# Schedule of the Spring School

## Monday, April 24

10:40 - 11:10 |
Registration & Welcome coffee |

11:10 - 11:20 |
Opening remarks |

11:20 - 12:00 |
Samuel Mimram: Introduction to Concurrency Theory through Algebraic Topology (part 1) |

12:00 - 13:30 |
Lunch break |

13:30 - 14:10 |
Samuel Mimram: Introduction to Concurrency Theory through Algebraic Topology (part 2) |

14:30 - 16:00 |
Ulrich Bauer: Algebraic perspectives of Persistence |

16:00 - 16:30 |
Tea and cake |

16:30 - 18:00 |
Gerard Ben Arous: Geometry of random Morse functions |

afterwards |
Reception |

## Tuesday, April 25

9:20 - 10:50 |
Martin Raussen: Topological and combinatorial models of directed path spaces |

10:50 - 11:20 |
Group photo and coffee break |

11:20 - 12:00 |
Daniel Hug: Introduction to Stochastic Geometry (part 1) |

12:00 - 13:30 |
Lunch break |

13:30 - 14:10 |
Daniel Hug: Introduction to Stochastic Geometry (part 2) |

14:30 - 16:00 |
Roy Meshulam: High dimensional Expanders |

16:00 - 16:30 |
Tea and cake |

16:30 - 17:30 |
Poster session |

## Wednesday, April 26

9:20 - 10:50 |
Daniel Hug: Point Processes in Spatial Statistics |

10:50 - 11:20 |
Coffee break |

11:20 - 12:50 |
Ran Levi: Neurotopology – a beginning |

12:50 - |
Lunch break, free time (excursion?) |

## Thursday, April 27

9:20 - 10:50 |
Maurice Herlihy: Introduction to Distributed Computing through Combinatorial Topology |

10:50 - 11:20 |
Coffee break |

11:20 - 12:00 |
Dmitry Kozlov: Topology of complexes arising in models for Distributed Computing (part 1) |

12:00 - 13:30 |
Lunch break |

13:30 - 14:10 |
Dmitry Kozlov: Topology of complexes arising in models for Distributed Computing (part 2) |

14:30 - 16:00 |
Ginestra Bianconi: Large Random Networks |

16:00 - 16:30 |
Tea and cake |

16:30 - 17:30 |
Software and problem session |

## Friday, April 28

9:20 - 10:50 |
Michael Kerber: Computational perspectives of Persistence |

10:50 - 11:20 |
Coffee break |

11:20 - 12:00 |
Frederic Chazal: Sampling and Topological Data Analysis (part 1) |

12:00 - 13:30 |
Lunch break |

13:30 - 14:10 |
Frederic Chazal: Sampling and Topological Data Analysis (part 2) |

14:30 - 16:00 |
Michael Farber: Topology of large random spaces |

16:00 - 16:30 |
Tea and cake, farewell |

# Abstracts

## Michael Kerber: Computational perspectives of Persistence

The computational pipeline of topological data analysis consists of three major steps:

(1) Deriving a multi-scale representation of the underlying data set

(2) Computing topological invariants of that representation

(3) Interpreting the outcome of (2) to draw conclusions about the data

Each of the three steps comes with its own algorithmic challenges. Obviously, the interest of tda in various applied fields asks for software to handle all these tasks efficiently, generally, and reliably. We will review standard techniques for all three steps, in particular:

@(1) how to generate filtrations of simplicial complexes out of point clouds and scalar fields,

@(2) how to compute persistent homology of a filtration efficiently using matrix reduction,

@(3) how to compare persistent diagrams efficiently.

My conference talk on Tuesday morning will be a continuation on the topic, presenting recent advances on the three steps of the pipeline.

## Martin Raussen: Topological and combinatorial models of directed path spaces

Concurrency theory in Computer Science studies effects that arise when several processors run simultaneously sharing common resources. It attempts to advise methods dealing with the "state space explosion problem" characterized by an exponentially growing number of execution paths; sometimes using models with a combinatorial/topological flavor. It is a common feature of these models that an execution corresponds to a directed path (d-path) in a (time-flow directed) state space, and that d-homotopies (preserving the directions) have equivalent computations as a result.

Getting to grips with the effects of the non-reversible time-flow is essential, and one needs to "twist" methods from ordinary algebraic topology in order to make them applicable. An essential task consists in inferring information about spaces of executions (directed paths) between two given states from information about the state space. The determination of path components is particularly important for applications.

I will discuss particular directed spaces arising from Higher Dimensional Automata (HDA). There are various methods identifying the homotopy type of the space of executions between two states in such an automaton with some finite complex: in simple cases as prodsimplicial complex – with products of simplices as building blocks – or as a configuration space living in a product of simplices. In several interesting cases, it is possible to give an explicit description of the homotopy type of the Alexander dual of such a configuration space and hence of the stable homotopy type of the corresponding trace space. This opens up for calculations of homology groups and of other topological invariants of some execution spaces.

We sketch a method recently devised by Ziemiański identifying – for a general HDA – a space of directed paths with a prodpermutahedral complex arising by glueing various permutahedra along their boundaries.

Joint work with L. Fajstrup (Aalborg), E. Goubault, E. Haucourt, S. Mimram (Éc. Polytechnique, Paris), R. Meshulam (Haifa) and K. Ziemiański (Warsaw).