We construct and study the asymptotic properties of "perfect t-embeddings" of uniformly weighted hexagon graphs. Hexagon graphs are subgraphs of the honeycomb lattice, and the corresponding dimer model is equivalent to the model of uniformly random lozenge tilings of the hexagon. We provide exact formulas describing the perfect t-embeddings of these graphs, and we use these to prove the convergence of naturally associated discrete surfaces (coming from the "origami maps") to a maximal surface in Minkowski space carrying the conformal structure of the limiting Gaussian free field (GFF). The emergence of such a maximal surface is predicted to hold for a large class of dimer models by Chelkak, Laslier, and Russkikh. In addition, we check all conditions of a theorem of Chelkak, Laslier, and Russkikh which uses perfect t-embeddings to prove convergence of height fluctuations to the GFF, and thus we complete give a new proof, via t-embeddings, of convergence to the GFF. This is based on joint work with Marianna Russkikh and Tomas Berggren.

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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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.

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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. 

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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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.

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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.

Zoom Link

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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.

Zoom Link

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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.

Zoom Link

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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.

Zoom Link

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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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