What is a mathematical proof? This question was at the heart of the debate which shook the foundations of mathematics at the beginning of the 20th century. Consequently, two opposing viewpoints emerged: one defending the new infinitary methods of set theory, and the other based on constructivism, the belief that mathematics should be limited to its constructive or computational content. Nowadays, these foundational attitudes largely reconciled and, while endorsing the usage of infinitary methods, mathematics in the new millennium acknowledges the deeper informative content of constructive proofs. This shift of paradigm is witnessed by the methods of proof mining, providing drastically sharper bounds in analysis by unwinding the constructive content of classical proofs. Yet, in recent years, an unsuspected driving force was reshaping the role of constructive mathematics: the need to asses the status of an expanding collection of computer-aided proofs of long-standing conjectures such as the four colour map problem. The quest for a solid base in which these could be formally verified motivated the development of the new constructive foundational programme univalent foundations, endorsed among others by Fields medallist Voevodsky. At the heart of this programme is the core of this research project: the idea that constructive phenomena could be interpreted through the lenses of classical geometric intuitions. Indeed one of the most successful approaches to constructive logic builds on the powerful methods of geometry to provide a semantics of intuitionistic logic in terms of order-topological structures called Esakia spaces that can be viewed as canonical models of constructive theories. The objective of the proposed project is to tackle the long-standing "representation problem" via recently introduced novel methods. This cornerstone of the geometric approach to intuitionistic logic asks to describe the class of posets that can be endowed with an Esakia topology. A solution to the representation problem would be a major breakthrough both in constructive logic and its applications in theoretical computer science.

When told that you may stay or go, you normally conclude that you may stay and you may go, contrary to the prescriptions of classical logic. This project investigates such cases of divergence between human and logical-mathematical reasoning and, challenging the canonical view, hypotheses that they are a straightforward consequence of a tendency in human cognition to neglect empty representations (horror vacui). We will develop logics which model this tendency and rigorously isolate its effects on deductions and interpretation; and experimentally test the precise predictions arising from these formalisations.

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Quexistentials are words that can be used both as question words and as existential quantifiers. An example is the Dutch word wat, which can be used as a question word meaning what but also as an existential quantifier meaning something. Many languages around the world have quexistentials. Investigating the syntactic, semantic, and prosodic properties of these items is important in order to come to a better understanding of the exact commonalities and differences between questions and existential statements. Prof. Iatridou (MIT) has been collaborating with Roelofsen and Hengeveld (University of Amsterdam) since 2016 to address this issue. This has led to significant new insights, but a number of crucial questions remain open. Prof. Iatridous visit is intended to allow for a period of intense collaboration in order to resolve these and to finalise a substantial publication on the topic. Highlights of this project will also be made accessible to the general public through an article in the online popular science magazine Kennislink.