### Online reading seminar for Zhang’s “bounded gaps between primes”

In a recent paper, Yitang Zhang has proven the following theorem: Theorem 1 (Bounded gaps between primes) There exists a natural number such that there are infinitely many pairs of distinct primes with . Zhang obtained the explicit value of for . A polymath project has been proposed to lower this value and also to […]

### Reading seminar 6: “Stable group theory and approximate subgroups”, by Ehud Hrushovski

This is the last reading seminar of this quarter for the Hrushovski paper. Anush Tserunyan continued working through her notes on stable theories. We introduced the key notion of non-forking extensions (in the context of stable theories, at least) of types when constants are added; these are extensions which are “as generic as possible” with […]

### Reading seminar 5: “Stable group theory and approximate subgroups”, by Ehud Hrushovski

After a one-week hiatus, we are resuming our reading seminar of the Hrushovski paper. This week, we are taking a break from the paper proper, and are instead focusing on the subject of stable theories (or more precisely, -stable theories), which form an important component of the general model-theoretic machinery that the Hrushovski paper uses. […]

### Reading seminar 4: “Stable group theory and approximate subgroups”, by Ehud Hrushovski

This week, Henry Towsner concluded his portion of reading seminar of the Hrushovski paper, by discussing (a weaker, simplified version of) main model-theoretic theorem (Theorem 3.4 of Hrushovski), and described how this theorem implied the combinatorial application in Corollary 1.2 of Hrushovski. The presentation here differs slightly from that in Hrushovski’s paper, for instance by […]

### Reading seminar 3: “Stable group theory and approximate subgroups”, by Ehud Hrushovski

This week, Henry Towsner continued some model-theoretic preliminaries for the reading seminar of the Hrushovski paper, particularly regarding the behaviour of wide types, leading up to the main model-theoretic theorem (Theorem 3.4 of Hrushovski) which in turn implies the various combinatorial applications (such as Corollary 1.2 of Hrushovski). Henry’s notes can be found here. A […]

### Reading seminar 2: “Stable group theory and approximate subgroups”, by Ehud Hrushovski

At UCLA we just concluded our third seminar in our reading of “Stable group theory and approximate subgroups” by Ehud Hrushovski. In this seminar, Isaac Goldbring made some more general remarks about universal saturated models (extending the discussion from the previous seminar), and then Henry Towsner gave some preliminaries on Kiesler measures, in preparation for […]

### Reading seminar: “Stable group theory and approximate subgroups”, by Ehud Hrushovski

One of my favorite open problems, which I have blogged about in the past, is that of establishing (or even correctly formulating) a non-commutative analogue of Freiman’s theorem. Roughly speaking, the question is this: given a finite set in a non-commutative group which is of small doubling in the sense that the product set is […]

### A reading seminar on density Hales-Jewett

As part of the polymath1 project, I would like to set up a reading seminar on this blog for the following three papers and notes: H. Furstenberg, Y. Katznelson, “A density version of the Hales-Jewett theorem for k=3“, Graph Theory and Combinatorics (Cambridge, 1988). Discrete Math. 75 (1989), no. 1-3, 227–241. R. McCutcheon, “The conclusion […]

### Polymath proposal: upper bounding the de Bruijn-Newman constant

Building on the interest expressed in the comments to this previous post, I am now formally proposing to initiate a “Polymath project” on the topic of obtaining new upper bounds on the de Bruijn-Newman constant . The purpose of this post is to describe the proposal and discuss the scope and parameters of the project. […]

### Estimation of the Type III sums

This is the final continuation of the online reading seminar of Zhang’s paper for the polymath8 project. (There are two other continuations; this previous post, which deals with the combinatorial aspects of the second part of Zhang’s paper, and this previous post, that covers the Type I and Type II sums.) The main purpose of […]