Please see the schedule below for the time and place of each talk.

Some seminars will take place on the Lausanne campus and others at Campus Biotech.

### Program

Date and place | Title | Speaker |
---|---|---|

04.10.2017 13h Campus Biotech B1.05 Videoconf room |
Witness complexes for time series analysis | Nikki Sanderson University of Colorado |

11.10.2017 15h Campus Biotech B1.05 Videoconf room |
Geometry, topology, and the structure of data | Jacek Brodzki University of Southampton |

20.10.2017 14h Campus Biotech B1.05 Videoconf room |
A survey of directed algebraic topology | Eric Goubault LIX, Ecole Polytechnique |

02.03.2018 14h Campus Biotech B1.05 Videoconf room |
Decoding of neural data using cohomological learning | Erik Rybakken NTNU |

13.03.2018 09h15 MA 12, EPFL |
Different approaches to the data skeletonization problem | Yury Elkin Liverpool |

28.03.2018 16h15 CM 9, EPFL |
Curvature-based analysis of complex networks | Melanie Weber Princeton |

08.05.2018 14h15 CM 9, EPFL |
Structure and evolution of topological brain scaffolds | Giovanni Petri ISI Foundation |

15.05.2018 14h15 GC A1 416, EPFL |
Emergent complex network geometry and frustrated synchronization | Ginestra Bianconi Queen Mary University London |

15.06.2018 14h Campus Biotech B1.05 Videoconf room |
Higher order partial least squares and an application to neuroscience | Luke Oeding Alabama |

05.07.2018 11h15 GR A3 31 |
Computational tools for handling simplicial complexes in real datasets | Alice Patania Indiana |

15.08.2018 14h15 MA 30 |
Evolution of the homology and related geometric properties of the Eden Growth Model | Erika Roldan Roa CIMAT |

### Abstracts

**Sanderson: **Time series analysis traditionally relies upon statistics and frequency analyses that make restrictive assumptions about the data – i.e. nonlinearity, non-stationarity. We believe topological data analysis (TDA) can be of benefit in these situations. 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 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. The 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 can ameliorate additional challenges that arise when dealing with non-uniform samples of strange attractors.

**Brodzki: ** Modern data is astonishing in its variety, and is far removed from anything that could be held in spreadsheet or analysed using traditional methods. Through intense research effort, topology has emerged as a source of novel methodology to provide insight into the structure of very complex, high dimensional data. It provided us with tools like persistent homology, which are used to compute numerical topological characteristics of the data. More recently, these methods have been augmented by geometric insights, which are valuable in capturing the structure of and relationships between complex shapes. In this talk, I will provide an overview of new techniques from topology and geometry and illustrate them on particular examples. One set of data was created through the study of CT scans of human lungs, and another addresses the problem of classification of three-dimensional shapes.

**Goubault: ** I will present in this talk some results obtained in directed algebraic topology, a theory of spaces with a preferred direction (typically caused by time being considered as « irreversible »), up to deformation. This has originally appeared in semantics and analysis of concurrent programs, but is natural in many contexts (fault-tolerant distributed systems, differential inclusions and control, rewriting systems, type theory…and even, physics and some other areas of mathematics). It has been a long struggle to find a way to build some of these results upon the vast body of knowledge of homotopy and homology theories, and we will try to show what has been achieved as well as what definitely has not, up to now. I will present joint work along the years with Martin Raussen, Lisbeth Fajstrup, Emmanuel Haucourt, Samuel Mimram, Jeremy Dubut, Jean Goubault-Larrecq…among others.

**Rybakken: ** We introduce a novel data-driven approach to discover and decode features in the neural code coming from large population neural recordings with minimal assumptions, using cohomological learning. We apply our approach to neural recordings of mice moving freely in a box, where we find a circular feature. We then observe that the decoded value corresponds well to the head direction of the mouse. Thus we capture head direction cells and decode the head direction from the neural population activity without having to process the behaviour of the mouse.

**Elkin: ** Given a point cloud in a Euclidean space we consider the problem of approximating the cloud by an embedded graph in the same space. We will review past approaches to related data skeletonization problems via Reeb graphs, Mapper algorithm and a 1-dimensional Homologically Persistent Skeleton. Then we will introduce a new approximate skeleton for detecting topological shapes of micelles in 3 dimensions. The talk is based on joint work with Vitaliy Kurlin at the University of Liverpool.

**Weber: ** Complex networks are popular means for studying a wide variety of systems across the social and natural sciences. Recent technological advances allow for a description of these systems on an unprecedented scale. However, due to the immense size and complexity of the resulting networks, efficient evaluation remains a data-analytic challenge. In a recent series of articles, we developed geometric tools for efficiently analyzing the structure and evolution of complex networks. The core component of our theory, a discrete Ricci curvature, translates central tools from differential geometry to the discrete realm. With these tools, we extend the commonly used node-based approach to include edge-based information such as edge weights and directionality for a more comprehensive network characterization.The analysis of a wide range of complex networks suggests connections between curvature and higher order network structure. Our results identify important structural features, including long-range connections of high curvature acting as bridges between major network components. Thus, curvature identifies the network’s core structure on which expensive network hypothesis testing and further network analysis becomes more feasible. We will discuss an application of curvature-based methods to networks constructed from fMRI data. Joint work with E. Saucan (Technion) and J. Jost (MPI MIS).

**Petri: ** Topology, one of the oldest branches of mathematics, provides an expressive and affordable language which is progressively pervading many areas of biology, computer science and physics. I will illustrate the type of novel insights that algebraic topological tools are yielding in the study of the functional, structural and genetic connectomes. At the pure functional level, I will how topological observables capture and distinguish variations in the mesoscopic functional organization in two case studies: i) between drug-induced altered brain states, and ii) between perceptual states and the corresponding mental images. Moving to the structural level, I will compare the homological features of structural and functional brain networks across a large age span and highlight the presence of dynamically coordinated compensation mechanisms, suggesting that functional topology is conserved over the depleting structural substrate. Finally, using brain gene expression data, I describe recent work on the construction of a topological genetic skeleton highlighting differences in structure and function of different genetic pathways within the brain.

**Bianconi: **Topology has been recently shown to be key to study neuronal dynamics and brain function. In this context experiments probing the dynamics of networks of neuronal cultures have revealed that this dynamics is strongly dependent on the network topology and in particular on the network dimensionality. However, this phenomenon has been so far mostly unexplored from the theoretical point of view. Here we reveal the rich interplay between network topology, emergent network geometry and synchronization of coupled oscillators in the context of a simplicial complex model of manifolds called Complex Network Manifold. The networks generated by this model combine small world properties (infinite Hausdorff dimension) and a high modular structure with finite and tunable spectral dimension. We show that the networks display frustrated synchronization for a wide range of the coupling strength of the oscillators, and that the synchronization properties are directly affected by the spectral dimension of the network.

**Oeding: ** Partial least squares (PLS) is a method to discover a functional dependence between two sets of variables X and Y. PLS attempts to maximize the covariance between X and Y by projecting both onto new subspaces. Higher order partial least squares (HOPLS) comes into play when the sets of variables have additional tensorial structure. Simultaneous optimization of subspace projections may be obtained by a multilinear singular value decomposition (MSVD). I’ll review PLS and SVD, and explain their higher order counterparts. Finally I’ll describe recent work with G. Deshpande, A. Cichocki, D. Rangaprakash, and X.P. Hu where we propose to use HOPLS and Tensor Decompositions to discover latent linkages between EEG and fMRI signals from the brain, and ultimately use this to drive Brain Computer Interfaces (BCI)’s with the low size, weight and power of EEG, but with the accuracy of fMRI.

**Patania: ** In this talk, I will present some computational tools that are useful when we want to think of real data as simplicial complexes. I will first present a homology preserving mapping between simplicial complexes that, under certain conditions that are often met in real datasets, can reduce the computational complexity of the computation of homology. To demonstrate its efficiency, I will apply it in a case study to both real simplicial datasets—co-authorship inferred from the arXiv—and random simplicial complexes. To this end, I will present a randomized model of simplicial complexes that reproduce some important properties of real systems, as well as the associated MCMC sampler. Finally, through a topological study of the arXiv dataset, I will how empirically how homological cycles can be an important part of the underlying community linking structure.

**Rolda Roa: ** In this talk, we study the persistent homology and related geometric properties of the evolution in time of a discrete-time stochastic process defined on the 2-dimensional regular square lattice. This process corresponds to a cell growth model called the Eden Growth Model (EGM). It can be described as follows: start with the cell square of the 2-dimensional regular square lattice of the plane that contains the origin; then make the cell structure grow by adding one cell at each time uniformly random to the perimeter. We give a characterization of the possible change in the rank of the first homology group of this process. Based on this result we have designed and implemented a new algorithm that computes the persistent homology associated to this stochastic process and that also keeps track of geometric features related to the homology. Also, we present obtained results of computational experiments performed with this algorithm, and we establish conjectures about the asymptotic behavior of the homology and other related geometric random variables. The EGM can be seen as a First Passage Percolation model after a proper time-scaling. This is the first time that tools and techniques from stochastic topology and topological data analysis are used to measure the evolution of the topology of the EGM and in general in FPP models.