![[TopOx] Tom Leinster: The many faces of magnitude @ToposInstitute](https://i.ytimg.com/vi/wxqfxRCGoOE/maxresdefault.jpg)
Uploaded February 2026 | Updated September 2026, 2 weeks ago
16th of February 2026. Slides available at https://topos.institute/events/topox/
The magnitude of a square matrix is the sum of all the entries of its inverse. This strange definition, suitably used, enables us to define the "magnitude" of many objects in different contexts across mathematics. All of them can be seen as measures of size. For example, the magnitude of a metric space combines classical quantities such volume, surface area, and dimension. The magnitude of a category is closely related to Euler characteristic. The magnitude of a graph is an invariant sharing features with the Tutte polynomial (but not a specialization of it). Magnitude also appears in the difficult problem of quantifying biological diversity: under certain circumstances, the greatest possible diversity of an ecosystem is exactly its magnitude. And there is now a theory of magnitude homology, which has the same relationship to magnitude as ordinary homology does to Euler characteristic. I will give an aerial view of this landscape.
16th of February 2026. Slides available at https://topos.institute/events/topox/
The magnitude of a square matrix is the sum of all the entries of its inverse. This strange definition, suitably used, enables us to define the "magnitude" of many objects in different contexts across mathematics. All of them can be seen as measures of size. For example, the magnitude of a metric space combines classical quantities such volume, surface area, and dimension. The magnitude of a category is closely related to Euler characteristic. The magnitude of a graph is an invariant sharing features with the Tutte polynomial (but not a specialization of it). Magnitude also appears in the difficult problem of quantifying biological diversity: under certain circumstances, the greatest possible diversity of an ecosystem is exactly its magnitude. And there is now a theory of magnitude homology, which has the same relationship to magnitude as ordinary homology does to Euler characteristic. I will give an aerial view of this landscape.
![[TopOx] Tom Leinster: The many faces of magnitude](https://i.ytimg.com/vi/wxqfxRCGoOE/mqdefault.jpg "[TopOx] Tom Leinster: The many faces of magnitude
16th of February 2026. Slides available at https://topos.institute/events/topox/
The magnitude of a square matrix is the sum of all the entries of its inverse. This strange definition, suitably used, enables us to define the magnitude of many objects in different contexts across mathematics. All of them can be seen as measures of size. For example, the magnitude of a metric space combines classical quantities such volume, surface area, and dimension. The magnitude of a category is closely related to Euler characteristic. The magnitude of a graph is an invariant sharing features with the Tutte polynomial (but not a specialization of it). Magnitude also appears in the difficult problem of quantifying biological diversity: under certain circumstances, the greatest possible diversity of an ecosystem is exactly its magnitude. And there is now a theory of magnitude homology, which has the same relationship to magnitude as ordinary homology does to Euler characteristic. I will give an aerial view of this landscape.")
![[Oxford Seminar] Matthew Daggit | Developing support for end-to-end verification of neural AI agents](https://i.ytimg.com/vi/xK8Kw-g0mNs/mqdefault.jpg "[Oxford Seminar] Matthew Daggit | Developing support for end-to-end verification of neural AI agents
Oxford Seminar, 1st of May 2025,
Full Title: Developing programming language support for end-to-end verification of neural AI agents
Over the past decade, there has been impressive progress in developing algorithms capable of formally verifying first-order, linear specifications about small-to-medium-sized neural networks. This has opened the door to formally verifying the correctness of neural AI agents operating in complex environments. However, due to the complexity of integrating these verification algorithms with other reasoning tools and the many moving parts involved, deploying them in practice has proven more challenging than anticipated.
In this talk, I will introduce Vehicle, a tool that my collaborators and I have been developing. Vehicle allows users to: (i) write specifications for AI agents in a high-level, domain-specific language; (ii) use these specifications to guide training; (iii) formally verify whether the resulting neural network satisfies the specification; and (iv) export verified specifications to interactive theorem provers where the agents can be reasoned in the context of their environment. In the talk I will highlight how we leverage type theory in unusual ways to improve the user experience, our use of real-valued logic semantics to interpret Boolean specifications, and finally some of the open theoretical challenges in the field.
Links:
https://github.com/vehicle-lang/vehicle")
![[DOTS Lectures] 6. Categories of Moore Machines](https://i.ytimg.com/vi/xMYGOzYhVVo/mqdefault.jpg "[DOTS Lectures] 6. Categories of Moore Machines
Part of a lecture series on the Double Operadic Theory of Systems (DOTS) presented by David Jaz Myers.")
![[2-torial] Owen tells Tim about elaborators for type theories [2/2]](https://i.ytimg.com/vi/xQSG_a_uyKk/mqdefault.jpg "[2-torial] Owen tells Tim about elaborators for type theories [2/2]
Accompanying code can be found on github: https://github.com/ToposInstitute/elaboratorial")
![[Berkeley Seminar] Aaron Huntley (Topos Institute) | Generalised coproduct completions](https://i.ytimg.com/vi/xa04dVG4fEs/mqdefault.jpg "[Berkeley Seminar] Aaron Huntley (Topos Institute) | Generalised coproduct completions
Title: Partial functions and other generalised coproduct completions
Abstract: The category of sets and functions is the free coproduct completion of the terminal category. In a double-categorical setting, Evan Patterson defined a new notion of double (co)product and used it to prove that the double category of sets, functions and spans is the free coproduct completion of the terminal double category. In this talk we both generalise and refine this notion. First, we extend Patterson’s original definition to include (co)products in (co)virtual double categories, which can capture examples such as coproducts in Span(C) even when C does not have pullbacks. Second, we isolate finer classes of double (co)products, denoted (L,R)-(co)products, where L and R are certain classes of functions. As an application, we exhibit the double category of sets, functions and partial functions as a free (L,R)-coproduct completion of the terminal double category.
Date: 9/1/2026")
![[Oxford Seminar] Adrián Puerto Aubel | A glance at Petri net theory](https://i.ytimg.com/vi/z5DXdfV8Fw0/mqdefault.jpg "[Oxford Seminar] Adrián Puerto Aubel | A glance at Petri net theory
Oxford Seminar, February 19 2026
Speaker: Adrián Puerto Aubel
Full Title: A glance at Petri net theory
Abstract: Petri nets are formal models of computing well-known for depicting true concurrency. Unlike automata, they overcome the state-space explosion problem by avoiding interleaving semantics. In this talk I will give an overview of the most prominent theoretical developments of more than 50 years of Petri net theory. This will range from the different expressions of formal semantics of these models as marking graphs, unfoldings, or event structures, to an overview of relevant problems defined on them and their complexities, with a particular focus on structural analysis techniques.")
![[DOTS Lectures] 2. More categories of systems](https://i.ytimg.com/vi/zv10Vgl3Shc/mqdefault.jpg "[DOTS Lectures] 2. More categories of systems
Part of a lecture series on the Double Operadic Theory of Systems (DOTS) presented by David Jaz Myers.")
