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Stochastic Topology and its Applications

We explore the topology of random combinatorial and geometric structures. Our research includes percolation models on lattices, configuration spaces, and random simplicial and cubical complexes. We also investigate extremal topological structures and their properties.

Open positions

There are open positions for a postdoc and a PhD student and also research projects for undergraduate and master's students from local universities that are studying mathematics, computer science or related areas (e.g., from Leipzig University). Write me a message if you are interested in any of these positions.

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Research

Our research group aims to explore and understand the topology of random combinatorial and geometric structures. These include percolation and first passage percolation models on lattices, configuration spaces, as well as a variety of random simplicial and cubical complexes [1,2]. We also aim to find structures exhibiting extremal topology and study their properties [3].
In many of our projects, we perform computational experiments to gain intuition about the topological behavior of a model and use them to suggest conjectures for future mathematical research (for an example, see our 3D Eden Model simulation below [2,4]). Towards that end, we develop and apply tools and techniques from applied and computational geometry and topology, such as TGDA*. We also use MCMCM-H** algorithms and other algorithms used for performing computational experiments in statistical mechanics.

* Topological and geometric Data Analysis (TDGA) is a recently developed branch of mathematics and computer science that provide state-of-the-art mathematical and computational techniques that have been applied to elucidate the structure of large data sets derived from different scientific areas such as astrophysics, image analysis, oncology, neuroscience, material science, and chemistry, among others.
** Markov Chain Monte Carlo Metropolis-Hasting

Selected publications

[1] Matthew Kahle; Elliot Paquette and Érika Roldán: Topology of random 2-dimensional cubical complexes
Forum of mathematics / Sigma, 9 (2021), e76
[2] Fedor Manin; Érika Roldán and Benjamin Schweinhart: Topology and local geometry of the Eden model
Discrete and computational geometry (accepted for publication)
[3] Greg Malen ; Fedor Manin and Érika Roldán: High-dimensional holeyominoes
The electronic journal of combinatorics, 29 (2022) 3, P3.15

Software

[4] Anna Krymova ; Fedor Manin ; Érika Roldán and Benjamin Schweinhart: Topology and Geometry of the Eden Model
Github

Members

Recent Group Publications