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Applied Topology Day

A day for Dutch researchers interested in applied topology to come together, exchange ideas and initiate collaborations.

Vrije Universiteit Amsterdam, 30th September

NU-building, 9th floor (Math department), seminar room next to the common area.

Tentative schedule

10:00 – 10:50 Coffee, arriving

10:50 – 11:00 Welcome 

11:00 – 11:45 Renee Hoekzema (VU Amsterdam)

11:45 – 12:30 Fabio Buccoliero (VU Amsterdam)

12:30 – 14:00 Lunch at the Basket

14:00 – 14:45 Rien van de Weijgaert (Groningen)

14:45 – 15:30 Roland van der Veen (Groningen)

15:30 – 15:45 Coffee

15:45 – 16:30 Jo Ellis-Monaghan (UvAmsterdam)

16:30 – 18:00 Borrel

(There will not be a hybrid or online version of the event.)

Organisers:

Magnus Botnan, Senja Barthel

Abstracts:

1) Renee Hoekzema, VU Amsterdam

Title: Multiscale methods for gene selection in single cell transcriptomics data 

Abstract:  Single cell transcriptomics is a revolutionary technique in biology that allows for the measurement of gene expression levels in many individual cells simultaneously. Analysis of these vast datasets reveals complex variation in expression patterns between cells. Current analytical methods assume that cell types are discrete. However, in practice there is also continuous variation between cells: subtypes of subtypes, differentiation pathways, responses to treatment, et cetera. The complexity found in modern single cell transcriptomics datasets calls for intricate methods to biologically interpret both discrete clusters as well as continuous variations. We propose topologically-inspired data analysis methods that identify coherent gene expression patterns on multiple scales. The multiscale methods consider discrete and continuous transcriptional patterns on equal footing. As well as finding new biologically meaningful genes, the methodology allows one to visualise and explore the space of gene expression patterns in the dataset.

2) Fabio Buccoliero, VU Amsterdam

Title: Cellular embeddings of spatial graphs

Abstract: Every graph can be embedded on a closed oriented surface and the genus range for the surface is known. If instead of an abstract graph an embedding of a graph is considered, it is still easy to find a closed oriented surface on which the spatial graph is embedded on; simply think of placing the graph on the boundary of a small tubular neighbourhood around it. But in general there is no good control over the complement of the graph in a surface, it could be a union of discs with any number of punctures. We are interested in finding surfaces for a spatial graph, such that the complement of the graph in the surface is a set of open discs, so called cellular embeddings. The talk presents the difficulties of the problem, a construction for such a surface for sufficiently simple spatial graphs, and finishes with an outlook on the next steps towards a more general solution. 

3) Rien van de Weygaert, Kapteyn Astronomical Institute, University of Groningen

Title: The Cosmic Web: Complexity, Connectivity and Persistent Topology of the largest structure in the Universe 

Abstract:  The Cosmic Web is the fundamental spatial organization of matter on scales of a few up to a hundred  millions of lightyears. Galaxies, intergalactic gas and dark matter have aggregated in a wispy weblike network of dense compact clusters, elongated filaments, and sheetlike walls, amidst large near-empty void regions. An important additional aspect of this mass distribution is that it is marked by substructure over a wide range of scales and densities. A unique aspect of the cosmic web is its connectivity, the way in which its various structural components are spatially organized in a weblike network. Persistent Topology has provided us with the mathematical foundation and concepts to describe and quantify this key aspect of the large scale cosmic matter distribution. over the past decades, we have quantified the outcome of computer simulations in terms of their betti numbers and persistence diagrams. Most of this has been based on the analysis of the multiscale spatial density distribution and/or the distance field in terms of alphashapes.In the final part of the presentation I will describe recent developments in which we have coupled the dynamics of cosmic web formation to the multiscale-persistent-topology of the large-scale tidal/deformation field.

4) Roland van der Veen, Groningen university

Title: Discretizing partial differential equations using algebraic topology

Abstract: The goal of this talk is to promote the interaction between the fields of numerical mathematics and algebraic topology. Many PDE of interest such as the Navier-Stokes equation can be formulated abstractlyin terms of deRham cohomology. Lifting this formalism to abstract (co)homologythe recent work of Lawrence, Ranade and Sullivan shows one can obtain a tower of attractive discretizations that might improve/augment the common practice of finite element methods.

5) Jo Ellis-Monaghan,  KdVI, University of Amsterdam

 Title: A little topology for DNA self-assembly

Abstract: Currently, scientists are engineering self-assembling DNA molecules to serve emergent applications in biomolecular computing, nanoelectronics, biosensors, drug delivery systems, and organic synthesis.  These emergent technologies in self-assembly are now generating fascinating and challenging  design problems, thus giving rise to a new branch of mathematics, DNA mathematics.  Often, the self-assembled objects, e.g. lattices or polyhedral skeletons, may be modeled as graphs.  Since these graphs represent physical objects in 3-space, low-dimensional topology plays an important role in DNA mathematics. We will explore both how topology informs DNA design problems and also how new problems in topological graph theory arise from the design problems.