We consider random graph models where graphs are generated by connecting not only pairs of nodes by edges but also larger subsets of
nodes by copies of small atomic subgraphs of arbitrary topology. More specifically we consider canonical and microcanonical ensembles
corresponding to constraints placed on the counts and distributions of atomic subgraphs and derive general expressions for the entropy of such
models. We also introduce a procedure that enables the distributions of multiple atomic subgraphs to be combined resulting in more coarse
grained models. As a result we obtain a general class of models that can be parametrized in terms of basic building blocks and their
distributions that includes many widely used models as special cases. These models include random graphs with arbitrary distributions of subgraphs (Karrer & Newman PRE 2010, Bollobas et al. RSA 2011), random hypergraphs, bipartite models, stochastic block models, models of multilayer networks and their degree corrected and directed versions. We show that the entropy expressions for all these models can be derived from a single expression that is characterized by the symmetry groups of their atomic subgraphs.

We will discuss some problems around independence of l in compatible systems of Galois representations, mostly focusing on the independence of l of algebraic monodromy groups. We will explain how these problems fit into the context of the Langlands program, and present results both in characteristic zero and in positive characteristic settings.

Understanding the size of the rational points on a curve of higher genus is one of the major open problems in the theory of Diophantine equations. In this talk I will discuss the related problem of understanding how close together rational points can get. I will also discuss the relation to the subject of (generalised) Wieferich primes.

The goal of this talk is to introduce a way to use the philosophy of Random Matrix Theory to understand, pose, and maybe even solve problems about p-adic matrices.

A long time ago, Grothendieck made some conjectures. This has resulted in some things.

Discussion of https://arxiv.org/abs/1911.07895

Discussion of https://arxiv.org/abs/1812.04716

Discussin of https://arxiv.org/abs/1908.09858

Real networks are highly clustered (large number of short cycles) in contrast with their random counterparts. The Erdős–Rényi model and the Configuration model will generate networks with a tree like structure, a feature rarely observed in real networks. This means that traditional random networks are a poor choice as null models for real networks. Can we do better than that? Maximum entropy random graph ensembles are the natural choice to generate such networks. By introducing a bias with respect to the number of short cycles in a degree constrained graph, we aim to get a random graph model with a tuneable number of short cycles [1,2]. Nevertheless, the story is not so simple. In the same way random unclustered graphs present undesired topology, highly clustered ones will do as well if one is not careful with the scaling of the control parameters relative to the system size. Additionally the techniques to generate and sample numerically from general biased degree constrained graph ensembles will also be discussed. The topological transition has an important impact on the computational cost to sample graphs from these ensembles. To take it one step further, a general approach using the eigenvalues of the adjacency matrix rather than just the number of short cycles will also be presented, [2].

[1] López, Fabián Aguirre, et al. "Exactly solvable random graph ensemble with extensively many short cycles." Journal of Physics A: Mathematical and Theoretical 51.8 (2018): 085101.
[2] López, Fabián Aguirre, and Anthony CC Coolen. "Imaginary replica analysis of loopy regular random graphs." Journal of Physics A: Mathematical and Theoretical 53.6 (2020): 065002.