Lecturer
- About
-
- Email Address
- markus.upmeier@abdn.ac.uk
- Telephone Number
- +44 (0)1224 272752
- Office Address
Department of Mathematics, University of Aberdeen, Elphinstone Rd, Aberdeen AB24 3UE, U.K.
Office: Fraser Noble Building, Room 157
- School/Department
- School of Natural and Computing Sciences
Biography
I am Lecturer in Mathematics at the University of Aberdeen. Before joining as faculty, I was a Simons Collaboration researcher at the University of Oxford. I completed my PhD in 2013 at the University of Göttingen, advised by Thomas Schick
Current activities
- Reading seminar on the Baez-Dolan cobordism hypothesis and infinity-categories
- I am the organizer of the Topology Seminar
Lecture Notes
- Research
-
Research Overview
I am interested in applications of algebraic topology, particularly of index theory and higher categories, to study infinite-dimensional spaces and moduli spaces in gauge theory and algebraic geometry.
Most of the mathematical questions that I work on arise at the interface with theoretical physics. I am also very interested in exploring topological methods that can be applied to other branches of science.
Current Research
My research applies homotopy theory to study the topology of various moduli spaces, including Yang-Mills instanton moduli spaces and moduli spaces of holomorphic curves. The applications range from open problems in algebraic geometry to the mathematical development of quantum invariants with values in K-theory and elliptic cohomology, as currently studied by physicists.
Fundamental topological questions about moduli spaces are whether they are orientable (leading to a virtual fundamental class in ordinary homology, and to classical Donaldson invariants), admit a spin structure (so that one may use the Dirac operator for quantization, and define a virtual fundamental class in K-homology), or further refinements to elliptic homology. The higher differential topology of moduli spaces (spin, string, and fivebrane structures) is controlled by higher categorical analogues of the Quillen determinant line bundle, and one of my research objectives is to construct these.
The systematic study of these virtual fundamental classes leads to vertex algebra structures, as invented in conformal field theory. They encode sophisticated symmetries, for example, a conformal vector induces a Virasoro action. My research in this area is about the connections to generalized cohomology and bordism theory.
Past Research
In the time following my PhD, I worked on generalized differential cohomology - a marriage of stable homotopy theory and gauge theory. Later, I got interested in extremal metrics and integrability problems in almost Kähler geometry.
- Publications
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Page 1 of 2 Results 1 to 10 of 17
Bordism categories and orientations of gauge theory moduli spaces
Working Papers: Preprint Papers- [ONLINE] DOI: https://doi.org/10.48550/arXiv.2312.10516
Bordism invariance of orientations and real APS index theory
Working Papers: Preprint Papers- [ONLINE] DOI: https://doi.org/10.48550/arXiv.2312.06818
- [OPEN ACCESS] http://aura.abdn.ac.uk/bitstream/2164/23854/1/bio.pdf
Canonical orientations for moduli spaces of G2-instantons with gauge group SU(m) or U(m)
Journal of Differential Geometry, vol. 124, no. 2, pp. 199-229Contributions to Journals: ArticlesVertex F-algebra structures on the complex oriented homology of H-spaces
Journal of Pure and Applied Algebra, vol. 226, no. 8, 107019Contributions to Journals: ArticlesA categorified excision principle for elliptic symbol families
Quarterly Journal of Mathematics, vol. 72, no. 3, pp. 1099–1132Contributions to Journals: Articles- [ONLINE] DOI: https://doi.org/10.1093/qmath/haaa063
Connections on central extensions, lifting gerbes, and finite-dimensional obstruction vanishing
Chapters in Books, Reports and Conference Proceedings: Conference Proceedings- [ONLINE] https://arxiv.org/abs/1910.10402
- [ONLINE] DOI: https://doi.org/10.1090/conm/766
- [ONLINE] AAM at University of Oxford Repository
On spin structures and orientations for gauge-theoretic moduli spaces
Advances in Mathematics, vol. 381, 107630Contributions to Journals: Articles- [ONLINE] DOI: https://doi.org/10.1016/j.aim.2021.107630
Orientation data for moduli spaces of coherent sheaves over Calabi-Yau 3-folds
Advances in Mathematics, vol. 381, 107627Contributions to Journals: Articles- [ONLINE] DOI: https://doi.org/10.1016/j.aim.2021.107627
Homological Lie brackets on moduli spaces and pushforward operations in twisted K-theory
Working Papers: Preprint Papers- [ONLINE] DOI: https://doi.org/10.48550/arXiv.2101.10990
Closed almost Kähler 4-manifolds of constant non-negative Hermitian holomorphic sectional curvature are Kähler
Tohoku Mathematical Journal, vol. 72, no. 4, pp. 581-594Contributions to Journals: Articles- [ONLINE] DOI: https://doi.org/10.2748/tmj.20191025