Machine learning models of natural language have experienced a rapid development in the past decade, exhibiting impressive results and becoming one of the main forces driving the current AI revolution. However, the mathematical foundations of current computational language models remain largely unknown. One promising avenue to address this issue lies at the crossroads of three simultaneous active lines of research. First, a categorical approach to syntactic data leading to a generalization of both word vector embeddings and formal concept analysis (Bradley, 2020; Bradley, Terilla, & Vlassopoulos, 2021; Bradley, Gastaldi, & Terilla, 2024). Second, a theory of computational types stemming from the linear realisability program in logic and computer science, where types are conceived as descriptors or classifiers naturally defined over dynamic processes (typically, untyped computational programs) thanks to an original formal notion of execution (Seiller, 2024). Finally, a structuralist reassessment of distributional principles suggests that a type-theoretical approach, associated with a generalized and linguistically meaningful notion of duality implicit in embedding spaces' algebraic principles, is the key to the intelligibility of current neural language models' generative power (Gastaldi, 2021; Gastaldi & Pellissier, 2021). Recent results show that the notion of type advanced by the linear realisability program formally coincides with the categorical objects governing the duality underlying linguistic data (i.e., the types are the elements of a profunctor nucleus), and that the corresponding geometric and logical structures interact in remarkable ways (Jarvis, 2025; Gastaldi, Jarvis, Seiller, & Terilla, 2026a, 2026b). The resulting setting thus provides an original and promising framework to build solid mathematical foundations for current developments in computational language modeling. This meeting will aim to determine the formal, conceptual, and empirical orientations necessary to fulfill that program in the upcoming years.

• Bradley, Tai-Danae (2020). "At the Interface of Algebra and Statistics". In: ArXiv abs/2004.05631.
• Bradley, Tai-Danae, Juan Luis Gastaldi, and John Terilla (Feb. 2024). "The Structure of Meaning in Language: Parallel Narratives in Linear Algebra and Category Theory". In: Notices of the American Mathematical Society. issn: 1088-9477.
• Bradley, Tai-Danae, John Terilla, and Yiannis Vlassopoulos (2021). An enriched category theory of language: from syntax to semantics. arXiv: 2106.07890 [math.CT].
• Gastaldi, Juan Luis (2021). "Why Can Computers Understand Natural Language?" In: Philosophy & Technology 34.1, pp. 149–214. issn: 2210-5441. doi: 10.1007/s13347-020-00393-9.
• Gastaldi, Juan Luis and Luc Pellissier (2021). "The Calculus of Language: Explicit Representation of Emergent Linguistic Structure through Type-Theoretical Paradigms". In: Interdisciplinary Science Reviews 46.4, pp. 569590. doi: 10.1080/03080188.2021.1890484.
• Gastaldi, Juan Luis, Jarvis, Samantha, Seiller, Thomas, and Terilla, John (2026a). A calculus of types in Isbell nuclei. arXiv:2606.03369 [cs.LO].
• Gastaldi, Juan Luis, Jarvis, Samantha, Seiller, Thomas, and Terilla, John (2026b). Projective metric geometry of tropical nuclei: gap matrices, event loci, and order chambers. arXiv: 2601.07900 [math.AG].
• Jarvis, Samantha K. (2025). "A Novel Closed Monoidal Structure on the Nucleus of a Profunctor". CUNY Academic Works. PhD thesis. New York, NY: The Graduate Center, City University of New York.
• Seiller, Thomas (2024). "Mathematical Informatics". Habilitation thesis. Sorbonne Paris Nord University.

Creative Commons BY 4.0

Juan Luis Gastaldi, Samantha Jarvis, Thomas Seiller, and John Terilla