David Spivak

Visiting Assistant Professor
University of Oregon
Department of Mathematics

Office: 317 Fenton Hall
Email: dspivak at uoregon


"Information is the new physics."

Using mathematics to model the physical universe has proven quite successful (for both math and physics). In the modern world, it has also become imperative that we find better ways to classify, organize, process, and understand information. For this, we should again turn to mathematics.

I think that category theory and topology have a lot to offer in this regard. In order to find somewhere to start, I've been working on aspects of computer science which are relevant to the goal of understanding information. The idea is that we already do have ways of classifying, organizing, and processing information; by modeling them mathematically, we can understand their conceptual underpinnings, which in turn can help us create new and better systems in the future.

The following technical proposal (which is not too technical) may help make this more clear and explicit:

Databases and Networks. This is a proposal for a grant I received from the ONR (N000140910466). In it, I propose to use category theory to study databases, networks, and learning. It is a more technical version of the proposal Networking academia I had earlier (unsuccessfully) submitted for a grant called "Science and the Human Condition" at the University of Oregon. The idea here is to study networks, as found in computer science, economics, linguistics, sociology, and biology, under a single mathematical framework. The document is written for a layperson.


Simplicial databases. A paper that categorifies databases. In this model, a "schema" for a database is a labeled simplicial set, whose n-simplicies correspond to tables with n+1 columns. We form a simplicial set by taking the union of such simplices, and doing so reflects the links between objects of differing types. The data itself is then a sheaf on the corresponding site. This category of databases is closed under limits and colimits and these functors correspond to common operations on databases, such as joins, selects, and unions. [Submitted to Mathematical Structures in Computer Science, and the arXiv.]
I gave a talk on this subject at the 2008 ATMCS meeting in Paris, and wrote a more brief survey of the ideas here. written out as a .pdf document.
A primitive version of this talk in which we only discuss the category of tables, was given as a colloquium in the University of Oregon computer science department. It can be found here.

Higher-dimensional models of networks. In this paper, I make a case that many networks involve interactions between more than two entities, and as such should not be modeled by graphs but by simplicial sets. I give a variety of other models and suggest ways to determine which model is most applicable for a given situation, and how to transform models from one type into another. The paper is written mainly for a CS audience familiar with the basics of category theory. It can also be found on the arXiv. A more mathematical paper on networks and information processing is in the works.

Metric realization of fuzzy simplicial sets. Like weighted graphs, fuzzy simplicial sets can be used to model networks. The idea is that the connection between a group of people is a number between 0 and 1. The linked paper does not discuss these real-world applications -- just some preliminary thoughts on the mathematics involved.