Research

Lennart Meier

Teaching



I am an assistant professor at the University of Utrecht, specializing in algebraic topology
and related areas of arithmetic geometry. Here is my CV.

My research is part of the Utrecht Geometry Center. We have now large group of algebraic
topologists, including Ieke Moerdijk, Gijs Heuts, Brice Le Grignou, Magdalena Kedziorek,
Mingcong Zeng and Hadrian Heine.

We have a joint topology seminar with Nijmegen, called TopICS and happening regularly on Fridays.

Contact information:

Mathematical Institute of Utrecht
Budapestlaan 6
Kamer 508
Utrecht, Nederland
E-Mail: f.l.m.meier at uu.nl


Teaching

Research

News:

  • There is a derived algebraic geometry seminar running in Utrecht, which pauses over summer, but will resume next semester.
  • I will be travelling July 30 to August 23 (in particular Oberwolfach and Shanghai)


  • My Zentralblatt Reviews    My MathSciNet Reviews

    Research interests

    I am an algebraic topologist with a special interest in the spectrum of topological modular forms TMF. I am also interested in several related
    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:

  • Understanding torus-equivariant TMF (joint with D. Gepner and T. Nikolaus)
  • 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)

  • Preprints and Publications

  • Rings of modular forms and a splitting of TMF0(7) (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, arXiv:1707.05986 (joint with Rosona Eldred, Gijs Heuts and Akhil Mathew)

  • 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 , 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)

  • Non-arXived Preprints

    Lecture notes and expository writing

    Theses

    Talks

    Academic Year 2019/20

  • Block 1, Elementary number theory
  • Fall, Mastermath Algebraic topology I
  • 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.

    Students

  • Currently I supervise the master theses of Joost van Geffen, Jorge Becerra and Jeroen van der Meer and the PhD thesis of Jack Davies.
  • I supervised the master thesis of Simone Fabbrizzi in Bonn on the computation of Brauer groups of moduli stacks of elliptic curves
  • I supervised four 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!

    Prerequisites

  • 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%.

    Lecture notes/Literature

    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.

    Topologie en meetkunde 2019

    General information

    This is a Blok 3 course meeting on Tuesdays and Thursdays. See the syllabus for details on meeting times. All further information can be found on blackboard.

    Content

    The general aim of the course will be to apply algebraic tools to topological questions. In more detail:

  • We will start with a discussion, what topology is about. Can we classify spaces up to homeomorphism? Can we apply topology in analysis or algebra? This will quickly lead to the notion of a homotopy, a central concept for our course.

  • Using this notion, we will define the fundamental group of a space. Our first aim is to calculate it for the circle using the basics of covering space theory. Later we will give a beautiful classification of coverings of a space in terms of the fundamental group.

  • We will describe a way to calculate the fundamental group of CW-complexes, i.e. spaces build from spheres and disks (which includes most manifolds). The central tool is the van Kampen theorem that allows us to compute the fundamental group of a union of two spaces.

  • In the middle of the course, we will prove our first classification results. More precisely, we will classify closed one-dimensional manifolds (which is easy) and closed two-dimensional manifolds.

  • In the last weeks we will introduce the concept of homology. This can be used to solve higher-dimensional problems the fundamental group cannot deal with. Moreover it provides a way to show the topological invariance of the Euler characteristic, a number that first arose in Eulers's theorem on polyhedra. This opens the road to further applications.

    Recommended sources

    Our main sources are the books Algebraic Topology by Hatcher and further books by Lee and Munkres. We will cover most material up to Chapter 2.2, but will not follow the book always too closely. The only part that is not covered in Hatcher's book is the classification of surfaces, for which I wrote lecture notes.

    Prerequisites

    Inleiding Topologie provides more than enough background for this course.
  •