Uploaded April 2026 | Updated September 2026, 2 weeks ago
Topos Institute Colloquium, 2nd of April 2026.
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In intuitionistic mathematics one encounters theorems that may seem strange, even laughable, to a classically trained mathematician. Yet, interpreted internally in toposes and other non-classical models, they carry genuine mathematical content. For example, Lawvere’s fixed-point theorem says that if there is a surjection $A \to B^A$, then every endomap on $B$ has a fixed point; classically the precondition forces $B$ to be a singleton. Likewise, a statement that is false in classical set theory can be meaningful internally: in the effective topos there exists a set that is neither finite nor infinite. In the talk we will try not to laugh, and instead explain how such claims acquire substance once the viewpoint expands beyond classical mathematics.
Topos Institute Colloquium, 2nd of April 2026.
———
In intuitionistic mathematics one encounters theorems that may seem strange, even laughable, to a classically trained mathematician. Yet, interpreted internally in toposes and other non-classical models, they carry genuine mathematical content. For example, Lawvere’s fixed-point theorem says that if there is a surjection $A \to B^A$, then every endomap on $B$ has a fixed point; classically the precondition forces $B$ to be a singleton. Likewise, a statement that is false in classical set theory can be meaningful internally: in the effective topos there exists a set that is neither finite nor infinite. In the talk we will try not to laugh, and instead explain how such claims acquire substance once the viewpoint expands beyond classical mathematics.
![[Berkeley Seminar] Valeria de Paiva | Classical and constructive logics together: Ecumenical systems](https://i.ytimg.com/vi/M1yFhKKDrpI/mqdefault.jpg "[Berkeley Seminar] Valeria de Paiva | Classical and constructive logics together: Ecumenical systems
Title Classical and constructive logics together: Ecumenical systems
Abstract Much has been said about intuitionistic and classical logical systems since Gentzens seminal work. Recently Prawitz, Dowek and others, have been discussing how to put together Gentzens systems for classical and intuitionistic logic into a single system. This has been called Ecumenical Logic. I will present an ecumenical sequent calculus and state some of its proof theoretical properties. This approach to an unified system, enabling both classical and intuitionistic features, should shed some light not only on the logics themselves, but also in the ways they can interoperate. This in turn should help enhance the interoperability of proof assistants, to enable seamless communication between them, as discussed in the recent PhD thesis of Emilie Grienenberger, a student of under Gilles Dowek.
https://topos.institute/events/berkeley-seminar/")
![[Berkeley Seminar] Corinthia Aberlé | Synthetic Mathematics, Logical Frameworks, Categorical Algebra](https://i.ytimg.com/vi/MjkWT6GkISI/mqdefault.jpg "[Berkeley Seminar] Corinthia Aberlé | Synthetic Mathematics, Logical Frameworks, Categorical Algebra
Title: Synthetic Mathematics, Logical Frameworks, and Categorical Algebra
Abstract: I’ll be talking about three topics near and dear to my heart: synthetic mathematics, interactive theorem proving, and categorical algebra. My aim is to give an overview of how these topics are intimately linked, culminating in a discussion of how they can be used in tandem to give an elegant formal approach to problems at the heart of formal logic and type theory. To this end, I will proceed through a series of questions:
What is synthetic mathematics? …and why should I care?
What is a logical framework? …and how can I use one to help me do math?
What is categorical algebra? …and what are its limits?
What do synthetic mathematics, logical frameworks, and categorical algebra have to do with each other?
Date: 2025-07-01
https://topos.institute/events/berkeley-seminar/")
![[2-torial] Tim tells Jason about Deformation Theory [1/3]](https://i.ytimg.com/vi/MoBTZNGIjLs/mqdefault.jpg "[2-torial] Tim tells Jason about Deformation Theory [1/3]
[2-torial] Tim tells Jason about Deformation Theory [1/3]")
![[2-torial] Joanna tells Jason about enhanced simplicial categories](https://i.ytimg.com/vi/N04GYP5Qv7Y/mqdefault.jpg "[2-torial] Joanna tells Jason about enhanced simplicial categories
[2-torial] Joanna tells Jason about enhanced simplicial categories")
![[Berkeley Seminar] Shaowei Lin | Safety by Shared Synthesis](https://i.ytimg.com/vi/N1qZhOU5OWo/mqdefault.jpg "[Berkeley Seminar] Shaowei Lin | Safety by Shared Synthesis
Title: Safety by Shared Synthesis
Abstract: Today, critical infrastructure is vulnerable to both malicious attacks and unintended failures, and these risks are expected to grow in the foreseeable future. Deploying formal verification (FV) across critical cyber physical systems would dramatically improve safety and security, but has historically been too costly to use outside the simplest or most critical subsystems. AI could allow widespread use of FV in years not decades, shifting cyber risks strongly in favor of defense. In this talk, I will outline our report with Atlas Computing on AI-enabled tools for scaling formal verification (https://atlascomputing.org/ai-assisted-fv-toolchain.pdf). I will also discuss some lessons that I learnt along the way, especially about shared synthesis - the collaborative construction of formal specifications, implementations and proofs.
https://topos.site/events/berkeley-seminar/")
![[Oxford Seminar] David Jaz Myers | Preservation of 2-algebraic structure by pseudo-functors](https://i.ytimg.com/vi/NIhGTjTPnlY/mqdefault.jpg "[Oxford Seminar] David Jaz Myers | Preservation of 2-algebraic structure by pseudo-functors
Oxford Seminar, 31st of January 2025
If universal algebra is the study of sets equipped with the structure of some operations satisfying universally quantified equations, then 2-algebra is the study of *categories* equipped with the structure of some operations, some natural transformations between these operations, and some equations between these. Examples of 2-algebras include symmetric monoidal categories and categories with finite products. Just as the functorial semantics (due to Lawvere) for algebra lets us interpret algebraic theories in categories other than sets (so long as they have finite products), the functorial semantics for 2-algebra (due to Power, Lack, and others) lets us interpret 2-algebraic structure in 2-categories other than Cat. For example, instead of symmetric monoidal categories, we can consider symmetric monoidal double categories. Any strict 2-functor that preserves finite products will push forward 2-algebraic structure. But 2-functors are often only functorial up to coherent isomorphism. What 2-algebraic structure do such cartesian pseudo-functors preserve? In this talk, well ask the question and put forward an answer: so long as the 2-algebraic theory is *flexible*, in the sense that it only asks for equations between transformations and not between operations, then it will be preserved by cartesian pseudo-functors.")
![[Oxford Seminar] Mirco Giacobbe | Neural certificates](https://i.ytimg.com/vi/O9T1zHj1cAA/mqdefault.jpg "[Oxford Seminar] Mirco Giacobbe | Neural certificates
Oxford Seminar, 13th of March 2025
Model checking aims to derive rigorous proofs for the correctness of
systems and has traditionally relied on symbolic reasoning methods. In
this talk, I will argue that model checking can also be effectively
addressed using machine learning too. I will present a realm of
approaches for formal verification that leverage neural networks to
represent correctness certificates of systems, known as neural
certificates. This approach trains certificates from synthetic
executions of the system and then validates them using symbolic
reasoning techniques. Building upon the observation that checking a
correctness certificate is much simpler than finding one, and that
neural networks are an appropriate representation for such
certificates, this results in a machine learning approach to model
checking that is entirely unsupervised, formally sound, and practically
effective. I will demonstrate the principles and experimental results
of this approach in safety assurance of software, probabilistic
systems, and control.
About the speaker:
Mirco Giacobbe is an Associate Professor at the University of Birmingham. He previously held research positions at the University of Oxford and Fondazione Bruno Kessler, and obtained his PhD at the Institute and Science and Technology Austria. His research interests lie between formal methods and artificial intelligence, where he develops automatic techniques to assure that algorithmic systems are safe and trustworthy.")
