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We identify the size of the largest connected component in a sub-critical inhomogeneous random graph with a kernel of preferential attachment type. The component is polynomial in the graph size with an explicitly given exponent, which is strictly larger than the exponent for the largest degree in the graph. This is in stark contrast to the behaviour of inhomogeneous random graphs with a kernel of rank one. The proof uses local approximation by branching random walks going beyond the weak local limit and large deviation results on killed branching random walks.

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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Since suggested by Tur\'an in 1941, determining the Tur\'an density of hypergraphs has been a notoriously difficult problem at the center of extremal combinatorics. Roughly speaking, the Tur\'an density $\pi(F)$ of a hypergraph $F$ is the threshold of the edge density above which large hypergraphs are guaranteed to contain a copy of $F$. Over the past decade, some progress was made in understanding the set of all Tur\'an densities, i.e.,

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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.

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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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.

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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