Hamilton’s quaternion number system $latex {\mathbb{H}}&fg=000000$ is a non-commutative extension of the complex numbers, consisting of numbers of the form $latex {t + xi + yj + zk}&f…

What's new

18.12.2024

On the distribution of eigenvalues of GUE and its minors at fixed index | What's new

I’ve just the arXiv the paper “On the distribution of eigenvalues of GUE and its minors at fixed index”. This is a somewhat technical paper establishing some estimates regarding o…

What's new

06.12.2024

AI for Math fund | What's new

Renaissance Philanthropy and XTX Markets have announced the launch of the AI for Math Fund, a new grant program supporting projects that apply AI and machine learning to mathematics, with a focus o…

What's new

28.11.2024

On several irrationality problems for Ahmes series | What's new

Vjeko Kovac and I have just uploaded to the arXiv my paper “On several irrationality problems for Ahmes series”. This paper resolves (or at least makes partial progress on) some open qu…

What's new

12.11.2024

Higher uniformity of arithmetic functions in short intervals II. Almost all intervals | What's new

Kaisa Matomäki, Maksym Radziwill, Fernando Xuancheng Shao, Joni Teräväinen, and myself have (finally) uploaded to the arXiv our paper “Higher uniformity of arithmetic functions…

What's new

09.11.2024

The elephant in the room | What's new

The day after the election, I found myself struggling with how to approach the complex analysis class I was teaching. Could I ignore the (almost literal) elephant in the room? Would my students be …

What's new

03.11.2024

Climbing the cosmic distance ladder: launching an instagram | What's new

I have been collaborating for many years with a long-time personal friend and polymath, Tanya Klowden, on a popular book on astronomy tentatively entitled “Climbing the cosmic distance ladder…

What's new

13.10.2024

The equational theories project: a brief tour | What's new

Almost three weeks ago, I proposed a collaborative project, combining the efforts of professional and amateur mathematicians, automatic theorem provers, AI tools, and the proof assistant language L…

What's new

26.09.2024

A pilot project in universal algebra to explore new ways to collaborate and use machine assistance? | What's new

Traditionally, mathematics research projects are conducted by a small number (typically one to five) of expert mathematicians, each of which are familiar enough with all aspects of the project that…

What's new

18.09.2024

Pointwise convergence of bilinear polynomial averages over the primes | What's new

Ben Krause, Hamed Mousavi, Joni Teräväinen, and I have just uploaded to the arXiv the paper “Pointwise convergence of bilinear polynomial averages over the primes”. This paper…

What's new

09.09.2024

A possible approach to finite time blowup for the Euler equations | What's new

Given a smooth compact Riemannian manifold $latex {M = (M,g)}&fg=000000$, the incompressible Euler equations can be written in abstract index notation as $latex \displaystyle \partial_t u^\alph…

What's new

04.09.2024

Planar point sets with forbidden four-point patterns and few distinct distances | What's new

I’ve just uploaded to the arXiv my paper “Planar point sets with forbidden $latex {4}&fg=000000$-point patterns and few distinct distance”. This (very) short paper was a bypro…

What's new

20.08.2024

Erdos problem #385, the parity problem, and Siegel zeroes | What's new

The Erdös problem site was created last year, and announced earlier this year on this blog. Every so often, I have taken a look at a random problem from the site for fun. A few times, I was ab…

What's new

10.08.2024

A result of Bui–Pratt–Zaharescu, and Erdös problem #437 | What's new

The following problem was posed by Erdös and Graham (and is listed as problem #437 on the Erdös problems website): Problem 1 Let $latex {1 \leq a_1 < \dots < a_k \leq x}&fg=0000…

What's new

03.08.2024

What are the odds, II: the Venezuelan presidential election | 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

09.07.2024

Dense sets of natural numbers with unusually large least common multiples | What's new

I’ve just uploaded to the arXiv my paper “Dense sets of natural numbers with unusually large least common multiples”. This short paper answers (in the negative) a somewhat obscure…

What's new

08.07.2024

A computation-outsourced discussion of zero density theorems for the Riemann zeta function | What's new

Many modern mathematical proofs are a combination of conceptual arguments and technical calculations. There is something of a tradeoff between the two: one can add more conceptual arguments to try …

What's new

03.07.2024

Salem prize now accepting nominations for 2024 | 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

23.06.2024

An abridged proof of Marton’s conjecture | What's new

[This post is dedicated to Luca Trevisan, who recently passed away due to cancer. Though far from his most significant contribution to the field, I would like to mention that, as with most of my ot…

What's new

21.05.2024

On product representations of squares | What's new

I’ve just uploaded to the arXiv my paper “On product representations of squares”. This short paper answers (in the negative) a (not very well known) question of Erdös. Namely…

What's new

26.04.2024

Erratum for “An inverse theorem for the Gowers U^{s+1}[N]-norm” | What's new

The purpose of this post is to report an erratum to the 2012 paper “An inverse theorem for the Gowers $latex {U^{s+1}[N]}&fg=000000$-norm” of Ben Green, myself, and Tamar Ziegler (p…

What's new

25.04.2024

Notes on the B+B+t theorem | What's new

A recent paper of Kra, Moreira, Richter, and Robertson established the following theorem, resolving a question of Erdös. Given a discrete amenable group $latex {G = (G,+)}&fg=000000$, and …

What's new

20.04.2024

Two announcements: AI for Math resources, and erdosproblems.com | 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

05.04.2024

Marton’s conjecture in abelian groups with bounded torsion | What's new

Tim Gowers, Ben Green, Freddie Manners, and I have just uploaded to the arXiv our paper “Marton’s conjecture in abelian groups with bounded torsion”. This paper fully resolves a c…

What's new

03.04.2024

AI Mathematical Olympiad – Progress Prize Competition now open | What's new

The first progress prize competition for the AI Mathematical Olympiad has now launched. (Disclosure: I am on the advisory committee for the prize.) This is a competition in which contestants submit…

What's new

18.03.2024

Talks at the JMM | What's new

Earlier this year, I gave a series of lectures at the Joint Mathematics Meetings at San Francisco. I am uploading here the slides for these talks: “Machine assisted proof” (Video here) …

What's new

11.12.2023

A generalized Cauchy-Schwarz inequality via the Gibbs variational formula | What's new

Let $latex {S}&fg=000000$ be a non-empty finite set. If $latex {X}&fg=000000$ is a random variable taking values in $latex {S}&fg=000000$, the Shannon entropy $latex {H[X]}&fg=00000…

What's new

06.12.2023

A slightly longer Lean 4 proof tour | What's new

In my previous post, I walked through the task of formally deducing one lemma from another in Lean 4. The deduction was deliberately chosen to be short and only showcased a small number of Lean tac…

What's new

19.11.2023

Formalizing the proof of PFR in Lean4 using Blueprint: a short tour | What's new

Since the release of my preprint with Tim, Ben, and Freddie proving the Polynomial Freiman-Ruzsa (PFR) conjecture over $latex {\mathbb F}_2$, I (together with Yael Dilles and Bhavik Mehta) have sta…

What's new

14.11.2023

On a conjecture of Marton | What's new

Tim Gowers, Ben Green, Freddie Manners, and I have just uploaded to the arXiv our paper “On a conjecture of Marton”. This paper establishes a version of the notorious Polynomial Freiman…

What's new

13.10.2023

A Maclaurin type inequality | What's new

I have just uploaded to the arXiv my paper “A Maclaurin type inequality”. This paper concerns a variant of the Maclaurin inequality for the elementary symmetric means $latex \displaysty…

What's new

13.10.2023

Bounding sums or integrals of non-negative quantities | What's new

A common task in analysis is to obtain bounds on sums $latex \displaystyle \sum_{n \in A} f(n)&fg=000000$ or integrals $latex \displaystyle \int_A f(x)\ dx &fg=000000$ where $latex {A}&…

What's new

19.09.2023

Undecidability of translational monotilings | What's new

Rachel Greenfeld and I have just uploaded to the arXiv our paper “Undecidability of translational monotilings”. This is a sequel to our previous paper in which we constructed a translat…

What's new

06.09.2023

Monotone non-decreasing sequences of the Euler totient function | What's new

I have just uploaded to the arXiv my paper “Monotone non-decreasing sequences of the Euler totient function”. This paper concerns the quantity $latex {M(x)}&fg=000000$, defined as t…

What's new

25.08.2023

Yoneda’s lemma as an identification of form and function: the case study of polynomials | What's new

As someone who had a relatively light graduate education in algebra, the import of Yoneda’s lemma in category theory has always eluded me somewhat; the statement and proof are simple enough, …

What's new

24.08.2023

A lower bound on the mean value of the Erd\”os-Hooley delta function | What's new

Kevin Ford, Dimitris Koukoulopolous and I have just uploaded to the arXiv our paper “A lower bound on the mean value of the Erdös-Hooley delta function”. This paper complements the…

What's new

15.08.2023

The convergence of an alternating series of Erdos, assuming the Hardy–Littlewood prime tuples conjecture | What's new

I have just uploaded to the arXiv my paper “The convergence of an alternating series of Erdös, assuming the Hardy–Littlewood prime tuples conjecture”. This paper concerns an …

What's new

30.06.2023

Adjoint Brascamp-Lieb inequalities | What's new

Jon Bennett and I have just uploaded to the arXiv our paper “Adjoint Brascamp-Lieb inequalities”. In this paper, we observe that the family of multilinear inequalities known as the Bras…

What's new

28.06.2023

Sumsets and entropy revisited | What's new

Ben Green, Freddie Manners and I have just uploaded to the arXiv our preprint “Sumsets and entropy revisited”. This paper uses entropy methods to attack the Polynomial Freiman-Ruzsa (PF…

What's new

21.06.2023

home›What's new›Science

What's new - Science

Quaternions and spherical trigonometry | What's new

On the distribution of eigenvalues of GUE and its minors at fixed index | What's new

AI for Math fund | What's new

On several irrationality problems for Ahmes series | What's new

Higher uniformity of arithmetic functions in short intervals II. Almost all intervals | What's new

The elephant in the room | What's new

Climbing the cosmic distance ladder: launching an instagram | What's new

The equational theories project: a brief tour | What's new

A pilot project in universal algebra to explore new ways to collaborate and use machine assistance? | What's new

Pointwise convergence of bilinear polynomial averages over the primes | What's new

A possible approach to finite time blowup for the Euler equations | What's new

Planar point sets with forbidden four-point patterns and few distinct distances | What's new

Erdos problem #385, the parity problem, and Siegel zeroes | What's new

A result of Bui–Pratt–Zaharescu, and Erdös problem #437 | What's new

What are the odds, II: the Venezuelan presidential election | What's new

Dense sets of natural numbers with unusually large least common multiples | What's new

A computation-outsourced discussion of zero density theorems for the Riemann zeta function | What's new

Salem prize now accepting nominations for 2024 | What's new

An abridged proof of Marton’s conjecture | What's new

On product representations of squares | What's new

Erratum for “An inverse theorem for the Gowers U^{s+1}[N]-norm” | What's new

Notes on the B+B+t theorem | What's new

Two announcements: AI for Math resources, and erdosproblems.com | What's new

Marton’s conjecture in abelian groups with bounded torsion | What's new

AI Mathematical Olympiad – Progress Prize Competition now open | What's new

Talks at the JMM | What's new

A generalized Cauchy-Schwarz inequality via the Gibbs variational formula | What's new

A slightly longer Lean 4 proof tour | What's new

Formalizing the proof of PFR in Lean4 using Blueprint: a short tour | What's new

On a conjecture of Marton | What's new

A Maclaurin type inequality | What's new

Bounding sums or integrals of non-negative quantities | What's new

Undecidability of translational monotilings | What's new

Monotone non-decreasing sequences of the Euler totient function | What's new

Yoneda’s lemma as an identification of form and function: the case study of polynomials | What's new

A lower bound on the mean value of the Erd\”os-Hooley delta function | What's new

The convergence of an alternating series of Erdos, assuming the Hardy–Littlewood prime tuples conjecture | What's new

Adjoint Brascamp-Lieb inequalities | What's new

Sumsets and entropy revisited | What's new

Salem Prize now accepting nominations | What's new

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