The Azimuth Project Applied Category Theory Seminar (Rev #12)

Idea

We’re going to have a seminar on applied category theory here at U. C. Riverside, starting in January 2019. We will make it easy to have discussions on the Azimuth Forum and Azimuth Blog. These will work best if you read the papers we’re talking about and then join these discussions. We will also try to videotape the talks, to make it easier for you to follow along.

Here’s the schedule of talks so far — more are coming later. The talks are on Tuesdays 3:34–5:00 pm in Room 268 of Skye Hall, the mathematics building at U. C. Riverside.

January 8, 2019: John Baez - Mathematics in the 21st century

Abstract. The global warming crisis is part of a bigger transformation in which humanity realizes that the Earth is a finite system and that our population, energy usage, and the like cannot continue to grow exponentially. If civilization survives this transformation, it will affect mathematics - and be affected by it - just as dramatically as the agricultural revolution or industrial revolution. We should get ready!

Watch the video while also looking at the talk slides. For more information click on the links in green.

Also try these slides and videos from related talks:

January 15, 2019: Jonathan Lorand - Problems in symplectic linear algebra

Jonathan Lorand is visiting U. C. Riverside to work with our group on applications of symplectic geometry to chemistry.

Abstract. In this talk we will look at various examples of classification problems in symplectic linear algebra: conjugacy classes in the symplectic group and its Lie algebra, linear lagrangian relations up to conjugation, tuples of (co)isotropic subspaces. I will explain how many such problems can be encoded using the theory of symplectic poset representations, and will discuss some general results of this theory. Finally, I will recast this discussion from a broader category-theoretic perspective.

Talk slides:

January 22, 2019: Christina Vasilakopoulou - Systems as wiring diagram algebras

Abstract. We will start by describing the monoidal category of labeled boxes and wiring diagrams and its induced operad. Various kinds of systems such as discrete and continuous dynamical systems have been expressed as algebras for that operad, namely lax monoidal functors into the category of categories. A major advantage of this approach is that systems can be composed to form a system of the same kind, completely determined by the specific way the composite systems are interconnected (‘wired’ together). We will then introduce a generalized system, called a machine, again as a wiring diagram algebra. On the one hand, this abstract concept is all-inclusive in the sense that discrete and continuous dynamical systems are sub-algebras; on the other hand, we can specify succinct categorical conditions for totality and/or determinism of systems that also adhere to the algebraic description.

Christina Vasilakopoulou’s talk will be based on this paper:

but she will focus more on the algebraic description (and conditions for deterministic/total systems) rather than the sheaf theoretic aspect of the input types. This work builds on earlier papers such as these:

January 29, 2019: Daniel Cicala - Social contagion modeled on random networks

Abstract. A social contagion may manifest as a cultural trend, a spreading opinion or idea or belief. In this talk, we explore a simple model of social contagion on a random network. We also look at the effect that network connectivity, edge distribution, and heterogeneity has on the diffusion of a contagion.

• Mason A. Porter and James P. Gleeson, Dynamical systems on networks: a tutorial.

• Duncan J. Watts, A simple model of global cascades on random networks.

February 5, 2019: Jade Master - Backprop as functor: a compositional perspective on supervised learning

Abstract. Fong, Spivak and Tuyéras have found a categorical framework in which gradient descent algorithms can be constructed in a compositional way. To explain this, we first give a brief introduction to backprogation and gradient descent. We then describe their monoidal category $Learn$, where the morphisms are given by abstract learning algorithms. Finally, we show how gradient descent can be realized as a monoidal functor from $Para$, the category of Euclidean spaces with differentiable parameterized functions between them, to $Learn$.

February 22, 2019: Christian Williams - The pi calculus: towards global computing

Abstract. Historically, code represents a sequence of instructions for a single machine. Each computer is its own world, and only interacts with others by sending and receiving data through external ports. As society becomes more interconnected, this paradigm becomes more inadequate - these virtually isolated nodes tend to form networks of great bottleneck and opacity. Communication is a fundamental and integral part of computing, and needs to be incorporated in the theory of computation.

To describe systems of interacting agents with dynamic interconnection, in 1980 Robin Milner invented the pi calculus: a formal language in which a term represents an open, evolving system of processes (or agents) which communicate over names (or channels). Because a computer is itself such a system, the pi calculus can be seen as a generalization of traditional computing languages; there is an embedding of lambda into pi - but there is an important change in focus: programming is less like controlling a machine and more like designing an ecosystem of autonomous organisms.

We review the basics of the pi calculus, and explore a variety of examples which demonstrate this new approach to programming. We will discuss some of the history of these ideas, called “process algebra”, and see exciting modern applications in blockchain and biology.

“… as we seriously address the problem of modelling mobile communicating systems we get a sense of completing a model which was previously incomplete; for we can now begin to describe what goes on outside a computer in the same terms as what goes on inside - i.e. in terms of interaction. Turning this observation inside-out, we may say that we inhabit a global computer, an informatic world which demands to be understood just as fundamentally as physicists understand the material world.” — Robin Milner