L6

Hermitian matrix models with non-trivial covariance will be introduced. The Kontsevich Model is the prime example, which was used to prove Witten's conjecture about the generating function of intersection numbers of the moduli space $\overline{\mathcal{M}}_{g,n}$. However, we will discuss these models in a different direction, namely as a quantum field theory. As a formal matrix model,  the correlation functions of these models have a unique combinatorial/perturbative interpretation in the sense of Feynman diagrams. In particular, the additional structure (in comparison to ordinary quantum field theories) gives the possibility to compute exact expressions, which are resummations of infinitely many Feynman diagrams. For the easiest topologies, these exact expressions (given by implicitly defined functions) will be presented and discussed. If time remains, higher topologies are discussed by a connection to Topological Recursion.

In this talk, we use tools from representation theory and symmetric function theory to compute correlations of eigenvalues of unitary invariant ensembles. This approach provides a route to write exact formulae for the correlations, which further allows us to extract large matrix asymptotics and study universal properties.

TBA

Free fermion chains are particularly simple exactly solvable models. Despite this, typically one can find closed expressions for physically important correlators only in certain asymptotic limits. For a particular class of chains, I will show that we can apply Day's formula and Gorodetsky's formula for Toeplitz determinants with rational generating function. This leads to simple closed expressions for determinantal order parameters and the characteristic polynomial of the correlation matrix. The latter result allows us to prove that the ground state of the chain has an exact matrix-product state representation.

For a family $A$ in $\{0,...,k\}^n$, its deletion shadow is the set obtained from $A$ by deleting from any of its vectors one coordinate. Given the size of $A$, how should we choose $A$ to minimise its deletion shadow? And what happens if instead we may delete only a coordinate that is zero? We discuss these problems, and give an exact solution to the second problem.

Given a tournament with $d{n \choose 3}$ cycles of length three, how many cycles of length four must there be? Linial and Morgenstern (2016) conjectured that the minimum is asymptotically attained by "blowing up" a transitive tournament and orienting the edges randomly within the parts. This is reminiscent of the tight examples for the famous Triangle and Clique Density Theorems of Razborov, Nikiforov and Reiher. We prove the conjecture for $d \geq \frac{1}{36}$ using spectral methods. We also show that the family of tight examples is more complex than expected and fully characterise it for $d \geq \frac{1}{16}$. Joint work with Timothy Chan, Andrzej Grzesik and Daniel Král'.

Consider all planar graphs on n vertices. What is the smallest graph that contains them all as induced subgraphs? We provide an explicit construction of such a graph on $n^{4/3+o(1)}$ vertices, which improves upon the previous best upper bound of $n^{2+o(1)}$, obtained in 2007 by Gavoille and Labourel.

In this talk, we will gently introduce the audience to the notion of so-called universal graphs (graphs containing all graphs of a given family as induced subgraphs), and devote some time to a key lemma in the proof. That lemma comes from a recent breakthrough by Dujmovic et al. regarding the structure of planar graphs, and has already many interesting consequences - we hope the audience will be able to derive more. This is based on joint work with Cyril Gavoille and Michal Pilipczuk.