In this programmatic talk, we will sketch both a conceptual and formal framework for reasoning about the notion of algorithm. This framework will arise from the analysis we will make of the relationships existing between the notion of algorithm and other similar (but still different) notions, like that of computation and that of program. We will first show that the Turing-Church thesis concerning effective computability is not sufficient to capture the notion of algorithm, as it identifies programs which are intensionally different. We will then show the limits of the existing models of computation in capturing some basic construction processes that we are willing to call algorithmic. In order to solve this problem, we propose a formalisation of the notion of model of computation on the base of which we claim that the notion of algorithm could eventually be analyzed. This approach centered around the dynamics of program execution, reconciles the more mechanical view of computation (such as formalized by Turing machines and automata) with the logical view - as it in particular stems from a generalization of Jean-Yves Girard's Geometry of Interaction programme.[-]

I will report on progress with Daniel Allcock on the ”Monstrous Proposal”, namely the conjecture: Complex hyperbolic 13-space, modulo a particular discrete group, and with orbifold structure changed in a simple way, has fundamental group equal to (MxM)(semidirect)2, where M is the Monster finite simple group. Our progress is a proof that this orbifold fundamental group has generators that satisfy defining relations for (MxM)(semidirect) 2. It follows that either the monstrous proposal is true, or else the orbifold fundamental group collapses to Z/2. The generators and relations are extremely natural from the complex hyperbolic perspective, keeping hopes high for the conjecture. [-]

A meromorphic differential on a Riemann surface is said to be real-normalized if all its periods are real. Real-normalized differentials on Riemann surfaces of given genus with prescribed orders of their poles form real orbifolds whose topology is closely related to that of moduli spaces of Riemann surfaces with marked points. We propose a combinatorial model for the real normalized differentials with a single order 2 pole and use it to analyze certain ergodic properties of the corresponding absolute period foliation. It is a joint work with Igor Krichever and Sergey Lando.[-]

4. The zonoid ring and the nonarchimedean world. In the last lecture I will explain a ring-theoretical interpretation of the computations in random algebraic geometry, using a recently discovered ring structure on special convex bodies. This leads to the construction of a probabilistic version of Schubert calculus. In the final part of the lecture I will export some of the ideas from the previous lectures to the case $\mathbb{K}=\mathbb{Q}_{p}$, leaving with some open questions.[-]

While message-passing neural networks (MPNNs) are the most popular architectures for graph learning, their expressive power is inherently limited. In order to gain increased expressive power while retaining efficiency, several recent works apply MPNNs to subgraphs of the original graph. As a starting point, the talk will introduce the Equivariant Subgraph Aggregation Networks (ESAN) architecture, which is a representative framework for this class of methods. In ESAN, each graph is represented as a set of subgraphs, selected according to a predefined policy. The sets of subgraphs are then processed using an equivariant architecture designed specifically for this purpose. I will then present a recent follow-up work that revisits the symmetry group suggested in ESAN and suggests that a more precise choice can be made if we restrict our attention to a specific popular family of subgraph selection policies. We will see that using this observation, one can make a direct connection between subgraph GNNs and Invariant Graph Networks (IGNs), thus providing new insights into subgraph GNNs' expressive power and design space.[-]