Asymmetric topology — the study of spaces where the distance from x to y need not equal the distance from y to x — is a small and active corner of mathematics. The week covers the core results (quasi-metric spaces, bipolar metrics, fixed-point theorems in asymmetric settings, the connections to formal balls and weightability) and the applications that justify the area's continued attention: denotational semantics in theoretical CS, similarity learning in ML, and approximation theory.
The school is academic in posture and tone. Mornings are lectures, afternoons are working sessions with problem sets, evenings are open for collaboration. We expect each participant to leave with at least one open question they intend to work on, and ideally with a co-author.
Program Overview
A week-long advanced research school co-hosted with a partner mathematics department, with a hybrid option for remote participants. Lead: Y. U. Gaba, who publishes regularly in this area and co-authors with several of the senior figures in asymmetric topology. Co-organizers: a senior topologist from the partner department (named four weeks before the school). Each cohort has invited speakers; AIRINA covers honoraria.
Program structure
- 5 working days. AM lectures, PM problem sessions, evenings open for collaboration.
- Cohort size. ~12 participants. Application-based.
- Materials. Reading list 6 weeks ahead. Pre-school problem set optional.
- Fellowships. Need-based fellowships for African graduate students.
Certificate
The school issues an academic attendance certificate. Participants who choose to do so may submit a written research note (5–10 pages) within three months of the school; AIRINA faculty provide written feedback. Several past notes have grown into joint papers with the unit. The school is academic, not graded — the value is in the conversations and the network.
Learning Outcomes
By the end of the program, participants will be able to:
- Define quasi-metric, bipolar metric, and partial metric spaces, and explain when each is the right setting for a given problem.
- Prove a fixed-point theorem in an asymmetric setting from scratch, identifying the role of completeness and the contraction condition.
- Connect asymmetric topology to denotational semantics via formal balls and the Smyth completion.
- Identify a concrete machine-learning problem (e.g., directed-similarity learning) where asymmetric tools give a sharper formulation than the symmetric default.
- Read one of the open papers in the area and write a one-page critical summary.
Program curriculum
Definitions, motivating examples, the standard topology induced by a quasi-metric. T0, T1, sober spaces in this setting. Selected results from Künzi's school and the Spanish tradition.
Definition (Mutlu & Gürdal and the subsequent literature), examples, properties. Two-pole completeness, fixed-point theorems for contractive maps. The director's published work in this area; current open problems.
The Banach principle restated; Ćirić, Reich, Kannan-type contractions in quasi-metric and bipolar settings. Proof techniques and the role of the asymmetric completeness conditions. Joint problem session.
The complexity quasi-metric of Schellekens; formal balls and the Smyth completion; denotational semantics of programming languages with effects. Where the asymmetry is essential rather than incidental.
Asymmetric similarity learning, KL-divergence-flavored embeddings, directed graph representations. Survey of open problems suitable for a one-year research project. Final talks: each participant presents a problem they intend to take home.
Who Should Attend
This research school is for doctoral students, postdocs, and mathematically-inclined researchers who want a deeper toolkit in asymmetric settings — with the conversations and collaborations that come from a small, focused cohort.
- Mathematics doctoral students and postdocs working in topology, analysis, or functional analysis.
- Theoretical computer scientists interested in domain theory, denotational semantics, or formal methods.
- Mathematically-inclined ML researchers wanting a deeper toolkit for asymmetric similarity / dissimilarity problems.
Prerequisites
- Topology. Graduate-level topology (point-set is essential; algebraic is helpful).
- Functional analysis. Through complete metric spaces and the contraction-mapping theorem.
- Reading. Comfort reading research mathematics. The week is paper-heavy.
Selection
Application-based. We ask for a CV, a one-page research statement, and (for graduate students) a letter from your advisor. Need-based fellowships available for African graduate students.
Brochure
The detailed school brochure (PDF, EN/FR) is sent on request — including day-by-day program, faculty and invited speakers, the reading list, and the application timeline.
To receive the current brochure, write to contact@airina.africa with "Asymmetric Topology — brochure request" in the subject. The brochure is updated each cohort; we send the version current at the time of your request.