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Tamar Ziegler and I have just uploaded to the arXiv our paper, “The inverse conjecture for the Gowers norm over finite fields via the correspondence principle“, submitted to Analysis & PDE.  As announced a few months ago in this blog post, this paper establishes (most of) the inverse conjecture for the Gowers norm from an ergodic theory analogue of this conjecture (in a forthcoming paper by Vitaly Bergelson, Tamar Ziegler, and myself, which should be ready shortly), using a variant of the Furstenberg correspondence principle.  Our papers were held up for a while due to some unexpected technical difficulties arising in the low characteristic case; as a consequence, our paper only establishes the full inverse conjecture in the high characteristic case p \geq k, and gives a partial result in the low characteristic case p < k.

In the rest of this post, I would like to describe the inverse conjecture (in both combinatorial and ergodic forms), and sketch how one deduces one from the other via the correspondence principle (together with two additional ingredients, namely a statistical sampling lemma and a local testability result for polynomials).

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I’ve just uploaded to the arXiv my joint paper with Tim Austin, “On the testability and repair of hereditary hypergraph properties“, which has been submitted to Random Structures and Algorithms. In this paper we prove some positive and negative results for the testability (and the local repairability) of various properties of directed or undirected graphs and hypergraphs, which can be either monochromatic or multicoloured.

The negative results have already been discussed in a previous posting of mine, so today I will focus on the positive results. The property testing results here are finitary results, but it turns out to be rather convenient to use a certain correspondence principle (the hypergraph version of the Furstenberg correspondence principle) to convert the question into one about exchangeable probability measures on spaces of hypergraphs (i.e. on random hypergraphs whose probability distribution is invariant under exchange of vertices). Such objects are also closely related to the”graphons” and “hypergraphons” that emerge as graph limits, as studied by Lovasz-Szegedy, Elek-Szegedy, and others. Somewhat amusingly, once one does so, it then becomes convenient to keep track of objects indexed by vertex sets and how they are exchanged via the language of category theory, and in particular using the concept of a natural transformation to describe such objects as exchangeable measures, graph colourings, and local modification rules. I will try to sketch out some of these connections, after describing the main positive results.

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This month I have been at the Institute for Advanced Study, participating in their semester program on additive combinatorics. Today I gave a talk on my forthcoming paper with Tim Austin on the property testing of graphs and hypergraphs (I hope to make a preprint available here soon). There has been an immense amount of progress on these topics recently, based in large part on the graph and hypergraph regularity lemmas; but we have discovered some surprising subtleties regarding these results, namely a distinction between undirected and directed graphs, between graphs and hypergraphs, between partite hypergraphs and non-partite hypergraphs, and between monotone hypergraph properties and hereditary ones.

For simplicity let us first work with (uncoloured, undirected, loop-free) graphs G = (V,E). In the subject of graph property testing, one is given a property {\mathcal P} which any given graph G may or may not have. For example, {\mathcal P} could be one of the following properties:

  1. G is planar.
  2. G is four-colourable.
  3. G has a number of edges equal to a power of two.
  4. G contains no triangles.
  5. G is bipartite.
  6. G is empty.
  7. G is a complete bipartite graph.

We assume that the labeling of the graph is irrelevant. More precisely, we assume that whenever two graphs G, G’ are isomorphic, that G satisfies {\mathcal P} if and only if G’ satisfies {\mathcal P}. For instance, all seven of the graph properties listed above are invariant under graph isomorphism.

We shall think of G as being very large (so |V| is large) and dense (so |E| \sim |V|^2). We are interested in obtaining some sort of test that can answer the question “does G satisfy {\mathcal P}?” with reasonable speed and reasonable accuracy. By “reasonable speed”, we mean that we will only make a bounded number of queries about the graph, i.e. we only look at a bounded number k of distinct vertices in V (selected at random) and base our test purely on how these vertices are connected to each other in E. (We will always assume that the number of vertices in V is at least k.) By “reasonable accuracy”, we will mean that we specify in advance some error tolerance \varepsilon > 0 and require the following:

  1. (No false negatives) If G indeed satisfies {\mathcal P}, then our test will always (correctly) accept G.
  2. (Few false positives in the \varepsilon-far case) If G fails to satisfy {\mathcal P}, and is furthermore \varepsilon-far from satisfying {\mathcal P} in the sense that one needs to add or remove at least \varepsilon |V|^2 edges in G before {\mathcal P} can be satisfied, then our test will (correctly) reject G with probability at least \varepsilon.

When a test with the above properties exists for each given \varepsilon > 0 (with the number of queried vertices k being allowed to depend on \varepsilon), we say that the graph property {\mathcal P} is testable with one-sided error. (The general notion of property testing was introduced by Rubinfeld and Sudan, and first studied for graph properties by Goldreich, Goldwasser, and Ron; see this web page of Goldreich for further references and discussion.) The rejection probability \varepsilon is not very important in this definition, since if one wants to improve the success rate of the algorithm one can simply run independent trials of that algorithm (selecting fresh random vertices each time) in order to increase the chance that G is correctly rejected. However, it is intuitively clear that one must allow some probability of failure, since one is only inspecting a small portion of the graph and so cannot say with complete certainty whether the entire graph has the property {\mathcal P} or not. For similar reasons, one cannot reasonably demand to have a low false positive rate for all graphs that fail to obey {\mathcal P}, since if the graph is only one edge modification away from obeying {\mathcal P}, this modification is extremely unlikely to be detected by only querying a small portion of the graph. This explains why we need to restrict attention to graphs that are \varepsilon-far from obeying {\mathcal P}.

An example should illustrate this definition. Consider for instance property 6 above (the property that G is empty). To test whether a graph is empty, one can perform the following obvious algorithm: take k vertices in G at random and check whether they have any edges at all between them. If they do, then the test of course rejects G as being non-empty, while if they don’t, the test accepts G as being empty. Clearly there are no false negatives in this test, and if k is large enough depending on \varepsilon one can easily see (from the law of large numbers) that we will have few false positives if G is \varepsilon-far from being empty (i.e. if it has at least \varepsilon |V|^2 vertices). So the property of being empty is testable with one-sided error.

On the other hand, it is intuitively obvious that property 3 (having an number of edges equal to a power of 2) is not testable with one-sided error.

So it is reasonable to ask: what types of graph properties are testable with one-sided error, and which ones are not?

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In this second lecture, I wish to talk about the dichotomy between structure and randomness as it manifests itself in four closely related areas of mathematics:

  • Combinatorial number theory, which seeks to find patterns in unstructured dense sets (or colourings) of integers;
  • Ergodic theory (or more specifically, multiple recurrence theory), which seeks to find patterns in positive-measure sets under the action of a discrete dynamical system on probability spaces (or more specifically, measure-preserving actions of the integers {\Bbb Z});
  • Graph theory, or more specifically the portion of this theory concerned with finding patterns in large unstructured dense graphs; and
  • Ergodic graph theory, which is a very new and undeveloped subject, which roughly speaking seems to be concerned with the patterns within a measure-preserving action of the infinite permutation group S_\infty, which is one of several models we have available to study infinite “limits” of graphs.

The two “discrete” (or “finitary”, or “quantitative”) fields of combinatorial number theory and graph theory happen to be related to each other, basically by using the Cayley graph construction; I will give an example of this shortly. The two “continuous” (or “infinitary”, or “qualitative”) fields of ergodic theory and ergodic graph theory are at present only related on the level of analogy and informal intuition, but hopefully some more systematic connections between them will appear soon.

On the other hand, we have some very rigorous connections between combinatorial number theory and ergodic theory, and also (more recently) between graph theory and ergodic graph theory, basically by the procedure of viewing the infinitary continuous setting as a limit of the finitary discrete setting. These two connections go by the names of the Furstenberg correspondence principle and the graph correspondence principle respectively. These principles allow one to tap the power of the infinitary world (for instance, the ability to take limits and perform completions or closures of objects) in order to establish results in the finitary world, or at least to take the intuition gained in the infinitary world and transfer it to a finitary setting. Conversely, the finitary world provides an excellent model setting to refine one’s understanding of infinitary objects, for instance by establishing quantitative analogues of “soft” results obtained in an infinitary manner. I will remark here that this best-of-both-worlds approach, borrowing from both the finitary and infinitary traditions of mathematics, was absolutely necessary for Ben Green and I in order to establish our result on long arithmetic progressions in the primes. In particular, the infinitary setting is excellent for being able to rigorously define and study concepts (such as structure or randomness) which are much “fuzzier” and harder to pin down exactly in the finitary world.

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