We will survey several cryptographic notions of computation over encrypted data (e.g., fully homomorphic encryption), as well as their instantiations from lattices. The talk will focus on deriving a simple equation at the core of all of these constructions.

Many dualities in mathematics arise from the inherent duality of ‘syntax’ and ‘semantics’ in logic. Classical Stone duality, for example, is the syntax-semantics duality for theories of classical propositional logic, with Boolean algebras encoding the syntax of a propositional theory. The logical perspective on these ‘syntax-semantics’ dualities gives both an intuitive understanding for why mathematicians (or at least logicians) would expect these dualities to hold in the first place, as well as a framework to generalise to nearby logics.

In recent years, new ‘non-commutative’ generalisations of Stone duality have been discovered, involving inverse monoids and étale groupoids. Interestingly, this branch of duality theory was developed in the absence of a logical description. In this talk, we describe a class of logical theories whose syntax-semantics duality is given by a version of non-commutative Stone duality. Rather than originating in an exotic fragment of logic, these are theories of first-order logic which share many of the same properties as the theory of vector spaces, suggesting that non-commutative Stone duality is not so distant from classical logic as one might expect.

In this talk I will present the status on quantum communications in Portugal. From the Laboratory to the operational network what has been done through the main actors in industry, academia and public institutions in Portugal. Starting with the development of national technology under a European Defence project, DISCRETION, to the deployment of the first EuroQCI (the European Quantum Communication Infrastructure) segment in Portugal, PTQCI. I will show that quantum communications is no longer a science project, but it is on the heart of sovereignty in Europe.

Identical quantum particles exhibit only two types of statistics: bosonic and fermionic. Theoretically, this restriction is commonly established through the symmetrizationpostulate or (anti)commutation constraints imposed on the algebra of creation and annihilation operators. The physical motivation for these axioms remains poorly understood, leading to various generalizations by modifying the mathematical formalism in somewhat arbitrary ways. In this talk, I will take an opposing route and show how to classify quantum particle statistics based on operationally well-motivated assumptions. Specifically, I will consider that a) the standard (complex) unitary dynamics defines the set of single-particle transformations, and b) phase transformations act locally in the space of multi-particle systems. In this way, a complete characterization, which includes bosons and fermions as basic statistics with minimal symmetry is being developed. Interestingly, we have discovered whole families of novel statistics (dubbed transtatistics) accompanied by hidden symmetries, generic degeneracy of ground states, and spontaneous symmetry breaking — effects that are (typically) absent in ordinary statistics.

Reference: N. M. Sánchez and B. Dakić, Quantum 8, 1473 (2024).

This presentation will report on a number of studies conducted in my laboratory, mainly on the ability of large language models to solve analogies between sentences. We show that the task does not necessarily require a large number of parameters for fairly formal analogies. By defining constraints and introducing a measure of analogy quality, we are able to collect a large number of more or less robust analogies between sentences. We show that analogy quality prevails over quantity in fine-tuning and demonstrate this in machine translation tasks with low-resource languages. We also mention experiments on solving arithmetic problems by answer verification (binary classification), multiple-choice questions (selection from candidate answers), and generation of answer. The results clearly show that binary classification is the easiest task, and that there is little correlation between the three tasks.