Past Combinatorial Theory Seminar

Given a fixed poset $\mathcal P$, we say that a family $\mathcal F$ of subsets of $[n]$ is $\mathcal P$-free if it does not contain an (induced) copy of $\mathcal P$. And we say that $F$ is $\mathcal P$-saturated if it is maximal $\mathcal P$-free. How small can a $\mathcal P$-saturated family be? The smallest such size is the induced saturation number of $\mathcal P$, $\text{sat}^*(n, \mathcal P)$. Even for very small posets, the question of the growth speed of $\text{sat}^*(n,\mathcal P)$ seems to be hard. We present background on this problem and some recent results.

The $n$-queens problem asks how many ways there are to place $n$ queens on an $n \times n$ chessboard so that no two queens can attack one another, and the toroidal $n$-queens problem asks the same question where the board is considered on the surface of a torus. Let $Q(n)$ denote the number of $n$-queens configurations on the classical board and $T(n)$ the number of toroidal $n$-queens configurations. The toroidal problem was first studied in 1918 by Pólya who showed that $T(n)>0$ if and only if $n \equiv 1,5 \mod 6$. Much more recently Luria showed that $T(n)\leq ((1+o(1))ne^{-3})^n$ and conjectured equality when $n \equiv 1,5 \mod 6$. We prove this conjecture, prior to which no non-trivial lower bounds were known to hold for all (sufficiently large) $n \equiv 1,5 \mod 6$. We also show that $Q(n)\geq((1+o(1))ne^{-3})^n$ for all $n \in \mathbb{N}$ which was independently proved by Luria and Simkin and, combined with our toroidal result, completely settles a conjecture of Rivin, Vardi and Zimmerman regarding both $Q(n)$ and $T(n)$. 

In this talk we'll discuss our methods used to prove these results. A crucial element of this is translating the problem to one of counting matchings in a $4$-partite $4$-uniform hypergraph. Our strategy combines a random greedy algorithm to count `almost' configurations with a complex absorbing strategy that uses ideas from the methods of randomised algebraic construction and iterative absorption.

This is joint work with Peter Keevash.

We study a family of evolving directed random graphs that includes the directed preferential model and the directed uniform attachment model. The directed preferential model is of particular interest since it is known to produce scale-free graphs with regularly varying in-degree distribution. We start by describing the local weak limits for our family of random graphs in terms of randomly stopped continuous-time branching processes, and then use these limits to establish the asymptotic behavior of the corresponding PageRank distribution. We show that the limiting PageRank distribution decays as a power-law in both models, which is surprising for the uniform attachment model where the in-degree distribution has exponential tails. And even for the preferential attachment model, where the power-law hypothesis suggests that PageRank should follow a power-law, our result shows that the two tail indexes are different, with the PageRank distribution having a heavier tail than the in-degree distribution.

Let $A$ be drawn uniformly at random from the set of all $n \times n$ symmetric matrices with entries in $\{-1,1\}$. We show that $A$ is singular with probability at most $e^{-cn}$ for some absolute constant $c>0$, thereby resolving a well-known conjecture. This is joint work with Marcelo Campos, Marcus Michelen and Julian Sahasrabudhe.
 

Henstock and Macbeath asked in 1953 whether the Brunn-Minkowski inequality can be generalized to nonabelian locally compact groups; questions in the same line were also asked by Hrushovski, McCrudden, and Tao. We obtain here such an inequality and prove that it is sharp for helix-free locally compact groups, which includes real linear algebraic groups, Nash groups, semisimple Lie groups with finite center, solvable Lie groups, etc. If time allows I will also discuss some applications of this result. (Joint with Chieu-Minh Tran and Ruixiang Zhang)

We introduce a new method for studying stochastic processes in random graphs controlled by degree information, involving combining enumeration techniques with an abstract second moment argument. We use it to constructively resolve a conjecture of Füredi from 1988: with high probability, the random graph G(n,1/2) admits a friendly bisection of its vertex set, i.e., a partition of its vertex set into two parts whose sizes differ by at most one in which n-o(n) vertices have at least as many neighbours in their own part as across. This work is joint with Asaf Ferber, Matthew Kwan, Bhargav Narayanan, and Mehtaab Sawhney.

I'll consider recent results concerning the stability of the classic Brunn-Minkowski inequality. In particular, I will focus on the linear stability for homothetic sets. Resolving a conjecture of Figalli and Jerison, we showed there are constants $C,d>0$ depending only on $n$ such that for every subset $A$ of $\mathbb{R}^n$ of positive measure, if $|(A+A)/2 - A| \leq d |A|$, then $|co(A) - A| \leq C |(A+A)/2 - A|$ where $co(A)$ is the convex hull of $A$. The talk is based on joint work with Hunter Spink and Marius Tiba.

Suppose there are $n$ students in a class. But assume that not everybody is friends with everyone else, and there is a graph which determines the friendship structure. What is the chance that there are two friends in this class, both with birthdays on October 12? More generally, given a simple labelled graph $G_n$ on $n$ vertices, color each vertex with one of $c=c_n$ colors chosen uniformly at random, independent from other vertices. We study the question: what is the number of monochromatic edges of color 1?

As it turns out, the limiting distribution has three parts, the first and second of which are quadratic and linear functions of a homogeneous Poisson point process, and the third component is an independent Poisson. In fact, we show that any distribution limit must belong to the closure of this class of random variables. As an application, we characterize exactly when the limiting distribution is a Poisson random variable.

This talk is based on joint work with Bhaswar Bhattacharya and Somabha Mukherjee.

This is joint work with Paul Bourgade and Benjamin McKenna (Courant Institute, NYU).
The elastic manifold is a paradigmatic representative of the class of disordered elastic systems. These models describe random surfaces with rugged shapes resulting from a competition between random spatial impurities (preferring disordered configurations), on the one hand, and elastic self-interactions (preferring ordered configurations), on the other. The elastic manifold model is interesting because it displays a depinning phase transition and has a long history as a testing ground for new approaches in statistical physics of disordered media, for example for fixed dimension by Fisher (1986) using functional renormalization group methods, and in the high-dimensional limit by Mézard and Parisi (1992) using the replica method.
We study the topology of the energy landscape of this model in the Mézard-Parisi setting, and compute the (annealed) topological complexity both of total critical points and of local minima. Our main result confirms the recent formulas by Fyodorov and Le Doussal (2020) and allows to identify the boundary between simple and glassy phases. The core argument relies on the analysis of the asymptotic behavior of large random determinants in the exponential scale.

Consider an $n$ by $n$ random matrix $A$ with i.i.d entries. In this talk, we discuss some results on the magnitude of the smallest singular value of $A$, and, in particular, the problem of estimating the singularity probability when the entries of $A$ are discrete.

We study the set $L(G)$ of cycle lengths that appear in a sparse binomial random graph $G(n,c/n)$ and in a random $d$-regular graph $G_{n,d}$. We show in particular that for most values of $c$, for $G$ drawn from $G(n,c/n)$ the set $L(G)$ contains typically an interval $[\omega(1), (1-o(1))L_{\max}(G)]$, where $L_{\max}(G)$ is the length of a longest cycle (the circumference) of $G$. For the case of random $d$-regular graphs, $d\geq 3$ fixed, we obtain an accurate asymptotic estimate for the probability of $L(G)$ to contain a full interval $[k,n]$ for a fixed $k\geq 3$. Similar results are obtained also for the supercritical case $G(n,(1+\epsilon)/n)$, and for random directed graphs.
A joint work with Yahav Alon and Eyal Lubetzky.

Percolation models were originally introduced to describe the propagation of a fluid in a random medium. In dimension two, the percolation properties of a model are encoded by so-called crossing probabilities (probabilities that certain rectangles are crossed from left to right). In the eighties, Russo, Seymour and Welsh obtained general bounds on crossing probabilities for Bernoulli percolation (the most studied percolation model, where edges of a lattice are independently erased with some given probability $1-p$). These inequalities rapidly became central tools to analyze the critical behavior of the model.
In this talk I will present a new result which extends the Russo-Seymour-Welsh theory to general percolation measures satisfying two properties: symmetry and positive correlation. This is a joint work with Laurin Köhler-Schindler.

We study the model of randomly perturbed dense graphs, which is the union of any $n$-vertex graph $G_\alpha$ with minimum degree at least $\alpha n$ and the binomial random graph $G(n,p)$. In this talk, we shall examine the following central question in this area: to determine when $G_\alpha \cup G(n,p)$ contains $H$-factors, i.e. spanning subgraphs consisting of vertex disjoint copies of the graph $H$. We offer several new sharp and stability results.
This is joint work with Julia Böttcher, Olaf Parczyk, and Jozef Skokan.

In this talk we will discuss some classical and recent results on local limits of random graphs. It is well known that the limiting object of the local structure of the classical Erdos-Renyi random graph is a Galton-Watson tree. This can nicely be formalised in the language of Benjamini-Schramm or Aldous-Steele local weak convergence. Regarding local limits of sparse random planar graphs, there is a smooth transition from a Galton-Watson tree to a Skeleton tree. This talk is based on joint work with Michael Missethan.

We present an exponential improvement to the lower bound on diagonal Ramsey numbers for any fixed number of colors greater than two.
This is a joint work with David Conlon.
 

How does a colony of ants find the shortest path between its nest and a source of food without any means of communication other than the pheromones each ant leave behind itself?
In this joint work with Daniel Kious (Bath) and Bruno Schapira (Marseille), we introduce a new probabilistic model for this phenomenon. In this model, the nest and the source of food are two marked nodes in a finite graph. Ants perform successive random walks from the nest to the food, and the distribution of the $n$th walk depends on the trajectories of the $(n-1)$ previous walks through some linear reinforcement mechanism.
Using stochastic approximation methods, couplings with Pólya urns, and the electric conductances method for random walks on graphs (which I will explain on some simple examples), we prove that, depending on the exact reinforcement rule, the ants may or may not always find the shortest path(s) between their nest and the source food.

We discuss the behavior of geodesics in the continuous models of random geometry known as the Brownian map and the Brownian plane. We say that a point $x$ is a geodesic star with $m$ arms if $x$ is the endpoint of $m$ disjoint geodesics. We prove that the set of all geodesic stars with $m$ arms has dimension $5-m$, for $m=1,2,3,4$. This complements recents results of Miller and Qian, who derived upper bounds for these dimensions.

How much does the chromatic number of the random graph $G(n, 1/2)$ vary? Shamir and Spencer proved that it is contained in some sequence of intervals of length about $n^{1/2}$. Alon improved this slightly to $n^{1/2} / \log n$. Until recently, however, no lower bounds on the fluctuations of the chromatic number of $G(n, 1/2)$ were known, even though the question was raised by Bollobás many years ago. I will talk about the main ideas needed to prove that, at least for infinitely many $n$, the chromatic number of $G(n, 1/2)$ is not concentrated on fewer than $n^{1/2-o(1)}$ consecutive values.
I will also discuss the Zigzag Conjecture, made recently by Bollobás, Heckel, Morris, Panagiotou, Riordan and Smith: this proposes that the correct concentration interval length 'zigzags' between $n^{1/4+o(1)}$ and $n^{1/2+o(1)}$, depending on $n$.
Joint work with Oliver Riordan.

The title of the talk corresponds to a family of interesting random processes, which includes lazy random walks on graphs and much beyond them. Often, a key step in analysing such processes is to estimate their spectral gaps (ie. the difference between two largest eigenvalues). It is thus of interest to understand what else about the chain we can know from the spectral gap. We will present a simple comparison idea that often gives us the best possible estimates. In particular, we re-obtain and improve upon several known results on hitting, meeting, and intersection times; return probabilities; and concentration inequalities for time averages. We then specialize to the graph setting, and obtain sharp inequalities in that setting. This talk is based on work that has been in progress for far too long with Yuval Peres.

In this talk we will consider the extremal values of the stationary distribution of the sparse directed configuration model. Under the assumption of linear $(2+\eta)$-moments on the in-degrees and of bounded out-degrees, we obtain tight comparisons between the maximum value of the stationary distribution and the maximum in-degree. Under the further assumption that the order statistics of the in-degrees have power-law behavior, we show that the upper tail of the stationary distribution also has power-law behavior with the same index. Moreover, these results extend to the PageRank scores of the model, thus confirming a version of the so-called power-law hypothesis. Joint work with Xing Shi Cai, Pietro Caputo and Matteo Quattropani.

The classical Turán problem asks: given a graph $H$, how many edges can an $4n$-vertex graph have while containing no isomorphic copy of $H$? By viewing $(k+1)$-uniform hypergraphs as $k$-dimensional simplicial complexes, we can ask a topological version (first posed by Nati Linial): given a $k$-dimensional simplicial complex $S$, how many facets can an $n$-vertex $k$-dimensional simplicial complex have while containing no homeomorphic copy of $S$? Until recently, little was known for $k > 2$. In this talk, we give an answer for general $k$, by way of dependent random choice and the combinatorial notion of a trace-bounded hypergraph. Joint work with Jason Long and Bhargav Narayanan.

Self-similar fragmentation processes are random models for particles that are subject to successive fragmentations. When the index of self-similarity is negative the fragmentations intensify as the masses of particles decrease. This leads to a shattering phenomenon, where the initial particle is entirely reduced to dust - a set of zero-mass particles - in finite time which is what we call the extinction time. Equivalently, these extinction times may be seen as heights of continuous compact rooted trees or scaling limits of heights of sequences of discrete trees. Our objective is to set up precise bounds for the large time asymptotics of the tail distributions of these extinction times.

A uniform spanning tree of $\mathbb{Z}^4$ can be thought of as the "uniform measure" on trees of $\mathbb{Z}^4$. The past of 0 in the uniform spanning tree is the finite component that is disconnected from infinity when 0 is deleted from the tree. We establish the logarithmic corrections to the probabilities that the past contains a path of length $n$, that it has volume at least $n$ and that it reaches the boundary of the box of side length $n$ around 0. Dimension 4 is the upper critical dimension for this model in the sense that in higher dimensions it exhibits "mean-field" critical behaviour. An important part of our proof is the study of the Newtonian capacity of a loop erased random walk in 4 dimensions. This is joint work with Tom Hutchcroft.

We prove that bounded-degree expanders with non-negative Ollivier-Ricci curvature do not exist, thereby solving a long-standing open problem suggested by Naor and Milman and publicized by Ollivier (2010). In fact, this remains true even if we allow for a vanishing proportion of large degrees, large eigenvalues, and negatively-curved edges. To establish this, we work directly at the level of Benjamini-Schramm limits. More precisely, we exploit the entropic characterization of the Liouville property on stationary random graphs to show that non-negative curvature and spectral expansion are incompatible 'at infinity'. We then transfer this result to finite graphs via local weak convergence and a relative compactness argument. We believe that this 'local weak limit' approach to mixing properties of Markov chains will have many other applications.

It is a basic fact of convexity that the volume of convex bodies is a polynomial, whose coefficients contain many familiar geometric parameters as special cases. A fundamental result of convex geometry, the Alexandrov-Fenchel inequality, states that these coefficients are log-concave. This proves to have striking connections with other areas of mathematics: for example, the appearance of log-concave sequences in many combinatorial problems may be understood as a consequence of the Alexandrov-Fenchel inequality and its algebraic analogues.

There is a long-standing problem surrounding the Alexandrov-Fenchel inequality that has remained open since the original works of Minkowski (1903) and Alexandrov (1937): in what cases is equality attained? In convexity, this question corresponds to the solution of certain unusual isoperimetric problems, whose extremal bodies turn out to be numerous and strikingly bizarre. In combinatorics, an answer to this question would provide nontrivial information on the type of log-concave sequences that can arise in combinatorial applications. In recent work with Y. Shenfeld, we succeeded to settle the equality cases completely in the setting of convex polytopes. I will aim to describe this result, and to illustrate its potential combinatorial implications through a question of Stanley on the combinatorics of partially ordered sets.

We investigate a graph theoretic analog of geodesic geometry. In a graph $G=(V,E)$ we consider a system of paths $P=\{P_{u,v}| u,v\in V\}$ where $P_{u,v}$ connects vertices $u$ and $v$. This system is consistent in that if vertices $y,z$ are in $P_{u,v}$, then the sub-path of $P_{u,v}$ between them coincides with $P_{y,z}$. A map $w:E\to(0,\infty)$ is said to induce $P$ if for every $u,v\in V$ the path $P_{u,v}$ is $w$-geodesic. We say that $G$ is metrizable if every consistent path system is induced by some such $w$. As we show, metrizable graphs are very rare, whereas there exist infinitely many 2-connected metrizable graphs.
 

I will introduce product structure theory of graphs and show how families of graphs that have such a structure admit short adjacency labeling scheme and small induced universal graphs. Time permitting, I will talk about another recent application of product structure theory, namely vertex ranking (coloring).

We consider the random directed graph $D(n,p)$ with vertex set $\{1,2,…,n\}$ in which each of the $n(n-1)$ possible directed edges is present independently with probability $p$. We are interested in the strongly connected components of this directed graph. A phase transition for the emergence of a giant strongly connected component is known to occur at $p = 1/n$, with critical window $p = 1/n + \lambda n-4/3$ for $\lambda \in \mathbb{R}$. We show that, within this critical window, the strongly connected components of $D(n,p)$, ranked in decreasing order of size and rescaled by $n-1/3$, converge in distribution to a sequence $(C_1,C_2,\ldots)$ of finite strongly connected directed multigraphs with edge lengths which are either 3-regular or loops. The convergence occurs in the sense of an $L^1$ sequence metric for which two directed multigraphs are close if there are compatible isomorphisms between their vertex and edge sets which roughly preserve the edge lengths. Our proofs rely on a depth-first exploration of the graph which enables us to relate the strongly connected components to a particular spanning forest of the undirected Erdős-Rényi random graph $G(n,p)$, whose scaling limit is well understood. We show that the limiting sequence $(C_1,C_2,\ldots)$ contains only finitely many components which are not loops. If we ignore the edge lengths, any fixed finite sequence of 3-regular strongly connected directed multigraphs occurs with positive probability.

The dimer model, a classical model of statistical mechanics, is the uniform distribution on perfect matchings of a graph. In two dimensions, one can define an associated height function which turns the model into a random surface (with specified boundary conditions). In the 1960s, Kasteleyn and Temperley/Fisher found an exact "solution" to the model, computing the correlations in terms of a matrix called the Kasteleyn matrix. This exact solvability was the starting point for the breakthrough work of Kenyon (2000) who proved that the centred height function converges to the Dirichlet (or zero boundary conditions) Gaussian free field. This was the first proof of conformal invariance in statistical mechanics.

In this talk, I will focus on a natural modification of the model where one allows the vertices on the boundary of the graph to remain unmatched: this is the so-called monomer-dimer model, or dimer model with free boundary conditions. The main result that we obtain is that the scaling limit of the height function of the monomer-dimer model in the upper half-plane is the Neumann (or free boundary conditions) Gaussian free field. Key to this result is a somewhat miraculous random walk representation for the inverse Kasteleyn matrix, which I hope to discuss.

Joint work with Marcin Lis (Vienna) and Wei Qian (Paris).

It is sometimes possible to represent a complicated polytope as a projection of a much simpler polytope. To quantify this phenomenon, the extension complexity of a polytope $P$ is defined to be the minimum number of facets in a (possibly higher-dimensional) polytope from which $P$ can be obtained as a (linear) projection. In this talk, we discuss some results on the extension complexity of random $d$-dimensional polytopes (obtained as convex hulls of random points on either on the unit sphere or in the unit ball), and on the extension complexity of polygons with all vertices on a common circle. Joint work with Matthew Kwan and Yufei Zhao

Let $X$ and $Y$ be two $n$-vertex graphs. Identify the vertices of $Y$ with $n$ people, any two of whom are either friends or strangers (according to the edges and non-edges in $Y$), and imagine that these people are standing one at each vertex of $X$. At each point in time, two friends standing at adjacent vertices of $X$ may swap places, but two strangers may not. The friends-and-strangers graph $FS(X,Y)$ has as its vertex set the collection of all configurations of people standing on the vertices of $X$, where two configurations are adjacent when they are related via a single friendly swap. This provides a common generalization for the famous 15-puzzle, transposition Cayley graphs of symmetric groups, and early work of Wilson and of Stanley.
I will describe several recent results and open problems addressing the extremal and typical aspects of the notion, focusing on the result that the threshold probability for connectedness of $FS(X,Y)$ for two independent binomial random graphs $X$ and $Y$ in $G(n,p)$ is $p=p(n)=n-1/2+o(1)$.
Joint work with Colin Defant and Noah Kravitz.

I will discuss how to construct cycles of many different lengths in graphs, in particular answering the following two problems on odd and even cycles. Erdős and Hajnal asked in 1981 whether the sum of the reciprocals of the odd cycle lengths in a graph diverges as the chromatic number increases, while, in 1984, Erdős asked whether there is a constant $C$ such that every graph with average degree at least $C$ contains a cycle whose length is a power of 2.

We generalize the fact that all graphs omitting a fixed finite bipartite graph can be uniformly approximated by rectangles (Alon-Fischer-Newman, Lovász-Szegedy), showing that hypergraphs omitting a fixed finite $(k+1)$-partite $(k+1)$-uniform hypergraph can be approximated by $k$-ary cylinder sets. In particular, in the decomposition given by hypergraph regularity one only needs the first $k$ levels: such hypergraphs can be approximated using sets of vertices, sets of pairs, and so on up to sets of $k$-tuples, and on most of the resulting $k$-ary cylinder sets, the density is either close to 0 or close to 1. Moreover, existence of such approximations uniformly under all measures on the vertices is a characterization. Our proof uses a combination of analytic, combinatorial and model-theoretic methods, and involves a certain higher arity generalization of the epsilon-net theorem from VC-theory.  Joint work with Henry Towsner.

The Schmidt subspace theorem is a far-reaching generalization of the Thue-Siegel-Roth theorem in diophantine approximation. In this talk I will give an interpretation of Schmidt's subspace theorem in terms of the dynamics of diagonal flows on homogeneous spaces and describe how the exceptional subspaces arise from certain rational Schubert varieties associated to the family of linear forms through the notion of Harder-Narasimhan filtration and an associated slope formalism. This geometric understanding opens the way to a natural generalization of Schmidt's theorem to the setting of diophantine approximation on submanifolds of $GL_d$, which is our main result. In turn this allows us to recover and generalize the main results of Kleinbock and Margulis regarding diophantine exponents of submanifolds. I will also mention an application to diophantine approximation on Lie groups. Joint work with Nicolas de Saxcé.

This talk will focus on the following two related problems:
    (1) What is the minimum number of edges in a graph containing all $n$-vertex planar graphs as subgraphs? A simple construction of Babai, Erdos, Chung, Graham, and Spencer (1982) has $O(n^{3/2})$ edges, which is the best known upper bound.
    (2) What is the minimum number of *vertices* in a graph containing all $n$-vertex planar graphs as *induced* subgraphs? Here steady progress has been achieved over the years, culminating in a $O(n^{4/3})$ bound due to Bonamy, Gavoille, and Pilipczuk (2019).
    As it turns out, a bound of $n^{1+o(1)}$ can be achieved for each of these two problems. The two constructions are somewhat different but are based on a common technique. In this talk I will first give a gentle introduction to the area and then sketch these constructions. The talk is based on joint works with Vida Dujmović, Louis Esperet, Cyril Gavoille, Piotr Micek, and Pat Morin.

What is fairness, and to what extent is it practically achievable? I'll talk about a simple mathematical model under which one might hope to understand such questions. Red and blue points occur as independent homogeneous Poisson processes of equal intensity in Euclidean space, and we try to match them to each other. We would like to minimize the sum of a some function (say, a power, $\gamma$) of the distances between matched pairs. This does not make sense, because the sum is infinite, so instead we satisfy ourselves with minimizing *locally*. If the points are interpreted as agents who would like to be matched as close as possible, the parameter $\gamma$ encodes a measure of fairness - large $\gamma$ means that we try to avoid occasional very bad outcomes (long edges), even if that means inconvenience to others - small $\gamma$ means everyone is in it for themselves.
    In dimension 1 we have a reasonably complete picture, with a phase transition at $\gamma=1$. For $\gamma<1$ there is a unique minimal matching, while for $\gamma>1$ there are multiple matchings but no stationary solution. In higher dimensions, even existence is not clear in all cases.

A spanning tree of $G$ is a subgraph of $G$ with the same vertex set as $G$ that is a tree. In 1981, McKay proved an asymptotic result regarding the number of spanning trees in random $k$-regular graphs, showing that the number of spanning trees $\kappa_1(G_n)$ in a random $k$-regular graph on $n$ vertices satisfies $\lim_{n \to \infty} (\kappa_1(G_n))^{1/n} = c_{1,k}$ in probability, where $c_{1,k} = (k-1)^{k-1} (k^2-2k)^{-(k-2)/2}$.

In this talk we will discuss a high-dimensional of the matching model for simplicial complexes, known as random Steiner complexes. In particular, we will prove a high-dimensional counterpart of McKay's result and discuss the local limit of such random complexes. 
Based on a joint work with Lior Tenenbaum. 

We study the minimum total weight of a disk triangulation using any number of vertices out of $\{1,..,n\}$ where the boundary is fixed and the $n \choose 3$ triangles have independent rate-1 exponential weights. We show that, with high probability, the minimum weight is equal to $(c+o(1))n-1/2\log n$ for an explicit constant $c$. Further, we prove that, with high probability, the minimum weights of a homological filling and a homotopical filling of the cycle $(123)$ are both attained by the minimum weight disk triangulation. We will discuss a related open problem concerning simple-connectivity of random simplicial complexes, where a similar phenomenon is conjectured to hold. Based on joint works with Itai Benjamini, Eyal Lubetzky, and Zur Luria.

In long-range percolation on $\mathbb{Z}^d$, each potential edge $\{x,y\}$ is included independently at random with probability roughly $\beta\|x-y\|-d-\alpha$, where $\alpha > 0$ controls how long-range the model is and $\beta > 0$ is an intensity parameter. The smaller $\alpha$ is, the easier it is for very long edges to appear. We are normally interested in fixing $\alpha$ and studying the phase transition that occurs as $\beta$ is increased and an infinite cluster emerges. Perhaps surprisingly, the phase transition for long-range percolation is much better understood than that of nearest neighbour percolation, at least when $\alpha$ is small: It is a theorem of Noam Berger that if $\alpha < d$ then the phase transition is continuous, meaning that there are no infinite clusters at the critical value of $\beta$. (Proving the analogous result for nearest neighbour percolation is a notorious open problem!) In my talk I will describe a new, quantitative proof of Berger's theorem that yields power-law upper bounds on the distribution of the cluster of the origin at criticality.
    As a part of this proof, I will describe a new universal inequality stating that on any graph, the maximum size of a percolation cluster is of the same order as its median with high probability. This inequality can also be used to give streamlined new proofs of various classical results on e.g. Erdős-Rényi random graphs, which I will hopefully have time to talk a little bit about also.

I will present work in progress with D. Gayet and F. Pouran (Grenoble) to establish Russo-Seymour-Welsh (RSW) estimates for 2d statistical mechanics models that do not satisfy the FKG inequality. RSW states that critical percolation has no characteristic length, in the sense that large rectangles are crossed by an open path with a probability that is bounded below by a function of their shape, but uniformly in their size; this ensures the polynomial decay of many relevant quantities and opens the way to deeper understanding of the critical features of the model. All the standard proofs of RSW rely on the FKG inequality, i.e. on the positive correlation between increasing events; we establish the stability of RSW under small perturbations that do not preserve FKG, which extends it for instance to the high-temperature anti-ferromagnetic Ising model.

A connected matching is a matching contained in a connected component. A well-known method due to Łuczak reduces problems about monochromatic paths and cycles in complete graphs to problems about monochromatic matchings in almost complete graphs. We show that these can be further reduced to problems about monochromatic connected matchings in complete graphs.
    
I will describe Łuczak's reduction, introduce the new reduction, and mention potential applications of the improved method.

Let $X$ be a random variable, taking values in $\{1,…,n\}$, with standard deviation $\sigma$ and let $f_X$ be its probability generating function. Pemantle conjectured that if $\sigma$ is large and $f_X$ has no roots close to 1 in the complex plane then $X$ must approximate a normal distribution. In this talk, I will discuss a complete resolution of Pemantle's conjecture. As an application, we resolve a conjecture of Ghosh, Liggett and Pemantle by proving a multivariate central limit theorem for, so called, strong Rayleigh distributions. I will also discuss how these sorts of results shed light on random variables that arise naturally in combinatorial settings. This talk is based on joint work with Marcus Michelen.

In 1943, Hadwiger conjectured that every graph with no Kt minor is $(t-1)$-colorable for every $t\geq 1$. In the 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t(\log t)^{1/2})$ and hence is $O(t(\log t)^{1/2)}$-colorable. Recently, Norin, Song and I showed that every graph with no $K_t$ minor is $O(t(\log t)^\beta)$-colorable for every $\beta > 1/4$, making the first improvement on the order of magnitude of the $O(t(\log t)^{1/2})$ bound. Here we show that every graph with no $K_t$ minor is $O(t (\log t)^\beta)$-colorable for every $\beta > 0$; more specifically, they are $O(t (\log \log t)^6)$-colorable.

We will be interested in the structure of random typical minimal factorizations of the n-cycle into transpositions, which are factorizations of $(1,\ldots,n)$ as a product of $n-1$ transpositions. We shall establish a phase transition when a certain amount of transpositions have been read one after the other. One of the main tools is a limit theorem for two-type Bienaymé-Galton-Watson trees conditioned on having given numbers of vertices of both types, which is of independent interest. This is joint work with Valentin Féray.

Let $r>3$ be an integer and consider the following game on the complete graph $K_n$ for $n$ a multiple of $r$: Two players, Maker and Breaker, alternately claim previously unclaimed edges of $K_n$ such that in each turn Maker claims one and Breaker claims $b$ edges. Maker wins if her graph contains a $K_r$-factor, that is a collection of $n/r$ vertex-disjoint copies of $K_r$, and Breaker wins otherwise. In other words, we consider the $b$-biased $K_r$-factor Maker-Breaker game. We show that the threshold bias for this game is of order $n^2/(r+2)$. This makes a step towards determining the threshold bias for making bounded-degree spanning graphs and extends a result of Allen, Böttcher, Kohayakawa, Naves and Person who resolved the case $r=3$ or $4$ up to a logarithmic factor.
    Joint work with Rajko Nenadov.

I will introduce recent work on the two- and three-dimensional uniform spanning trees (USTs) that establish the laws of these random objects converge under rescaling in a space whose elements are measured, rooted real trees, continuously embedded into Euclidean space. (In the three-dimensional case, the scaling result is currently only known along a particular scaling sequence.) I will also discuss various properties of the intrinsic metrics and measures of the limiting spaces, including their Hausdorff dimension, as well as the scaling limits of the random walks on the two- and three-dimensional USTs. In the talk, I will attempt to emphasise where the differences lie between the two cases, and in particular the additional challenges that arise when it comes to the three-dimensional model.
    The two-dimensional results are joint with Martin Barlow (UBC) and Takashi Kumagai (Kyoto). The three-dimensional results are joint with Omer Angel (UBC) and Sarai Hernandez-Torres (UBC).

A hereditary graph property is a class of finite graphs closed under isomorphism and induced subgraphs. Given a hereditary graph property $H$, the speed of $H$ is the function which sends an integer n to the number of distinct elements in $H$ with underlying set $\{1,...,n\}$. Not just any function can occur as the speed of hereditary graph property. Specifically, there are discrete "jumps" in the possible speeds. Study of these jumps began with work of Scheinerman and Zito in the 90's, and culminated in a series of papers from the 2000's by Balogh, Bollobás, and Weinreich, in which essentially all possible speeds of a hereditary graph property were characterized. In contrast to this, many aspects of this problem in the hypergraph setting remained unknown. In this talk we present new hypergraph analogues of many of the jumps from the graph setting, specifically those involving the polynomial, exponential, and factorial speeds. The jumps in the factorial range turned out to have surprising connections to the model theoretic notion of mutual algebricity, which we also discuss. This is joint work with Chris Laskowski.

Let $G_n$ be a sequence of finite, simple, connected, regular graphs with degrees tending to infinity and let $T_n$ be a uniformly drawn spanning tree of $G_n$. In joint work with Yuval Peres we show that the local limit of $T_n$ is the $\text{Poisson}(1)$ branching process conditioned to survive forever (that is, the asymptotic frequency of the appearance of any small subtree is given by the branching process). The proof is based on electric network theory and I hope to show most of it.

Liouville quantum gravity (LQG) is a theory of random fractal surfaces with origin in the physics literature in the 1980s. Most literature is about LQG with matter central charge $c\in (-\infty,1]$. We study a discretization of LQG which makes sense for all $c\in (-\infty,25)$. Based on a joint work with Gwynne, Pfeffer, and Remy.