Master's programs and master's projects

We are welcoming master's students. To do your master's project at CTA², you need to enroll in the mathematics 2-years master's program of the VU Amsterdam, which starts every September and February. The entry requirements are a bachelor's degree in mathematics or closely related fields. The research for the master's project is undertaken during the second year of study. You choose your supervisor during the first year of study.

The master's program requires to follow some classes. Currently offered courses to choose from that are of most interest for researchers in geometry and topology are:

  • Algebraic Geometry 1, 2, advanced
  • Elliptic curves
  • Algebraic Topology 1, 2
  • Topological data analysis
  • Homotopy type theory
  • Differential Geometry
  • Riemannian surfaces
  • Symplectic geometry
  • Poisson geometry
  • Graph symmetries and Combinatoric design
  • Category theory
  • Topos theory

The topic for the master thesis is chosen in discussion with the supervisor. The following list is a compilation of suggested master projects at CTA². 

Project: Studying knots formed viral DNA – The family of Twist-wrap knots
Supervisor: Senja Barthel
Short Description: The formation of knot types that appear in viral DNA are studied, allowing to test models of DNA packing. The knots form a special class that are related to a particular class of braids, whose properties can be studied in their own right.

Project: Testing persistent homological descriptors for materials' property prediction.
Supervisor: Senja Barthel
Short Description: Several choices of generating persistent barcodes form chemical structures are compared with respect to their ability to predict properties of nanoporous materials (e.g. gas uptake, accessible surface area).

Project: Are all knotless leveled spatial graphs primitive?
Supervisor: Senja Barthel
Short Description: A leveled spatial graph a leveled spatial graph consists of an unknotted cycle with subgraphs attached to it, arranged so that the subgraphs lie on stacked discs whose common boundary is the cycle. A spatial graph is knotless if it does not contain a nontrivial knot. It is primitive, if contracting any spanning tree results in a trivial (i.e. can be drawn on the plane without crossings) bouquet of circles. The question is whether all knotless leveled spatial graphs are primitive.

Project: Tight asymptotic bounds in Topological Data analysis
Supervisor: M.B. Botnan
Short Description: Persistent homology is a central tool in topological data analysis which offers a multiscale view of data. The standard algorithm runs in cubic time (input size is the number of simplices) and it is easy to see that the algorithm takes no more than cubic time. However, an example by Morozov shows that this bound is asymptotically tight. In this project, we shall study Morozov’s construction to construct new examples in single and multi-parameter persistent homology. 

Project: Generalizing symplectic geometry from 1 to 2 dimensions
Supervisor: Oliver Fabert
Short description: Geodesics (= locally shortest paths) on Riemannian manifolds can be studied in the framework of symplectic geometry. Generalizing from one-dimensional geodesics to two-dimensional minimal surfaces (= soap films), it is the goal of this thesis to explore the corresponding two-dimensional generalization of symplectic geometry.

Project: Eilenberg–MacLane spaces via configuration spaces
Supervisor: Inbar Klang
Short description: Spaces of configurations of points in a manifold show up in many fields of mathematics, including algebraic topology, knot theory, and mathematical physics. For example, they can be used to model Eilenberg–MacLane spaces, which represent cohomology. This project aims to investigate how structures on cohomology manifest in the configuration space models of Eilenberg–MacLane spaces.

Project: Turán problem and eigenvalues of hypergraphs
Supervisor: Raffaella Mulas
Short description: Given integers n>= k > r >= 2, the Turán problem consists of determining or estimating the largest integer t such that there exists an r-uniform hypergraph on n vertices and t edges that does not contain any complete r-uniform hypergraph on k vertices as a sub-hypergraph. In this project, we aim to investigate the spectral properties of optimal solutions to this problem and its variants.

Project: Higher orientability
Supervisor: Thomas Rot
Short Description: A manifold is orientable if it is possible to consistently choose an orientation of the tangent bundle at each point. An example of an orientable manifold is the two dimensional sphere, while the Mobius strip is a non-orientable manifold. There are generalizations of this concept such as spin, string, 5-brane. These concepts are well studied in the classical literature. Recently Renee Hoekzema introduced a different kind of higher orientability. For this Hoekzema introduced a new class of spaces, whose topological properties will be studied in this project.