• Dmitry Feichtner-Kozlov (University of Bremen) - Applied and Computational Topology
  Abstract: We will give a brief introduction to the subject of Applied and Computational Topology. The survey of the subject's main ideas and tools will be complemented with applications to discrete mathematics and to theoretical distributed computing.

We will conclude with stating an open problem in combinatorial topology which is related to the complexity of the Weak Symmetry Breaking distributed task.

• Eugenia Saorin Gomez (University of Bremen) -  Common projections of convex bodies and inequalities within the L_p Brunn Minkowski theory
  Abstract: Please download the abstract here.

• Eva-Maria Feichtner (University of Bremen) - Bergman complexes of matroids
  Abstract: Bergman complexes are intertwined with the theory of hyperplane arrangements in various ways. They figure as tropicalizations of arrangement complements and, at the same time, provide combinatorial core structure for certain arrangement compactifications. They add to the collection of abstract (simplicial, resp. polyhedral) complexes associated with matroids, which makes them interesting objects from the viewpoint of algebraic and topological combinatorics. We review old and new results on these beautiful complexes. 

• John Harvey (Cardiff University) - The influence of sample density, topology and curvature on persistent homology
  Abstract: Bubenik, Hull, Patel and Whitten (2020) showed that, in theory, persistent homology can be used to detect the curvature of a metric space. At first glance, this idea appears to conflict with Bobrowski and Skraba’s conjecture (2022) that there is a ‘universal’ distribution for persistent homology. This is reconciled by observing that the power of persistent homology in this setting declines as the sample size increases. In this talk I will present work with Yim which investigates both experimentally and theoretically how sample density and the topology of the support of the sample all interact with curvature to make this a challenging task.

• Martin Henk (TU Berlin) - Notes on upper bounds for the volume of projection bodies
  Abstract: We show that the volume of the projection body $\Pi(Z)$ of an n-dimensional zonotope Z with n + 1 generators and unit volume is always exactly $2^n$. Moreover, we point out that an upper bound on the volume of  $\Pi(K)$ of a centrally symmetric n-dimensional convex body K of unit volume  is at least $2^n\cdot (9/8)^⌊n/3⌋$.

• Ambrose Yim (Cardiff University) - Morse reduction of poset-graded chain complexes via algebraic Morse theory
  Abstract: The Morse-Conley complex is a central tool for homological theories for dynamical systems and information compression in topological data analysis. Given a poset-graded chain complex of vector spaces, a Conley complex is the minimal chain-homotopic reduction of the initial complex that respects the poset grading.  In this work, we give a purely algebraic derivation of the Conley complex using algebraic Morse theory, shedding new light on current algorithmic and combinatorial approaches. We show how this algebraic perspective also yields efficient algorithms for computing the Conley complex.  This talk features joint work with Álvaro Torras Casas and Ulrich Pennig. 

• Linus Wiegmann (University of Bremen) - On inequalities in the elliptic Brunn-Minkowski theory
  Abstract: Please download the abstract here.