The ALT Group has a diverse set of projects underway or in preparation across various domains. Here is a breakdown of some of them:

Enriched categories and algebraic structures: The group is investigating a variety of themes in these areas, focusing on the algebraic behaviour of Ord-enriched categories and the categorical behaviour of Ord-enriched algebras and some of their generalisations, like (probabilistic) metric groups/monoids. In addition, we will continue with our research program on normed categories (as enriched categories) and their applications to analysis, geometry, and probability theory.

2-categories: We plan to introduce a calculus of lax fractions in the context of 2-categories, generalizing the classical one of Gabriel-Zisman, and develop a logic of Kan-injectivity for 2-categories, extending previous work in the Ord-enriched case.

Pointfree Topology: A main difference between pointfree topology and classical topology is that subobject lattices in the former are much more complicated. We plan to continue investigating them, in particular some special subclasses that have played an important role in recent research, namely the lattices of joins of closed sublocales (the pointfree version of closed subspaces), and fitted sublocales (the pointfree version of saturated subspaces), and their behaviour under separation properties of the locale.

Algebra and Computer Science: We intend to identify Schutzenberger groups of more symbolic dynamical systems, obtain further results on profinite codes, show Cerny's conjecture for meaningful classes of automata, investigate the tameness of free pro-aperiodic monoids, and prove the locality of some pseudovarieties of monoids. Other topics: (higher-order) computability and complexity, (higher) categories and functional analysis.

Grothendieck descent theory: Continuation of its study in internal and generalised enriched categories.

These projects involve collaborators from York Univ. (Toronto), Braunschweig, Milano, Cape Town, Utrecht, Stellenbosch Univ., Charles Univ. (Prague), Birmingham, Bilbao, CUNY, Amsterdam, and the universities of Aveiro, Algarve, and Porto in Portugal.