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January 2, 2022

Adjoint School 2022

Posted by John Baez

Every year since 2018 we’ve been having annual courses on applied category theory where you can do research with experts. It’s called the Adjoint School.

You can apply to be a student at the 2022 Adjoint School now, and applications are due January 29th! Go here:

Read on for more about how this works!

The school will be run online from February to June, 2022, and then — coronavirus permitting — there will be in-person research at the University of Strathclyde in Glasgow, Scotland the week of July 11–15, 2022. This is also the location of the applied category theory conference ACT2022.

The 2022 Adjoint School is organized by Angeline Aguinaldo, Elena Di Lavore, Sophie Libkind, and David Jaz Myers. You can read more about how it works here:

There are four topics to work on, and you can see descriptions of them below.

Who should apply?

Anyone, from anywhere in the world, who is interested in applying category-theoretic methods to problems outside of pure mathematics. This is emphatically not restricted to math students, but one should be comfortable working with mathematics. Knowledge of basic category-theoretic language — the definition of monoidal category for example — is encouraged.

We will consider advanced undergraduates, PhD students, post-docs, as well as people working outside of academia. Members of groups which are underrepresented in the mathematics and computer science communities are especially encouraged to apply.

Also check out our inclusivity statement.

Topic 1: Compositional Thermodynamics

Mentors: Spencer Breiner and Joe Moeller

TA: Owen Lynch

Description: Thermodynamics is the study of the relationships between heat, energy, work, and matter. In category theory, we model flows in physical systems using string diagrams, allowing us to formalize physical axioms as diagrammatic equations. The goal of this project is to establish such a compositional framework for thermodynamical networks. A first goal will be to formalize the laws of thermodynamics in categorical terms. Depending on the background and interest of the participants, further topics may include the Carnot and Otto engines, more realistic modeling for real-world systems, and software implementation within the AlgebraicJulia library.

Readings:

Topic 2: Fuzzy Type Theory for Opinion Dynamics

Mentor: Paige North

TA: Hans Reiss

Description: When working in type theory (or most logics), one is interested in proving propositions by constructing witnesses to their incontrovertible truth. In the real world, however, we can often only hope to understand how likely something is to be true, and we look for evidence that something is true. For example, when a doctor is trying to determine if a patient has a certain condition, they might ask certain questions and perform certain tests, each of which constitutes a piece of evidence that the patient does or does not have that condition. This suggests that a fuzzy version of type theory might be appropriate for capturing and analyzing real-world situations. In this project, we will explore the space of fuzzy type theories which can be used to reason about the fuzzy propositions of disease and similar dynamics.

Readings:

Topic 3: A Compositional Theory of Timed and Probabilistic Processes: CospanSpan(Graph)

Mentor: Nicoletta Sabadini

TA: Mario Román

9 days until applications are due! Nicoletta Sabadini has come out with her project description:

Description: Span(Graph), introduced by Katis, Sabadini and Walters as a categorical algebra for automata with interfaces, provides, in a very intuitive way, a compositional description of hierarchical networks of interacting components with fixed topology. The algebra also provides a calculus of connectors, with an elegant description of signal broadcasting. In particular, the operations of “parallel with communication” (that allows components to evolve simultaneously, like connected gears), and “non-sequential feedback” (not considered in Kleene’s algebra for classical automata) are fundamental in modelling complex distributed systems such as biological systems. Similarly, the dual algebra Cospan(Graph) allows us to compose systems sequentially. Hence, the combined algebra CospanSpan(Graph), which extends Kleene’s algebra for classical automata, is a general algebra for reconfigurable networks of interacting components. Still, some very interesting aspects and possible applications of this model deserve a better understanding:

  • How can timed actions and probability be combined in CospanSpan(Graph)?

  • If not, can we describe time-varying probability in a compositional setting?

  • Which is the possible role of “parallel with communication” in understanding causality?

Readings:

Topic 4: Algebraic Structures in Logic and Relations

Mentor: Filippo Bonchi

Description: Fox’s theorem provides a bridge between structures defined by universal properties (products in a category) and structures specified by algebraic means (comonoids in a symmetric monoidal category). Such a theorem has recently received a renewed interest as the algebraic structures allows for reasoning in terms of string diagrams. While the universal properties underlying logical theories have been extensively studied in categorical logic, their algebraic counterparts have been the objects of fewer investigations. This raises a natural question: can we capture the universal content of logical theories algebraically? In other words, what are the ‘Fox theorems’ for logic? In this project, we attempt to answer to this question by taking as starting point Cartesian bicategories which serves as algebraic setting for regular logic.

Readings:

• Aurelio Carboni and R. F. C. Walters, Cartesian bicategories I.

• Filippo Bonchi, Jens Seeber and Pawel Sobocinski, Graphical conjunctive queries.

• Filippo Bonchi, Dusko Pavlovic and Pawel Sobocinski, Functorial semantics for relational theories.

Posted at January 2, 2022 6:20 AM UTC

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Re: Adjoint School 2022

9 days until applications are due! Nicoletta Sabadini has come out with her project description:

Description: Span(Graph), introduced by Katis, Sabadini and Walters as a categorical algebra for automata with interfaces, provides, in a very intuitive way, a compositional description of hierarchical networks of interacting components with fixed topology. The algebra also provides a calculus of connectors, with an elegant description of signal broadcasting. In particular, the operations of “parallel with communication” (that allows components to evolve simultaneously, like connected gears), and “non-sequential feedback” (not considered in Kleene’s algebra for classical automata) are fundamental in modelling complex distributed systems such as biological systems. Similarly, the dual algebra Cospan(Graph) allows us to compose systems sequentially. Hence, the combined algebra CospanSpan(Graph), which extends Kleene’s algebra for classical automata, is a general algebra for reconfigurable networks of interacting components. Still, some very interesting aspects and possible applications of this model deserve a better understanding:

  • How can timed actions and probability be combined in CospanSpan(Graph)?

  • If not, can we describe time-varying probability in a compositional setting?

  • Which is the possible role of “parallel with communication” in understanding causality?

Readings:

Posted by: John Baez on January 21, 2022 12:32 AM | Permalink | Reply to this

Re: Adjoint School 2022

The fourth mentor for the Adjoint School of applied category theory, Filippo Bonchi, has finally announced his course.

Due to the delay, the deadline for applying—and choosing which course you want!—has been pushed back to February 4. Apply here:

http://adjointschool.com/apply.html


Topic 4: Algebraic Structures in Logic and Relations

Mentor: Filippo Bonchi

Description: Fox’s theorem provides a bridge between structures defined by universal properties (products in a category) and structures specified by algebraic means (comonoids in a symmetric monoidal category). Such a theorem has recently received a renewed interest as the algebraic structures allows for reasoning in terms of string diagrams. While the universal properties underlying logical theories have been extensively studied in categorical logic, their algebraic counterparts have been the objects of fewer investigations. This raises a natural question: can we capture the universal content of logical theories algebraically? In other words, what are the ‘Fox theorems’ for logic? In this project, we attempt to answer to this question by taking as starting point Cartesian bicategories which serves as algebraic setting for regular logic.

Readings:

• Aurelio Carboni and R. F. C. Walters, Cartesian bicategories I.

• Filippo Bonchi, Jens Seeber and Pawel Sobocinski, Graphical conjunctive queries.

• Filippo Bonchi, Dusko Pavlovic and Pawel Sobocinski, Functorial semantics for relational theories.

Posted by: John Baez on January 28, 2022 6:12 PM | Permalink | Reply to this

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