Applied category theory: Information structures and modular systems
Event details
| Date | 15.09.2015 › 18.09.2015 |
| Hour | 10:15 › 12:00 |
| Speaker | David Spivak (MIT) |
| Location |
CM 10 (T, Th, F) & CM 113 (W)
|
| Category | Conferences - Seminars |
Over the past 75 years, category theory has organized and standardized much of the discipline of pure mathematics. In this 8-hour mini-course, we will consider how and in what respects it may have a similar effect outside of math, namely in science and industry. Indeed, category theory's success stems from its ability to connect, characterize, and translate between, disparate subjects.
We will begin with a short, non-standard review of category theory, where we will discuss its 35-year role in computer science, as the semantics of functional programming languages. This will take us through the definition of symmetric monoidal categories, which can be thought of as governing the arithmetic of parallel and serial composition. After the review, we will turn our attention to information-bearing structures, such as databases and ontologies. We will see how querying a database is a special case of translating data from one structure to another, which itself can be modeled by the category theoretic notion of Kan extensions.
In the second part of the course, we will discuss the notion of modularity---from data flow diagrams, to hierarchical protein materials, to connected arrangements of dynamical systems---and model each of these modular systems with operads and their algebras. Operads are a category-theoretic framework that controls how a higher-level system can be formed as an arrangement of interacting components. For example, we will spell out a case in which an operad models hives made up of agents, which interact by sending signals to each other. These signals control the internal states of each agent, and, in turn, the agents' internal states determine the communication pattern connecting agents in the hive. The operadic nature of this model amounts to a formal sense in which the hive itself forms an agent like any other. Finally, we will consider databases, programs, and matrices again, this time from the perspective of modular systems.
The audience for this mini-course is meant to include both mathematicians and scientists. One goal is to increase communication between scientific disciplines and pure math, a subject to which I consider category theory as a primary gateway. I hope that this mini-course will foster collaborations between mathematicians and other researchers.
We will begin with a short, non-standard review of category theory, where we will discuss its 35-year role in computer science, as the semantics of functional programming languages. This will take us through the definition of symmetric monoidal categories, which can be thought of as governing the arithmetic of parallel and serial composition. After the review, we will turn our attention to information-bearing structures, such as databases and ontologies. We will see how querying a database is a special case of translating data from one structure to another, which itself can be modeled by the category theoretic notion of Kan extensions.
In the second part of the course, we will discuss the notion of modularity---from data flow diagrams, to hierarchical protein materials, to connected arrangements of dynamical systems---and model each of these modular systems with operads and their algebras. Operads are a category-theoretic framework that controls how a higher-level system can be formed as an arrangement of interacting components. For example, we will spell out a case in which an operad models hives made up of agents, which interact by sending signals to each other. These signals control the internal states of each agent, and, in turn, the agents' internal states determine the communication pattern connecting agents in the hive. The operadic nature of this model amounts to a formal sense in which the hive itself forms an agent like any other. Finally, we will consider databases, programs, and matrices again, this time from the perspective of modular systems.
The audience for this mini-course is meant to include both mathematicians and scientists. One goal is to increase communication between scientific disciplines and pure math, a subject to which I consider category theory as a primary gateway. I hope that this mini-course will foster collaborations between mathematicians and other researchers.
Practical information
- Informed public
- Free
Organizer
- Kathryn Hess