![[Oxford Seminar] Virginie Debauche | The Path-Complete Formalism for Switched Systems @ToposInstitute](https://i.ytimg.com/vi/cckhqPmiVuw/maxresdefault.jpg)
Uploaded April 2025 | Updated September 2026, 2 weeks ago
Oxford Seminar, 17th of April 2025
TITLE: The Path-Complete Formalism for Switched Systems: Stability and Beyond
ABSTRACT: Switched systems play a crucial role in modern engineering thanks to their ability to capture complex behaviours involving transitions between different operational modes. However, analyzing their stability remains a challenging task due to the intricate interplay between discrete switching and dynamics. This complexity calls for sophisticated mathematical tools. While Lyapunov theory remains a cornerstone of stability analysis, traditional methods often fall short when applied to switched systems, prompting ongoing efforts to extend the theory to better accommodate their unique characteristics.
This talk presents the framework of path-complete Lyapunov functions, which offers a fresh perspective by integrating combinatorial structures to represent switching behaviour. Specifically, a path-complete Lyapunov function comprises two components: a combinatorial element, represented by an automaton (a directed graph) that encodes admissible switching sequences, and an algebraic element, consisting of a collection of Lyapunov functions—one for each node in the graph. The graph edges govern how these Lyapunov pieces interact and decrease across transitions. This framework is particularly appealing for the analysis of switched systems because it allows for the construction of tailored, nonstandard, and less conservative stability criteria, all while mitigating the combinatorial complexity that often burdens classical optimization techniques.
While originally introduced for stability analysis, the path-complete Lyapunov framework has been recently extended to encompass constrained switching systems, stabilization via switching sequence and control Lyapunov function design, and safety through path-complete barrier functions. This talk will highlight these recent developments and their unifying role within the framework.
Oxford Seminar, 17th of April 2025
TITLE: The Path-Complete Formalism for Switched Systems: Stability and Beyond
ABSTRACT: Switched systems play a crucial role in modern engineering thanks to their ability to capture complex behaviours involving transitions between different operational modes. However, analyzing their stability remains a challenging task due to the intricate interplay between discrete switching and dynamics. This complexity calls for sophisticated mathematical tools. While Lyapunov theory remains a cornerstone of stability analysis, traditional methods often fall short when applied to switched systems, prompting ongoing efforts to extend the theory to better accommodate their unique characteristics.
This talk presents the framework of path-complete Lyapunov functions, which offers a fresh perspective by integrating combinatorial structures to represent switching behaviour. Specifically, a path-complete Lyapunov function comprises two components: a combinatorial element, represented by an automaton (a directed graph) that encodes admissible switching sequences, and an algebraic element, consisting of a collection of Lyapunov functions—one for each node in the graph. The graph edges govern how these Lyapunov pieces interact and decrease across transitions. This framework is particularly appealing for the analysis of switched systems because it allows for the construction of tailored, nonstandard, and less conservative stability criteria, all while mitigating the combinatorial complexity that often burdens classical optimization techniques.
While originally introduced for stability analysis, the path-complete Lyapunov framework has been recently extended to encompass constrained switching systems, stabilization via switching sequence and control Lyapunov function design, and safety through path-complete barrier functions. This talk will highlight these recent developments and their unifying role within the framework.
![[Oxford Seminar] Virginie Debauche | The Path-Complete Formalism for Switched Systems](https://i.ytimg.com/vi/cckhqPmiVuw/mqdefault.jpg "[Oxford Seminar] Virginie Debauche | The Path-Complete Formalism for Switched Systems
Oxford Seminar, 17th of April 2025
TITLE: The Path-Complete Formalism for Switched Systems: Stability and Beyond
ABSTRACT: Switched systems play a crucial role in modern engineering thanks to their ability to capture complex behaviours involving transitions between different operational modes. However, analyzing their stability remains a challenging task due to the intricate interplay between discrete switching and dynamics. This complexity calls for sophisticated mathematical tools. While Lyapunov theory remains a cornerstone of stability analysis, traditional methods often fall short when applied to switched systems, prompting ongoing efforts to extend the theory to better accommodate their unique characteristics.
This talk presents the framework of path-complete Lyapunov functions, which offers a fresh perspective by integrating combinatorial structures to represent switching behaviour. Specifically, a path-complete Lyapunov function comprises two components: a combinatorial element, represented by an automaton (a directed graph) that encodes admissible switching sequences, and an algebraic element, consisting of a collection of Lyapunov functions—one for each node in the graph. The graph edges govern how these Lyapunov pieces interact and decrease across transitions. This framework is particularly appealing for the analysis of switched systems because it allows for the construction of tailored, nonstandard, and less conservative stability criteria, all while mitigating the combinatorial complexity that often burdens classical optimization techniques.
While originally introduced for stability analysis, the path-complete Lyapunov framework has been recently extended to encompass constrained switching systems, stabilization via switching sequence and control Lyapunov function design, and safety through path-complete barrier functions. This talk will highlight these recent developments and their unifying role within the framework.")
![[Oxford Seminar] Owen Lynch | An introduction to the geolog project](https://i.ytimg.com/vi/d6xMeIxsCM4/mqdefault.jpg "[Oxford Seminar] Owen Lynch | An introduction to the geolog project
Oxford Seminar, April 2 2026
Speaker: Owen Lynch
Full Title: An introduction to the geolog project
Abstract: Within the ARIA program, a number of researchers are working on a project called geolog. Geolog has some ambitious goals and some controversial theses; in this talk we will give an overview of these goals and theses alongside an intro to some of the math and computer science that we are using to attempt to achieve the goals and confirm the theses.")
![[2-torial] Categorical algebraic geometry, Part 2](https://i.ytimg.com/vi/dZKDEhOhz8E/mqdefault.jpg "[2-torial] Categorical algebraic geometry, Part 2
2-torial, July 28 2026
You can find accompanying notes and other information on this video at https://topos.institute/work/lectures/2-torials/2026-07-28_hosgood_geometry/
Speaker: Tim Hosgood
There are many approaches to algebraic geometry, and many different ways to arrive at the subject. Here were going to take an incredibly specific and biased approach: what if you already love 2-categories and want to be able to say the phrase fpqc sheaf as quickly as possible, but not /too/ quickly? In this series of exercises we will build towards an understanding of *relative algebraic geometry*, which allows us to work in arbitrary (nice) symmetric monoidal categories.")
![[DOTS Lectures] 9. Behavioural modules of systems](https://i.ytimg.com/vi/de20OAaAqW0/mqdefault.jpg "[DOTS Lectures] 9. Behavioural modules of systems
Part of a lecture series on the Double Operadic Theory of Systems (DOTS) presented by David Jaz Myers.
Some material from these lectures can be found in Davids book on categorical systems theory:
https://www.davidjaz.com/Papers/DynamicalBook.pdf")
![[Oxford Seminar] Elizabeth Amelia | AI for precision oncology](https://i.ytimg.com/vi/eDZuVRE-XVU/mqdefault.jpg "[Oxford Seminar] Elizabeth Amelia | AI for precision oncology
Oxford Seminar, 29th of May 2025")
![[TopOx] Jakub Opršal: Homotopy theory in the complexity of homomorphism problems](https://i.ytimg.com/vi/fU2pZbrE2aA/mqdefault.jpg "[TopOx] Jakub Opršal: Homotopy theory in the complexity of homomorphism problems
27th of November 2025. Slides available at https://topos.institute/events/topox/
I will talk about an emerging connection between homotopy theory and computational complexity of discrete problems. I will outline a theorem stating that contractibility is necessary for tractability (assumin P ≠ NP) in the realm of finite-template constraint satisfaction problems (CSPs).
There are many ways the CSP can be formulated. One of them is as a homomorphism problem: given two relational structures A and B, decide whether there is a homomorphism from A to B. We usually study a restricted version of this problem where B is fixed, e.g., if B is the k-clique graph K_k, the problem is the same as k-colouring of a given graph A. If B is finite (and of finite signature), such a problem is called finite-template. Famously, finite-template CSPs exhibit a P vs NP-complete dichotomy as proved independently by Bulatov and Zhuk in 2017.
The main theorem of the talk states a sufficient condition for NP-completeness in terms of the topology of ‘solution spaces’ and provides both all necessary hardness for the Bulatov–Zhuk dichotomy and also a new proof of an earlier Hell–Nešetřil dichotomy of graph homomorphism problems.")
![[Oxford Seminar] Khyathi Komalan | Adult Brainrot: Mandela Effect, Misinformation & Conspiracies](https://i.ytimg.com/vi/gk0jJtjNQGI/mqdefault.jpg "[Oxford Seminar] Khyathi Komalan | Adult Brainrot: Mandela Effect, Misinformation & Conspiracies
Oxford Seminar, 25th of September 2025
Abstract: The internet is a strange place — one moment youre
convinced the Starbucks logo never had a star on top, the next youre
knee-deep in a rumor that spreads everywhere at once, and before long
youre staring at a series of pictures that prove the moon landing
was fake. These phenomena may look like pure nonsense, but they also
show how perception, memory, and belief are shaped by our
experiences, the hidden biases behind what we perceive, and the
difficulty of stitching multiple pieces of information, sometimes
conflicting, into a coherent story that fits the most into ones
worldview.
In this talk I introduce adult brainrot as a categorical framework
for these inconsistencies. I will show how categorical models of
quantum processes can capture order-sensitive memory and
indistinguishable sources of misinformation, and how categorical
notions of centers help explain why people’s theories sometimes fail
to form a consistent picture of reality. The aim is to argue that
quantum categorical structures offer a natural mathematical framework
for modeling the inconsistencies and oddities of human perception and
belief.")
