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The Topos Institute Colloquium is an expository virtual talk series on topics relevant to the Topos mission and community. We usually meet over Zoom on Thursdays at 17:00 UTC, but this may vary depending on the speaker's time-zone. Talks are recorded, and remain available on our YouTube channel.
If you wish to be subscribed to the mailing list, then simply send an email to seminars+subscribe@topos.institute. For any queries about the colloquium, contact tim+colloquium@topos.institute.
The colloquium centres around following four themes, with many talks fitting multiple themes.
Topos aims to shape technology for public benefit via creating and deploying a new mathematical systems science. What does this mean, and how do we do it? How does technology shape people’s lives today? What are the risks that future technologies may bring? What is the role of mathematicians and computer scientists in shaping how their work has been deployed? What lessons can we learn from the past and present about successes and failures? This theme aims to spark discussion among mathematicians and computer scientists about these questions, and bring them into contact with experts on these subjects.
Thinking clearly about today’s most pressing scientific, technical, and societal questions requires finding the right abstractions. Category theory can help with this by offering design principles for structuring how we account for phenomena in a specific domain, as well as how we translate problems and solutions between different domains. This theme aims to highlight recent developments in applied category theory, in domains such as computation, neuroscience, physics, artificial intelligence, game theory, and robotics.
Like many parts of pure mathematics, results that are initially seen as purely theoretical may not be ripe for application until much later. At Topos we foster the entire pipeline, from the creation of elegant theory to the development of its effective application. The goal of this theme is to foster the pure side of this pipeline, to take in the most beautiful results in category theory, logic, type theory, and related fields, as well as to scout for not-yet-categorical work that appears ripe for "categorification".
Beyond its intrinsic interest, the purpose of applied mathematics is to provide a foundation for new capabilities and technologies that benefit the public. In this theme, we highlight work, at the intersection of research and engineering, that transfers ideas from applied category theory and other parts of mathematics into viable technologies, with an emphasis on software systems and tools. Topics may include scientific modeling, functional programming, differential programming, probabilistic programming, quantum computing, formal verification, and software and systems engineering.
To see the talks from previous years, use the links to the archives at the top of the page.
Robert Paré (11th of January)
As a tool for studying the structure of endofunctors F of Set, we introduce the difference operator△
△[F](X) = F(X + 1) \ F(X).
This is analogous to the classical difference operator for real valued functions, a discrete form of the derivative.
The \ above is set difference and can't be expected to be functorial, but it is for a large class of functors, the taut functors of Manes, which include polynomial functors and many more.
We obtain combinatorial versions of classical identities, often "improved". Many examples will be given.
The talk should be accessible to everyone. The only prerequisite is some very basic category theory.
John Cartmell (18th of January)
I will discuss significant aspects of a theory of data and what may be achieved by representing data specifications as sketches of Range Categories with additional structure. I will discuss the distinction between relational and non-relational physical data specifications and contrast physical and logical data specifications. I will discuss goodness criteria for such specifications and define some specific criteria which generalise the classic relational goodness criteria i.e. the so called normal forms of Codd, Fagin, Ling and Goh and others.
My goal for a fully elaborated Mathematical Theory of Data is to effect a change in what is considered best practice for the way in which data is specified and programmed as as to enable best practice to be shifted from being at the level that data is physically represented and communicated to being at the more abstract level of its logical structure.
Juan Pablo Vigneaux (25th of January)
This talk will discuss the cohomological aspects of information functions within the framework of information cohomology (first introduced by Baudot and Bennequin in 2015). Several known functionals can be identified as cohomology classes in this framework, including the Shannon entropy of discrete probability measures and the differential entropy and underlying dimension of continuous measures. I’ll try to provide an accessible overview of the foundations of the theory, which should require only a basic familiarity with category theory and homological algebra, and survey the main known results. Finally, I'll discuss some perspectives and open problems: firstly in connection with Renyi's information dimension and other (possibly geometric) invariants of laws taking values on manifolds, and secondly with notions of entropy for categories (akin to Leinster's diversity of metric spaces).
Susan Niefield (2nd of February)
In his 1973 paper (TAC Reprints, 2002), Lawvere observed that a metric space Y is a category enriched in the extended reals, and showed that Y is Cauchy complete if and only if every bimodule (i.e., profunctor) with codomain Y has a right adjoint. More recently, Paré (2021) considered adjoints and Cauchy completeness in double categories, and showed that an (S,R)-bimodule M has a right adjoint in the double category of commutative rings if and only if it is finitely generated and projective as an S-module. It is well known that the latter property characterizes the existence of a left adjoint to tensoring with M on the category of S-modules, and this was generalized to rigs and quantales in a 2017 paper by Wood and the speaker.
This talk consists of two parts. First, after recalling the relevant definitions, we present examples of Cauchy complete objects in some "familiar" double categories. Second, we incorporate the two above mentioned projectivity results into a version of the 2017 theorem with Wood which we then apply to (not-necessarily commutative) rings, rigs, and quantales.
André Joyal (15th of February)
Whitman's theory of free lattices can be extended to free lattices enriched over a quantales, to free bicomplete enriched categories and even to free bicomplete enriched oo-categories. It has applications to the semantic of Linear Logic (Hongde Hu and J.)
Joachim Kock (22nd of February)
In their simplest form, polynomial functors are endofunctors of the category of sets built from sums and products. At first they can be considered a categorification of the notion of polynomial function, but it has turned out the theory of polynomial functors is more generally an efficient toolbox for dealing with induction, nesting, and substitution. The talk will highlight some of these aspects in combinatorics, logic, and homotopy theory.
Steven Clontz (28th of March)
The National Science Foundation defines "cyberinfrastructure" as "the hardware, software, networks, data and people that underpin today's advanced computing technology", particularly technologies that advance scientific discovery. In particular, this infrastructure is incomplete without its "people", leading some to prefer the terminology "sociotechnical infrastructure" to emphasize the importance of how these technologies connect human researchers, and how human researchers in turn use and develop these technologies in order to create new knowledge. In mathematics research, even theoretical mathematics, we use many technologies, and engage with many different communities, but there is little scholarship on the ad hoc research infrastructure itself that we implicitly rely on from day to day. This talk will provide an overview of some of the work I've done as part of my spring 2024 sabbatical dedicated to the research and development of improved sociotechnical infrastructure for mathematics research.
Po-Shen Loh (4th of April)
One of the central challenges of beyond-standard-curriculum instruction (such as "gifted" education) is how to achieve equitably-distributed scale. Making matters worse, generative AI such as ChatGPT is increasingly adept at solving standard curricular tasks, so it is urgent to scalably deliver teaching that goes beyond current standards. Fortunately, there is an area close to math which devises solutions in which problems solve themselves even through self-serving human behavior: Game Theory.
The speaker will describe his recent work, which uses Game Theory to create a novel alignment of incentives, which concurrently solves pain points in disparate sectors. At the heart of the innovation is a new, mutually-beneficial cooperation between high school math stars and professionally trained actors and comedians. This creates a highly scalable community of extraordinary coaches with sufficient capacity to teach large numbers of middle schoolers seeking to learn critical thinking and creative analytical problem solving (https://live.poshenloh.com). At the same time, it creates a new pathway for high school math stars to significantly strengthen their emotional intelligence. The whole program is conducted virtually, so it reaches through geographical barriers. The speaker will also share his experience extending this work to build talent development pipelines in underprivileged communities, identifying and supporting highly motivated middle school students who otherwise did not have access to coaching.
Edward Lee (18th of April)
Mathematical models can yield certainty, as can probabilistic models where the probabilities degenerate. The field of formal methods emphasizes developing such certainty about engineering designs. In safety critical systems, such certainty is highly valued and, in some cases, even required by regulatory bodies. But achieving reasonable performance for sufficiently complex environments appears to require the use of AI technologies, which resist such certainty. This talk suggests that certainty and intelligence may be fundamentally incompatible.