In this talk, we will present some initial work on planning using topological abstraction. We will consider a multi-pursuer multi-evader problem as a case study to ground the discussion. We will describe how we can cast the abstraction problem as a topological problem and by leveraging sheaf theoretic methods develop a framework to search for strategies that lead to evader capture. We will show how the sheaf can be leveraged to encode the belief state of the problem and how the capture event can be formalized through topological property of the sheaf.
Recent data sparsification strategies in topological data analysis such as Graph Induced Complex and sparsified Rips complex give rise to a sequence of simplicial complexes connected by simplicial maps rather than inclusions. As a result, the need for computing topological persistence under such maps arises. We propose a practical algorithm for computing such persistence in Z2-homology.
Efficient handling of sparse data is a key challenge in Computer Science. Binary convolutions, such as the Fast Fourier Transform or theWalsh Transform are a useful tool in many applications and are efficiently solved. In the last decade, several problems required efficient solution of sparse binary convolutions.
Computers crash, hang, succumb to viruses, run buggy programs, and harbor spyware. By contrast, mathematics is free of all imperfection. Why are imperfect computational devices so vital for the future of mathematics?