Ulrich Bauer, Institute of Science and Technology of Austria
Persistent homology is an algebraic tool for measuring topological features of shapes and functions. Given a filtration of a topological space (a nested sequence of subspaces), it determines the lifespan of homological features (connected components, tunnels, voids) within the filtration.
The course will cover the fundamental ideas of persistent homology, fundamental results such as the stability of persistence diagrams, and motivating applications such as homology inference of shapes from point clouds and topological simplification of scalar functions.
Bruno Benedetti, Freie Unversität Berlin, Germany
Discrete Morse theory
Morse theory studies smooth surfaces or manifolds by looking at generic real-valued functions defined on them. Morse theory was developed in 1927 by Marston Morse; it represents "the most important single contribution to mathematics by an American mathematician", according to Fields laureate Steven Smale.
Discrete Morse theory is a way to simplify a triangulated surface/manifold (or even an arbitrary simplicial complex) maintaining some of its topological properties, like homotopy and homology groups. The theory was developed by Forman in 1998, and unlike the smooth counterpart, it is quite elementary.
In this short course, we plan to cover (1) basic definitions, (2) relations between smooth and discrete Morse vectors, (3) obstructions from elementary knot theory and (4) computational approaches.
Ivan Bratko, University of Ljubljana
Qualitative analysis, modelling and simulation of dynamic systems
Typically, dynamic systems are modelled and analysed quantitatively, for example with differential equations, and system simulation is carried out numerically. In contrast to this, qualitative reasoning and modelling is an area of Artificial Intelligence where modelling, reasoning and simulation are done qualitatively, without numbers. In such an approach, the velocity of a car may be “high”, or “low”, etc., and it may be increasing or decreasing. In this lecture, an approach to qualitative modelling and simulation based on qualitative differential equations will be introduced. It will be shown how a qualitative model of a systems can be automatically learned from observed quantitative behaviours of the system. The techniques will be illustrated by applications to robot learning, and to reconstructing human operator’s skill of controlling a crane.
Jeff Erickson, University of Illinois at Urbana-Champaign, USA
Basic algorithms for surface-embedded graphs
For many classical algorithmic graph problems, faster algorithms are known for graphs that have additional structure. This short course will survey some important algorithmic techniques for graphs that can be drawn in the plane or other surfaces without crossing edges. The course will introduce several fundamental mathematical tools, including Euler's formula, rotation systems, duality, tree-cotree decompositions, the combinatorial Gauss-Bonnet theorem, homotopy, homology, covering spaces, balanced separators, and treewidth, as well as applications of these tools for computing minimum spanning trees, shortest paths, minimum cuts, and approximation solutions for several NP-hard problems.
Massimo Ferri, Università di Bologna, Italy
Mathematics, Shape, Computer Vision
Abstract: This will be an excursus of many aspects of mathematics – in particular of geometry and topology – which are applied in the fields of shape analysis and computer vision.
Mathematical subjects to be touched: gradients, critical points, transforms, distances, transformation groups, persistent homology.
Applications: contour extraction, alignment detection, shape from X, shape retrieval, automatic diagnosis.
Sara Kališnik, Stanford University
The Classification of Hepatic Lesions Using Multidimensional Persistent Homology
This presentation will build on work on hepatic lesions conducted by Aaron Adcock, Daniel Rubin, Gunnar Carlsson at Stanford University. The authors develop a methodology for classifying these lesions using multidimensional persistent homology and a matching metric (called the bottleneck distance), as well as a support vector machine. I will conclude by presenting their classification results on a dataset of 132 lesions that have been outlined and annotated by radiologists.
Danica Kragić, KTH Royal Institute of Technology, Sweden
Data analysis in integration of perception and action
A robot system needs to autonomously acquire new knowledge through interaction with the environment. The knowledge can be acquired only if suitable perception-action capabilities are present: a robotic system has to be able to detect, attend to and manipulate objects in the environment as well as interact with people and other robots. We present our work in the area of vision based sensing and control with specific objectives on attention, segmentation, multisensory control and learning with a focus on what and how to represent to achieve intelligent behaviour in robots.
Bojan Mohar, Simon Fraser University, Canada, and IMFM, Slovenia
Beyond Planarity of Graphs
Importance of planar graphs and some more general classes of graphs in mathematics, computer science and applications will be discussed.
Florian Pokorny, KTH Royal Institute of Technology, Sweden
Topological Concepts and Robotic Manipulation
The manipulation of complicated or deformable objects poses a challenge to current robotic systems. We will discuss recent efforts undertaken at KTH which attempt to adapt ideas from topology for applications in robotic grasping and machine learning. We will present some of the research questions and challenges that one faces when using topological ideas in the real world, and we will discuss how approximately shortest homology generators, winding numbers and Gauss linking integrals can be used in robotics.
Marinka Žitnik, UL FRI, Slovenia
Topological Methods in Machine Learning
Fast growth in the amount of data emerging from studies across various scientific disciplines and engineering requires alternative approaches to understand large and complex data sets in order to turn data into useful knowledge. Topological methods are making an increasing contribution in revealing patterns and shapes of high-dimensional data sets. Ideas, such as studying the shapes in a coordinate free ways, compressed representations and invariance to data deformations are important when one is dealing with large data sets. In this talk we consider which key concepts make topological methods appropriate for data analysis and survey some machine learning techniques proposed in the literature, which exploit them. We illustrate their utility with examples from computational biology, text classification and data visualization.