorations of the Erdös problem website in recent months, with a vague emerging plan to locate a suitable problem that might be suitable for some combination of a crowdsourced “Polymath” style project and/or a test case for emerging AI tools. The question below was one potential candidate; however, upon reviewi...| What's new
William Banks, Kevin Ford, and I have just uploaded to the arXiv our paper “Large prime gaps and probabilistic models”. In this paper we introduce a random model to help understand the …| What's new
We continue the discussion of sieve theory from Notes 4, but now specialise to the case of the linear sieve in which the sieve dimension $latex {\kappa}&fg=000000$ is equal to $latex {1}&fg…| What's new
We now move away from the world of multiplicative prime number theory covered in Notes 1 and Notes 2, and enter the wider, and complementary, world of non-multiplicative prime number theory, in whi…| What's new
Kevin Ford, Ben Green, Sergei Konyagin, James Maynard, and I have just uploaded to the arXiv our paper “Long gaps between primes”. This is a followup work to our two previous papers (di…| What's new
The parity problem is a notorious problem in sieve theory: this theory was invented in order to count prime patterns of various types (e.g. twin primes), but despite superb success in obtaining upp…| What's new
In a previous blog post, I discussed how, from a Bayesian perspective, learning about some new information $latex {E}&fg=000000$ can update one’s perceived odds $latex {{\mathbb P}(H_1) /…| What's new
This post contains two unrelated announcements. Firstly, I would like to promote a useful list of resources for AI in Mathematics, that was initiated by Talia Ringer (with the crowdsourced assistan…| What's new
