Speaker: Professor Ran Levi, University of Aberdeen.

Title: Neuro-Topology: An interaction between topology and neuroscience.

Abstract: While algebraic topology is now well established as an applicable branch of mathematics, its emergence in neuroscience is surprisingly recent. In this talk I will present a summary of an ongoing joint project with mathematician and neuroscientists. I will start with some basic facts on neuroscience and the digital reconstruction of a rat’s neocortex by the Blue Brain Project in EPFL. I will then explain how data emerging from this reconstruction can be mapped into abstract graphs that in turn give rise to certain mathematical objects in the realm of algebraic and combinatorial topology. Following a short introduction to some of the basic tools of algebraic topology, I will explain how they can potentially be used in the context of neuroscience. Having set up the scene, I will proceed by presenting the results of an ongoing collaboration with the Blue Brain Project team. In particular I shall demonstrate how the topological techniques give new insights on the behaviour of neural systems and inspire new directions in neuroscience research.

Speaker: Wojtek Chachólski from KTH (Stockholm)

Title: What is persistence?

Abstract: In this talk I will describe what persistence means for us (TDA group at KTH in Stockholm), how to measure it, and how to attempt to make statistical conclusions using it. Our proposed solution is equally suitable for one and multi parameter situations. It can be used to illustrate why understanding correlations is a hard problem as we show that for correlations our persistence signatures are in general NP hard to calculate. This is in contrast with single measurement situations, where calculating the signatures requires basically a linear time. I will present several illustrations of how our invariant can be used for effective clustering. The talk is aimed at general audience.

The TDA group at KTH in Stockholm consists of O. Gafvert, R. Ramanujam, H. Riihimaki, and myself.

Speaker: Nikki Sanderson, University of Colorado, Boulder

Title: Witness Complexes for Time Series Analysis

Abstract: Time series analysis relying upon statistics and frequency analyses makes restrictive assumptions about the underlying data - i.e. nonlinearity, non-stationarity. We believe topological data analysis (TDA) can be of benefit in these situations. Yet time series do not necessarily have interesting topology in their own right. The process of delay coordinate reconstruction "unfolds" a scalar time-series into a point cloud in R ^m . We can then compute the persistent homology of the reconstructed data to obtain a topological signature. With the ultimate goal of online regime shift detection in mind, we choose to use the witness complex - a sparse simplicial complex - for these computations. Topologically accurate delay reconstruction requires appropriate choices for the dimension m and time delay. We introduce novel witness relations that incorporate time and improve the robustness of the resulting homology with respect to choice of delay. These new relations seek to inhibit data points from witnessing landmarks traveling in dissimilar directions, as these can create false connections. We explore how these relations simultaneously address challenges that arise when dealing with non-uniform samples of strange attractors from chaotic dynamical systems.