## Final Report Summary - BAG (Benjamini-Schramm approximation of Groups and Graphings)

Large networks have appeared everywhere in the last decades: in our life (internet and social networks, like the facebook), multiple sciences (from molecular biology to neural networks and statistical physics) and in many subfields of mathematics (graph theory, probability theory, group theory, dynamics). The basic, novel challenge when facing such a large network is how to obtain information about it. The primary way is sampling. This reveals the local statistics of a sparse graph, the distribution of r-balls for every r. A graph property is forcible if there is a local statistics such that every graph with this statistics (or close to it) has the property. The project aims to answer the following general problems:

Which properties are forcible?

Given a forcible coloring property of a graph (graphing) can we find a proper coloring efficiently?

Kun has spent six years in North America. He has become widely recognized among American mathematicians and computer scientists. He returned to Europe after this and managed to rebuild his research connections. He has a perfect research profile in order to get more acquainted with the groups at the Renyi Institute open to Kun's interdisciplinary research line. The scientist in charge will be Gabor Tardos, a leading figure in discrete mathematics.

The proposal is at the crossroads of the following three fields:

1. Sparse graphs and computer science

2. Dynamics and measured group theory

3. Graph limit theory

The main objectives regarding transfer of knowledge are to revitalize the connection of the institute to modern computer science by Kun's presence, especially to two popular topics: Constraint Satisfaction Problems and the study of pseudorandom structures related to derandomization, to connect Kun's research in an organic way to already existing research directions at the institute and to integrate Kun in the mathematical life of Hungary and the ERA.

The most important result of the project is the proof of Bowen's conjecture on Kazhdan groups by Kun. This shows, in particular, that expansion properties of large networks can be enforced by local conditions. This is a promising tool to solve the main problem of the field, the construction of a nonsofic (inapproximable) group. The result has several applications from ergodic theory to topology and graph theory reproving the theorems of Freedman and Hastings, Naor, and L. M. Lovasz.

Kun has submitted the paper.

Kun and Szegedy gave an analytic approach to the famous dichotomy conjecture for Constraint Satisfaction Problems. This is a classical topic for interaction between algebra, analysis and computer science. The paper is already published.

Kun and Dadush have given a novel deterministic algorithm for one of the most basic geometric problems, the Approximate Closest Vector Problem. Their approach is based on a random sparsification that can be derandomized using the ideas on pseudorandomness: this is another application to computer science in this project. The paper is accepted.

Kun has improved the Gaboriau-Lyons solution to the dynamical von Neumann problem on non-amenable groups. His result was achieved via a measurable version of the Lovasz Local Lemma. This is the first application of the celebrated Moser-Tardos algorithm in a measurable setting. He has also solved Monod's problem on geometric random subgroups. The paper is submitted.

Abert, Csikvari, Frenkel and Kun prove that the normalized logarithm of the number of matchings in a graph is (locally) estimable. Researchers from many areas paid attention to this paper. It is already published.

Kun has been very active in the Hungarian mathematical community. Besides giving talks to professors at many Hungarian universities he also gave talks to students on his research topic. Moreover, he has become the organizer of the mathematics seminar of the most important college in sciences, the Bolyai College of the Eotvos University. He has given a couple of talks to highschool students, too, and organizes an annual career workshop for PhD students in mathematics at the Eotvos University.

Kun's reintegration in the Hungarian mathematical community is the best shown by the Junior Prize of the Hungary Academy of Sciences received by Kun in 2014. Altogether, the project has been very successful both from the scientific and the dissemination perspective.

See the project homepage for details: http://www.renyi.hu/bag.html

Which properties are forcible?

Given a forcible coloring property of a graph (graphing) can we find a proper coloring efficiently?

Kun has spent six years in North America. He has become widely recognized among American mathematicians and computer scientists. He returned to Europe after this and managed to rebuild his research connections. He has a perfect research profile in order to get more acquainted with the groups at the Renyi Institute open to Kun's interdisciplinary research line. The scientist in charge will be Gabor Tardos, a leading figure in discrete mathematics.

The proposal is at the crossroads of the following three fields:

1. Sparse graphs and computer science

2. Dynamics and measured group theory

3. Graph limit theory

The main objectives regarding transfer of knowledge are to revitalize the connection of the institute to modern computer science by Kun's presence, especially to two popular topics: Constraint Satisfaction Problems and the study of pseudorandom structures related to derandomization, to connect Kun's research in an organic way to already existing research directions at the institute and to integrate Kun in the mathematical life of Hungary and the ERA.

The most important result of the project is the proof of Bowen's conjecture on Kazhdan groups by Kun. This shows, in particular, that expansion properties of large networks can be enforced by local conditions. This is a promising tool to solve the main problem of the field, the construction of a nonsofic (inapproximable) group. The result has several applications from ergodic theory to topology and graph theory reproving the theorems of Freedman and Hastings, Naor, and L. M. Lovasz.

Kun has submitted the paper.

Kun and Szegedy gave an analytic approach to the famous dichotomy conjecture for Constraint Satisfaction Problems. This is a classical topic for interaction between algebra, analysis and computer science. The paper is already published.

Kun and Dadush have given a novel deterministic algorithm for one of the most basic geometric problems, the Approximate Closest Vector Problem. Their approach is based on a random sparsification that can be derandomized using the ideas on pseudorandomness: this is another application to computer science in this project. The paper is accepted.

Kun has improved the Gaboriau-Lyons solution to the dynamical von Neumann problem on non-amenable groups. His result was achieved via a measurable version of the Lovasz Local Lemma. This is the first application of the celebrated Moser-Tardos algorithm in a measurable setting. He has also solved Monod's problem on geometric random subgroups. The paper is submitted.

Abert, Csikvari, Frenkel and Kun prove that the normalized logarithm of the number of matchings in a graph is (locally) estimable. Researchers from many areas paid attention to this paper. It is already published.

Kun has been very active in the Hungarian mathematical community. Besides giving talks to professors at many Hungarian universities he also gave talks to students on his research topic. Moreover, he has become the organizer of the mathematics seminar of the most important college in sciences, the Bolyai College of the Eotvos University. He has given a couple of talks to highschool students, too, and organizes an annual career workshop for PhD students in mathematics at the Eotvos University.

Kun's reintegration in the Hungarian mathematical community is the best shown by the Junior Prize of the Hungary Academy of Sciences received by Kun in 2014. Altogether, the project has been very successful both from the scientific and the dissemination perspective.

See the project homepage for details: http://www.renyi.hu/bag.html