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 (last updated July 2022).

My research is part of the Utrecht Geometry Center. We have now large group of algebraic topologists, including Ieke Moerdijk, Gijs Heuts, Guy Boyde, Tobias Lenz, Niall Taggart, Miguel Barata, Max Blans, Yuqing Shi and my PhD students Sven van Nigtevecht and Jaco Ruit.

We have a joint topology seminar with Nijmegen, called TopICS and a Learning Seminar on Goodwillie Calculus. See also the UGC page of the Algebraic Topology Group.

Contact information:

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


Teaching

Research

News:

  • October 2022: My student Jack Davies has defended his thesis and has started a postdoc in Bonn. Congratulations!

  • October 2022: My student Jaco Ruit has posted a nice paper about pasting theorem for higher categories: A pasting theorem for iterated Segal spaces. Congratulations!

  • September 2022: My student Sven van Nigtevecht posted a nice paper about K-theory cochains and unstable homotopy theory at height $1$: The K-theory cochains of H-spaces and height 1 chromatic homotopy theory. Congratulations!

  • In May 2022, we have had a Spring School on the Interactions of Algebraic Topology and Field theories.

  • November 2021: My student Jaco Ruit has recently posted together with Fabian Hebestreit and Gijs Heuts a nice paper on the arXiv: A short proof of the straightening theorem. Congratulations!

  • Beginning with September 1 (2021), I welcome Sven van Nigtevecht as my new PhD student.

  • June 2021: My student Jack Davies has recently posted two nice papers on the arXiv: Elliptic cohomology is unique up to homotopy and Constructing and calculating Adams operations on topological modular forms. Congratulations!

  • November 2020: My proposal Understanding symmetries of spaces via modular forms has been awarded a VIDI grant from the NWO. Advertisements for one postdoc position will follow in autumn 2021.

  • Beginning December 1 (2020), I welcome Jaco Ruit as my new PhD student.

  • July 2020: My proposal The interplay of orientations and symmetry has been awarded an ENW-KLEIN grant from the NWO.


    • 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 stable and unstable chromatic homotopy theory and (derived) algebraic geometry.
    Currently I am thinking mostly about various aspects of equivariant homotopy theory, topological modular forms and Brauer groups.

    Preprints and Publications

  • Picard sheaves, local Brauer groups, and topological modular forms, arXiv:2210.15743 (joint with Ben Antieau and Vesna Stojanoska)

  • Connective Models for Topological Modular Forms of Level n, arXiv:2104.12649 (accepted at Algebraic & Geometric Topology)

  • Norms of Eilenberg-Mac Lane Spectra and Real Bordism, arXiv:2008.04963 (joint with XiaoLin Danny Shi and Mingcong Zeng) (accepted at Advances in Mathemaics)

  • On equivariant topological modular forms, arXiv:2004.10254 (joint with David Gepner)

  • Purity in chromatically localized algebraic K-theory, arXiv:2001.10425 (joint with Markus Land, Akhil Mathew and Georg Tamme)

  • Rings of modular forms and a splitting of TMF0(7) (joint with Viktoriya Ozornova), Selecta Mathematica, February 2020, 26:7 (arXiv)

  • A Whitehead theorem for periodic homotopy groups (joint with Tobias Barthel and Gijs Heuts), Israel Journal of Mathematics volume 241 (2021), 1–16 (arXiv)

  • Topological modular forms with level structures: decompositions and duality, Trans. Amer. Math. Soc. 375 (2022), 1305-1355 (arXiv)

  • Decomposition results for rings of modular forms, arXiv:1710.03461 (accepted at Documenta Mathematica)

  • 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 (joint with Ben Antieau), Algebra and Number Theory Vol. 14 (2020), No. 9, 2295–2333 (arXiv)

  • 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'
          (joint with Niko Naumann and Justin Noel), Journal of the EMS Volume 22, Issue 4, 2020, pp. 1149–1200 (arXiv)

  • 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 , 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

  • Computation of weight 1 cusp forms (2018)

  • Brauer groups of stacks via coarse moduli spaces (2017)

  • Relatively free TMF-modules (2017)

  • Forms of BPR<n> (2017)

  • Homotopy Colimits of Relative Categories (2014) (Notes)


    • Lecture notes and expository writing

  • The classification of surfaces (2019)

  • Introduction to Stable Homotopy Theory (2018)

  • Lecture notes on elliptic genera, elliptic cohomology and TMF (2017, updated later)

  • The moduli stack of elliptic curves (joint with Viktoriya Ozornova) (2017)

  • Hilbert Manifold Models for Mapping Spaces (2009)

    • Theses

  • My PhD Thesis - United Elliptic Homology (2012)

  • My Diploma Thesis - A Geometric View on String Topology (2009)

    • Talks

  • Chromatic localizations of algebraic K-theory (Online Talk at CRM, 2021)

  • Equivariant topological modular forms (ECHT, 2021)

  • Elliptic cohomology of level n (OATS, 2020)

  • The chromatic behaviour of algebraic K-theory (eAKTS, 2020)

  • Equivariant elliptic cohomology with integral coefficients (Online Talk at Perimeter Institute, 2020)

  • The chromatic behaviour of algebraic K-theory (Online Talk at MSRI, 2020)

  • Chromatic localizations of algebraic K-theory (video, Banff 2020)

  • Topological modular forms with level (video, Cambridge 2018)

  • Homotopy of Real K-theory and its duals (ECHT, 2017)

  • TMF with level structure: Decompositions and Duality (video, Banff 2016)

  • Fibrancy of (Relative) Categories (YTM 2014, Copenhagen)

  • Modules over real K-Theory and TMF (YTM 2012, Copenhagen)

  • Vereinigte elliptische Homologie (Verteidigungsvortrag)

    • Academic Year 2022/23

  • Block 2, Inleiding Topologie
  • Spring, Algebraic Topology II (jointly with Steffen Sagave)

    • Academic Year 2021/22

  • Block 2, Inleiding Topologie
  • Spring, Algebraic Topology II (jointly with Steffen Sagave)
  • Spring, Master seminar on the immersion conjecture and cobordism (jointly with Max Blans)

    • Academic Year 2020/21

  • Block 1, Bewijzen in de wiskunde
  • Block 2, Inleiding Topologie
  • Fall, Algebraic Topology I (jointly with Gijs Heuts)
  • Spring, Master seminar on spectral sequences and spectra (jointly with Gijs Heuts)

    • 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.

    Students

  • Currently I supervise the PhD thesis of Sven van Nigtevecht and Jaco Ruit and the master theses of Ryan Quinn (Equivariant Localization Theorems), Leon Goertz, Bouke Jansen and Antonie de Potter.

  • In the past I have supervised the master theses of Simone Fabbrizzi (Brauer group of moduli stacks of elliptic curves), Joost van Geffen (The oriented cobordism ring), Jorge Becerra (K-theory with Reality), Jeroen van der Meer (KU-local stable homotopy theory), Abe ten Voorde (Cyclic algebras over local fields arising from elliptic curves), Pascal Sitbon (Equivariant bordism), Divya Ghanshani (Homology theories in cofibration categories) and Christiaan van den Brink (Collapsing theorem for Delaunay complexes in non-general position and symmetry).

  • Moreover I have supervised or am supervising bachelor theses on the topics of de Rham cohomology, applications of combinatorial topology to concurrent computation, homology with local coefficients, genus formulae for modular curves, framed bordism, principal bundles, elliptic curves, path integrals, the representation theory of SO(3) and SU(2) in relation with spin, Clifford algebras, braid groups, monads, Haskell, the Aharonov--Bohm effect, and the Stone representation theorem.

  • Generally I am happy to supervise bachelor theses in many parts of pure mathematics (including relations to theoretical physics), but especially in the area of topology. If you are a student interested in a thesis or reading project, please contact me!

    • Prerequisites

  • 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%.

    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.
    •