I am an algebraic topologist with a special interest in the spectrum of topological modular forms TMF. I am also interested in several related Understanding torus-equivariant TMF (joint with D. Gepner)
Towards the Brauer group of TMF (joint with B. Antieau and V. Stojanoska)
Unstable periodic homotopy theory (joint with G. Heuts)
Stacks associated with connective tmf with level structures (joint with V. Ozornova)
Galois theory in equivariant context (joint with M. Kedziorek and V. Ozornova)
Periodic localizations of algebraic K-theory (M. Land and G. Tamme)
The dual Steenrod algebra spectrum and hyperreal cobordism (M. Zeng)
and unrelated questions in abstract homotopy theory, stable and unstable chromatic homotopy theory and (derived) algebraic geometry.
Currently I am thinking mostly about the Bousfield-Kuhn functor, TMF with level structures, equivariant TMF and its applications to elliptic
genera and Brauer groups. This includes the following collaborations:
Preprints and Publications
Rings of modular forms and a splitting of TMF0(7), arXiv:1812.04425 (joint with Viktoriya Ozornova)
A Whitehead theorem for periodic homotopy groups, arXiv:1811.04030 (joint with Tobias Barthel and Gijs Heuts)
Topological modular forms with level structures: decompositions and duality, arXiv:1806.06709
Decomposition results for rings of modular forms, arXiv:1710.03461
Monadicity of the Bousfield-Kuhn functor (joint with Rosona Eldred, Gijs Heuts and Akhil Mathew), Proc. Amer. Math. Soc. 147 (2019), 1789-1796 (arXiv)
The Brauer group of the moduli stack of elliptic curves, arXiv:1608.00851 (joint with Ben Antieau)
Gorenstein duality for Real spectra (joint with John Greenlees), Algebraic & Geometric Topology 17 (2017), 3547-3619 , (arXiv\>, Erratum to the published version)
Appendix B: 'Descent for higher real K-theories'
to 'Descent in algebraic K-theory and a conjecture of Ausoni-Rognes', arXiv:1606.03328
(joint with Niko Naumann and Justin Noel, 2016)
The C2-spectrum Tmf1(3) and its invertible modules (joint with Mike Hill), Algebraic & Geometric Topology 17 (2017) 1953-2011 (arXiv)
Fibration Categories are Fibrant Relative Categories, arXiv:1503.02036 , Algebraic & Geometric Topology 16 (2016), 3271-3300. (arXiv)
Fibrancy of Partial Model Categories (joint with Viktoriya Ozornova, 2015) Homology, Homotopy and Appl Volume 17.2 (2015), 53-80 (arXiv)
Affineness and Chromatic Homotopy Theory (joint with Akhil Mathew) J Topology (2015) 8 (2): 476-528 (arXiv, Erratum to the published version)
Vector Bundles on the Moduli Stack of Elliptic Curves, Journal of Algebra Volume 428 (2015), 425-456. (arXiv)
Hilbert Manifolds, Bulletin of the Manifold Atlas (2014, expository)
Spectral Sequences in String Topology, Algebraic & Geometric Topology 11 (2011), 2829-2860. (arXiv)
Lecture notes and expository writing
Academic Year 2019/20
Block 1, Elementaire Getaltheorie (all information on blackboard)
Fall, Mastermath Algebraic topology I
Fall, Orientation on Mathematical Research (WISM102)
Academic Year 2018/19
Block 3, Topologie en Meetkunde
Spring, Mastermath Algebraic Topology II (jointly with Gijs Heuts)
Spring, Master seminar on a rational homotopy theory (jointly with Gijs Heuts)
Academic Year 2017/18
Block 3, Topologie en Meetkunde
Spring, Master seminar seminar on topological K-theory (jointly with Gijs Heuts)
Previously, courses at the University of Virginia and the University of Bonn on linear algebra, multivariable calculus, algebra, homology theories and elliptic cohomology
Currently I supervise the PhD thesis of Jack Davies.
I supervised the master thesis of Simone Fabbrizzi (Brauer group of moduli stacks of elliptic curves), Joost van Geffen (The oriented cobordism ring), Jorge Becerra (K-theory with Reality) and Jeroen van der Meer (KU-local stable homotopy theory)
I supervised bachelor theses so far on the topics of de Rham cohomology, applications of combinatorial topology to concurrent computation, homology with local coefficients and genus formulae of modular curves. Generally I am happy to supervise bachelor theses in topology or on topics related to modular forms or elliptic curves.
If you are student interested in a thesis or reading project, please contact me!
Basic knowledge about topological spaces
Knowledge about basic constructions with vector spaces and abelian groups.
Some familiarity with categories and functors is also helpful. For those who haven't seen this before, the Intensive course on Categories and Modules at the start of the term is recommended.
Aim of the course
This course is an introduction to Algebraic Topology. Its main topic is the study of homology groups of topological spaces and their applications. These homology groups provide algebraic invariants of topological spaces which can be computed in many examples of interest. They are a bit like the fundamental group, only that they are abelian and also better adapted to study higher-dimensional phenomena.
In the first part of the course we will construct the singular homology groups of topological spaces and establish their basic properties, such as homotopy invariance and long exact sequences. In the second part of the course we will introduce CW-complexes. These provide a useful class of topological spaces with favorable properties, and we will explain how the homology of CW-complexes can be computed using cellular homology. We will also discuss some basic concepts from homotopy theory, in particular the generalization of the fundamental group to arbitrary homotopy groups.
Rules about homework/exam
There will be regular hand-in homework sets throughout the course and there will be a written exam at the end.
The score from the homework assignments will count as a bonus for the final examination: if the grade from the written exam is at least 5.0 and the average grade from the homework assignments is higher, then the written exam counts 75% and the homework assignments count 25%.
We will follow the lecture notes of Steffen Sagave
Additional literature includes
Bredon: Topology and Geometry
Hatcher: Algebraic Topology
For more information see the ELO page.