- Algebra & Computation
- Algebraic & Geometric Complexity Theory
- Computational Complexity
- General Theory of Complex Systems
- Networks (comparison, isomorphism, dynamics of, dynamics on)
- Phase transitions in NP-complete problems
- Symmetry in mathematics and complex systems

Omidyar Postdoctoral Fellow

Good theories have the power to shed light on subjects previously shrouded in dark alleys, complicated formulae, and mysterious phenomena. There really ought to be a theory of complex systems, and there isn't. At least, not yet. Although he recognizes that there is unlikely to be a theory of complex systems with the same predictive power as quantum mechanics has for fundamental particles or general relativity has for planetary motions, Joshua A. Grochow strives towards unifying ideas that can help us understand a variety of complex systems from many different fields. He thinks that this may involve deeper mathematics than has typically been used so far to understand complex systems, and possibly even mathematics that hasn't been invented yet.

Joshua sees computational complexity - the study of the fundamental limitations of the power of algorithms - as an excellent starting point from which to build a theory of complex systems. Individual algorithms can behave in just as complicated a manner as nearly any system out there; indeed, most complex systems are studied by computer simulations, which are really just the output of some algorithm. Yet, compared to many complex systems such as those coming from biology and society, the mathematical theory of algorithms is fairly well-developed. There is also no modelling problem in the theory of algorithms: if a mathematical result says that the economy will behave a certain way, then maybe it will, or maybe the mathematical model was missing something; but if a mathematical result says that an algorithm will behave a certain way, then it actually behaves that way on a real-world computer and in real-world robots. Much of Joshua's research so far has been in using deep mathematical ideas from areas such as geometry and symmetry (technically: algebraic geometry, group theory, and representation theory) to better understand the behavior and limitations of algorithms. At SFI, he has started to bring some of this deep mathematics to bear on more general complex systems.