C4

I will talk about intersection theory over any globally valued field and how it is connected to some model-theoretic problems.

I will describe a non-commutative version of the Zariski topology and explain how to use it to produce a functorial spectrum for all derived rings. If time permits I will give some examples and show how a weak form of Gelfand duality for non-commutative rings can be deduced from this. This work is in collaboration with Simone Murro and Matteo Capoferri.

Networks provide a powerful language to model interacting systems by representing their units as nodes and the interactions between them as links. Interactions can be connotated in several ways, such as binary/weighted, undirected/directed, etc. In the present talk, we focus on the positive/negative connotation - modelling trust/distrust, alliance/enmity, friendship/conflict, etc. - by considering the so-called signed networks. Rooted in the psychological framework of the balance theory, the study of signed networks has found application in fields as different as biology, ecology, economics. Here, we approach it from the perspective of statistical physics by extending the framework of Exponential Random Graph Models to the class of binary un/directed signed networks and employing it to assess the significance of frustrated patterns in real-world networks. As our results reveal, it critically depends on i) the considered system and ii) the employed benchmark. For what concerns binary directed networks, instead, we explore the relationship between frustration and reciprocity and suggest an alternative interpretation of balance in the light of directionality. Finally,  leveraging the ERGMs framework, we propose an unsupervised algorithm to obtain statistically validated projections of bipartite signed networks, according to which any, two nodes sharing a statistically significant number of concordant (discordant) motifs are connected by a positive (negative) edge, and we investigate signed structures at the mesoscopic scale by evaluating the tendency of a configuration to be either `traditionally' or `relaxedly' balanced.

While the brain has long been conceptualized as a network of neurons connected by synapses, attempts to describe the connectome using established models in network science have yielded conflicting outcomes, leaving the architecture of neural networks unresolved. Here, we analyze eight experimentally mapped connectomes, finding that the degree and the strength distribution of the underlying networks cannot be described by random nor scale-free models. Rather, the node degrees and strengths are well approximated by lognormal distributions, whose emergence lacks a mechanistic model in the context of networks. Acknowledging the fact that the brain is a physical network, whose architecture is driven by the spatially extended nature of its neurons, we analytically derive the multiplicative process responsible for the lognormal neuron length distribution, arriving to a series of empirically falsifiable predictions and testable relationships that govern the degree and the strength of individual neurons. The lognormal network characterizing the connectome represents a novel architecture for network science, that bridges critical gaps between neural structure and function, with unique implications for brain dynamics, robustness, and synchronization.

Our Random Walkers Induced temporal Graphs (RWIG) model generates temporal graph sequences based on M independent, random walkers that traverse an underlying graph as a function of time. Co-location of walkers at a given node and time defines an individual-level contact. RWIG is shown to be a realistic model for temporal human contact graphs.   

A key idea is that a random walk on a Markov graph executes the Markov process. Each of the M walkers traverses the same set of nodes (= states in the Markov graph), but with own transition probabilities (in discrete time) or rates (in continuous time). Hence, the Markov transition probability matrix P_j reflects the policy of motion of walker w_j. RWIG is analytically feasible: we derive closed form solutions for the probability distribution of contact graphs.

Usually, human mobility networks are inferred through measurements of timeseries of contacts between individuals. We also discuss this “inverse RWIG problem”, which aims to determine the parameters in RWIG (i.e. the set of probability transfer matrices P₁, P₂, ..., P_M and the initial probability state vectors s₁[0], ...,s_M[0] of walkers w₁,w₂, ...,w_M in discrete time), given a timeseries of contact graphs.

This talk is based on the article:
Almasan, A.-D., Shvydun, S., Scholtes, I. and P. Van Mieghem, 2025, "Generating Temporal Contact Graphs Using Random Walkers", IEEE Transactions on Network Science and Engineering, to appear.

A base-g Niven number is an integer divisible by the sum of its digits in base-g. We show that any sufficiently large integer can be written as the sum of three base-3 Niven numbers, and comment on the extension to other bases. This is an application of the circle method, which we use to count the number of ways an integer can be written as the sum of three integers with fixed, near-average, digit sum. 

A generalisation of Wiles’ famous modularity theorem, the Fontaine-Mazur conjecture, predicts that two dimensional representations of the absolute Galois group of the rationals, with a few specific properties, exactly correspond to those representations coming from classical modular forms. Under some mild hypotheses, this is now a theorem of Kisin. In this talk, I will explain how one can p-adically interpolate the objects on both sides of this correspondence to construct an eigensurface and “trianguline” Galois deformation space, as well as outline a new approach to proving a theorem of Emerton, that these spaces are often isomorphic.

The class of henselian valued fields with non-discrete value group is not well-understood. In 2018, Koenigsmann conjectured that a list of seven natural axioms describes a complete axiomatisation of $\mathbb{Q}_p^{ab}$, the maximal extension of the $p$-adic numbers $\mathbb{Q}_p$ with abelian Galois group, which is an example of such a valued field. Informed by the recent work of Jahnke-Kartas on the model theory of perfectoid fields, we formulate an eighth axiom (the discriminant property) that is not a consequence of the other seven. Revisiting work by Koenigsmann (the Galois characterisation of $\mathbb{Q}_p$) and Jahnke-Kartas, we give a uniform treatment of their underlying method. In particular, we highlight how this method yields short, non-standard model-theoretic proofs of known results (e.g. finite extensions of perfectoid fields are perfectoid).

The security of many widely used communication systems hinges on the presumed difficulty of factoring integers or computing discrete logarithms. However, Shor's celebrated algorithm from 1994 demonstrated that quantum computers can perform these tasks in polynomial time. In 2023, Regev proposed an even faster quantum algorithm for factoring integers. Unfortunately, the correctness of his new method is conditional on an ad hoc number-theoretic conjecture. Using tools from analytic number theory, we establish a result in the direction of Regev's conjecture. This enables us to design a provably correct quantum algorithm for factoring and solving the discrete logarithm problem, whose efficiency is comparable to Regev's approach. In this talk, we will give an accessible account of these developments.

Measurement-Based Quantum Computation (MBQC) is a model of quantum computation driven by measurements instead of unitary gates.   In 2D it is capable of supporting universal quantum computations.   Interestingly, while all measurements are local, the computational output involves non local observables.   We will use the simpler case of 1D MBQC to illustrate how these features can be captured by ideas from gauge theory and extended TQFT. We will also explain  MBQC from the perspective of the extended Hilbert space construction in gauge theories, in which the entanglement edge modes play the role of the logical qubit.