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Various graph properties can exist for "local" or for "global" reasons. For example, some graphs have high chromatic number because they contain a large clique, so their high chromatic number is witnessed locally. But other graphs have high chromatic number and high girth, and in these graphs, high chromatic number is witnessed only globally. 

These two examples indicate that high chromatic number can appear as either a local or a global feature of a graph. With good definitions of "local" and "global," we could ask for a local-global analysis of any graph property: given a graph exhibiting a certain property, is this property a local or a global feature of this graph? 

In this talk, I will discuss a recent framework for understanding local vs global structure of graphs, focusing on decompositions that represent global structure. The talk will include recent results and also ideas, questions, and open problems related to the implications of this model of global structure. My goal is to foster discussion and exchange of ideas during this talk, so feel free to come ready to share your thoughts and reactions! 

This talk is based on joint work with Reinhard Diestel and Paul Knappe.

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In this talk, we will discuss several extremal problems involving concepts like MaxCut, minimum semidefinite rank, the Lovász theta function, and the extension complexity of polytopes. We will show how a bipartite generalization of Alon and Szegedy’s nearly orthogonal vectors implies strong bounds for these problems. Some of the results that will be presented are in joint work with Letzter and Sudakov, or Janzer and Sudakov.

The theory of combinatorial limits is a rapidly developing area of mathematics, which provides analytic tools to study large combinatorial objects (e.g., graphs representing social networks). These analytic methods have led to new ways to cope with notoriously difficult extremal combinatorics questions and established new links between analysis, combinatorics, ergodic theory, group theory, probability theory and statistics. The theory was also the subject of the 2021 Abel Prize lecture of Lovász entitled "Continuous limits of finite structures".

The course will present basic concepts of the theory of combinatorial limits related to various combinatorial objects such as graphs, permutations, and hypergraphs, and discuss analytic representations of their limits. We will discuss how the theory of combinatorial limits is related to regularity decompositions and how its analytic tools can be applied to various problems in computer science and mathematics, in particular, in extremal combinatorics where Razborov's flag algebra method has led to advances on long-standing open problems (with solutions of the Erdős-Rademacher Problem and the Erdős Pentagon Problem being among the first results obtained using the method). We will demonstrate how the flag algebra arguments can be applied both directly and in a computer-assisted way, including non-asymptotic settings, e.g., to compute particular Ramsey numbers.

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Random multiplicative functions are objects studied by mainly number theorists and probabilists in recent years. I will give an introduction and update on this rapidly developing area. Some open questions have a strong probabilistic combinatorics flavor that might be of interest to people in the audience. 

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M-Lipschitz functions on sufficiently good "expander" graphs typically exhibit small fluctuations, giving sharp bounds on the typical range of such functions, assuming M is not too large. We prove that the same
conclusion holds under a relaxed expansion condition and for larger M, using a combination of Sapozhenko's graph container methods and entropy methods. In this talk, I aim to discuss our result and some context, some
elements of the proof, and some open problems. This is joint work with Lina Li and Jinyoung Park.

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For a bipartite graph $H$, its linear threshold is the smallest real number $\sigma$ such that every bipartite graph $G = (U \sqcup V, E)$ with unbalanced parts $|V| \gtrsim |U|^\sigma$ and without a copy of $H$ must have a linear number of edges $|E| \lesssim |V|$. We prove that the linear threshold of the complete bipartite subdivision graph $K_{s,t}'$ is at most $\sigma_s = 2 - 1/s$. Moreover, we show that any $\sigma < \sigma_s$ is less than the linear threshold of $K_{s,t}'$ for sufficiently large $t$ (depending on $s$ and $\sigma$). In this talk, I will discuss the proof of this result and some consequences in incidence geometry. Joint work with Lili Ködmön and Anqi Li.

Pipe dreams are tiling models originally introduced to study objects related to the Schubert calculus and K-theory of the Grassmannian. They can also be viewed as ensembles of random lattice walks with various interaction constraints. In our quest to understand what the maximal and typical algebraic objects look like, we revealed some interesting permutons. The proofs use the theory of the Totally Asymmetric Simple Exclusion Process (TASEP). Deeper connections with free fermion 6 vertex models and domino tilings of the Aztec diamond and Alternating Sign Matrices allow us to describe the extreme cases of the original algebraic problem. This is based on joint work with A. H. Morales, L. Petrov, D. Yeliussizov.

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The theory of combinatorial limits is a rapidly developing area of mathematics, which provides analytic tools to study large combinatorial objects (e.g., graphs representing social networks). These analytic methods have led to new ways to cope with notoriously difficult extremal combinatorics questions and established new links between analysis, combinatorics, ergodic theory, group theory, probability theory and statistics. The theory was also the subject of the 2021 Abel Prize lecture of Lovász entitled "Continuous limits of finite structures".

The course will present basic concepts of the theory of combinatorial limits related to various combinatorial objects such as graphs, permutations, and hypergraphs, and discuss analytic representations of their limits. We will discuss how the theory of combinatorial limits is related to regularity decompositions and how its analytic tools can be applied to various problems in computer science and mathematics, in particular, in extremal combinatorics where Razborov's flag algebra method has led to advances on long-standing open problems (with solutions of the Erdős-Rademacher Problem and the Erdős Pentagon Problem being among the first results obtained using the method). We will demonstrate how the flag algebra arguments can be applied both directly and in a computer-assisted way, including non-asymptotic settings, e.g., to compute particular Ramsey numbers.

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The disordered ferromagnet is a disordered version of the ferromagnetic Ising model in which the coupling constants are quenched random, chosen independently from a distribution on the non-negative reals. A ground configuration is an infinite-volume configuration whose energy cannot be reduced by finite modifications. It is a long-standing challenge to ascertain whether the disordered ferromagnet on the Z^D lattice admits non-constant ground configurations. When D=2, the problem is equivalent to the existence of bigeodesics in first-passage percolation, so a negative answer is expected. We provide a positive answer in dimensions D>=4, when the distribution of the coupling constants is sufficiently concentrated.

The talk will discuss the problem and its background, and present ideas from the proof. No previous familiarity with the topic will be assumed. Based on joint work of with Shoni Gilboa and Ron Peled.

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The theory of combinatorial limits is a rapidly developing area of mathematics, which provides analytic tools to study large combinatorial objects (e.g., graphs representing social networks). These analytic methods have led to new ways to cope with notoriously difficult extremal combinatorics questions and established new links between analysis, combinatorics, ergodic theory, group theory, probability theory and statistics. The theory was also the subject of the 2021 Abel Prize lecture of Lovász entitled "Continuous limits of finite structures".

The course will present basic concepts of the theory of combinatorial limits related to various combinatorial objects such as graphs, permutations, and hypergraphs, and discuss analytic representations of their limits. We will discuss how the theory of combinatorial limits is related to regularity decompositions and how its analytic tools can be applied to various problems in computer science and mathematics, in particular, in extremal combinatorics where Razborov's flag algebra method has led to advances on long-standing open problems (with solutions of the Erdős-Rademacher Problem and the Erdős Pentagon Problem being among the first results obtained using the method). We will demonstrate how the flag algebra arguments can be applied both directly and in a computer-assisted way, including non-asymptotic settings, e.g., to compute particular Ramsey numbers.

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A temporal random geometric graph is a random geometric graph in which all edges are endowed with a uniformly random time-stamp, representing the time of interaction between vertices. In such graphs, paths with increasing time stamps indicate the propagation of information. We determine a threshold for the existence of monotone increasing paths between all pairs of vertices in temporal random geometric graphs. The results reveal that temporal connectivity appears at a significantly larger edge density than simple connectivity of the underlying random geometric graph. This is in contrast with Erdős-Rényi random graphs in which the thresholds for temporal connectivity and simple connectivity are of the same order of magnitude. Our results hold for a family of "soft" random geometric graphs as well as the standard random geometric graph. 

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The theory of combinatorial limits is a rapidly developing area of mathematics, which provides analytic tools to study large combinatorial objects (e.g., graphs representing social networks). These analytic methods have led to new ways to cope with notoriously difficult extremal combinatorics questions and established new links between analysis, combinatorics, ergodic theory, group theory, probability theory and statistics. The theory was also the subject of the 2021 Abel Prize lecture of Lovász entitled "Continuous limits of finite structures".

The course will present basic concepts of the theory of combinatorial limits related to various combinatorial objects such as graphs, permutations, and hypergraphs, and discuss analytic representations of their limits. We will discuss how the theory of combinatorial limits is related to regularity decompositions and how its analytic tools can be applied to various problems in computer science and mathematics, in particular, in extremal combinatorics where Razborov's flag algebra method has led to advances on long-standing open problems (with solutions of the Erdős-Rademacher Problem and the Erdős Pentagon Problem being among the first results obtained using the method). We will demonstrate how the flag algebra arguments can be applied both directly and in a computer-assisted way, including non-asymptotic settings, e.g., to compute particular Ramsey numbers.

The theory of combinatorial limits is a rapidly developing area of mathematics, which provides analytic tools to study large combinatorial objects (e.g., graphs representing social networks). These analytic methods have led to new ways to cope with notoriously difficult extremal combinatorics questions and established new links between analysis, combinatorics, ergodic theory, group theory, probability theory and statistics. The theory was also the subject of the 2021 Abel Prize lecture of Lovász entitled "Continuous limits of finite structures".

The course will present basic concepts of the theory of combinatorial limits related to various combinatorial objects such as graphs, permutations, and hypergraphs, and discuss analytic representations of their limits. We will discuss how the theory of combinatorial limits is related to regularity decompositions and how its analytic tools can be applied to various problems in computer science and mathematics, in particular, in extremal combinatorics where Razborov's flag algebra method has led to advances on long-standing open problems (with solutions of the Erdős-Rademacher Problem and the Erdős Pentagon Problem being among the first results obtained using the method). We will demonstrate how the flag algebra arguments can be applied both directly and in a computer-assisted way, including non-asymptotic settings, e.g., to compute particular Ramsey numbers.

The theory of combinatorial limits is a rapidly developing area of mathematics, which provides analytic tools to study large combinatorial objects (e.g., graphs representing social networks). These analytic methods have led to new ways to cope with notoriously difficult extremal combinatorics questions and established new links between analysis, combinatorics, ergodic theory, group theory, probability theory and statistics. The theory was also the subject of the 2021 Abel Prize lecture of Lovász entitled "Continuous limits of finite structures".

The course will present basic concepts of the theory of combinatorial limits related to various combinatorial objects such as graphs, permutations, and hypergraphs, and discuss analytic representations of their limits. We will discuss how the theory of combinatorial limits is related to regularity decompositions and how its analytic tools can be applied to various problems in computer science and mathematics, in particular, in extremal combinatorics where Razborov's flag algebra method has led to advances on long-standing open problems (with solutions of the Erdős-Rademacher Problem and the Erdős Pentagon Problem being among the first results obtained using the method). We will demonstrate how the flag algebra arguments can be applied both directly and in a computer-assisted way, including non-asymptotic settings, e.g., to compute particular Ramsey numbers.

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The theory of combinatorial limits is a rapidly developing area of mathematics, which provides analytic tools to study large combinatorial objects (e.g., graphs representing social networks). These analytic methods have led to new ways to cope with notoriously difficult extremal combinatorics questions and established new links between analysis, combinatorics, ergodic theory, group theory, probability theory and statistics. The theory was also the subject of the 2021 Abel Prize lecture of Lovász entitled "Continuous limits of finite structures".

The course will present basic concepts of the theory of combinatorial limits related to various combinatorial objects such as graphs, permutations, and hypergraphs, and discuss analytic representations of their limits. We will discuss how the theory of combinatorial limits is related to regularity decompositions and how its analytic tools can be applied to various problems in computer science and mathematics, in particular, in extremal combinatorics where Razborov's flag algebra method has led to advances on long-standing open problems (with solutions of the Erdős-Rademacher Problem and the Erdős Pentagon Problem being among the first results obtained using the method). We will demonstrate how the flag algebra arguments can be applied both directly and in a computer-assisted way, including non-asymptotic settings, e.g., to compute particular Ramsey numbers.

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The theory of combinatorial limits is a rapidly developing area of mathematics, which provides analytic tools to study large combinatorial objects (e.g., graphs representing social networks). These analytic methods have led to new ways to cope with notoriously difficult extremal combinatorics questions and established new links between analysis, combinatorics, ergodic theory, group theory, probability theory and statistics. The theory was also the subject of the 2021 Abel Prize lecture of Lovász entitled "Continuous limits of finite structures".

The course will present basic concepts of the theory of combinatorial limits related to various combinatorial objects such as graphs, permutations, and hypergraphs, and discuss analytic representations of their limits. We will discuss how the theory of combinatorial limits is related to regularity decompositions and how its analytic tools can be applied to various problems in computer science and mathematics, in particular, in extremal combinatorics where Razborov's flag algebra method has led to advances on long-standing open problems (with solutions of the Erdős-Rademacher Problem and the Erdős Pentagon Problem being among the first results obtained using the method). We will demonstrate how the flag algebra arguments can be applied both directly and in a computer-assisted way, including non-asymptotic settings, e.g., to compute particular Ramsey numbers.

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The theory of combinatorial limits is a rapidly developing area of mathematics, which provides analytic tools to study large combinatorial objects (e.g., graphs representing social networks). These analytic methods have led to new ways to cope with notoriously difficult extremal combinatorics questions and established new links between analysis, combinatorics, ergodic theory, group theory, probability theory and statistics. The theory was also the subject of the 2021 Abel Prize lecture of Lovász entitled "Continuous limits of finite structures".

The course will present basic concepts of the theory of combinatorial limits related to various combinatorial objects such as graphs, permutations, and hypergraphs, and discuss analytic representations of their limits. We will discuss how the theory of combinatorial limits is related to regularity decompositions and how its analytic tools can be applied to various problems in computer science and mathematics, in particular, in extremal combinatorics where Razborov's flag algebra method has led to advances on long-standing open problems (with solutions of the Erdős-Rademacher Problem and the Erdős Pentagon Problem being among the first results obtained using the method). We will demonstrate how the flag algebra arguments can be applied both directly and in a computer-assisted way, including non-asymptotic settings, e.g., to compute particular Ramsey numbers.

The theory of combinatorial limits is a rapidly developing area of mathematics, which provides analytic tools to study large combinatorial objects (e.g., graphs representing social networks). These analytic methods have led to new ways to cope with notoriously difficult extremal combinatorics questions and established new links between analysis, combinatorics, ergodic theory, group theory, probability theory and statistics. The theory was also the subject of the 2021 Abel Prize lecture of Lovász entitled "Continuous limits of finite structures".

The course will present basic concepts of the theory of combinatorial limits related to various combinatorial objects such as graphs, permutations, and hypergraphs, and discuss analytic representations of their limits. We will discuss how the theory of combinatorial limits is related to regularity decompositions and how its analytic tools can be applied to various problems in computer science and mathematics, in particular, in extremal combinatorics where Razborov's flag algebra method has led to advances on long-standing open problems (with solutions of the Erdős-Rademacher Problem and the Erdős Pentagon Problem being among the first results obtained using the method). We will demonstrate how the flag algebra arguments can be applied both directly and in a computer-assisted way, including non-asymptotic settings, e.g., to compute particular Ramsey numbers.

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