Dr Markus Upmeier
I joined the University of Aberdeen this year as a permanent Lecturer, following four years as a Simons Collaboration researcher at the University of Oxford and teaching at St. Anne's College. Previously, I was visiting assistant professor (Akad. Rat a.Z.) at the University of Augsburg. I completed my PhD in 2013 at the University of Göttingen, advised by Thomas Schick.
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.
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.
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.
Non-course Teaching Responsibilities
- Two MX4023 4th year projects on "Lie groups and representation theory"
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Vertex F-algebra structures on the complex oriented homology of H-spacesJournal of Pure and Applied Algebra, vol. 226, no. 8, 107019Contributions to Journals: Articles
Canonical orientations for moduli spaces of G_2-instantons with gauge group SU(m) or U(m)Journal of Differential GeometryContributions to Journals: Articles
A categorified excision principle for elliptic symbol familiesQuarterly Journal of Mathematics, vol. 72, no. 3, pp. 1099–1132Contributions to Journals: Articles
Connections on central extensions, lifting gerbes, and finite-dimensional obstruction vanishingChapters in Books, Reports and Conference Proceedings: Conference Proceedings
On spin structures and orientations for gauge-theoretic moduli spacesAdvances in Mathematics, vol. 381, 107630Contributions to Journals: Articles
Orientation data for moduli spaces of coherent sheaves over Calabi-Yau 3-foldsAdvances in Mathematics, vol. 381, 107627Contributions to Journals: Articles
Closed almost Kähler 4-manifolds of constant non-negative Hermitian holomorphic sectional curvature are KählerTohoku Mathematical Journal, vol. 72, no. 4, pp. 581-594Contributions to Journals: Articles
Integrability theorems and conformally constant Chern scalar curvature metrics in almost Hermitian geometryCommunications in Analysis and Geometry, vol. 28, no. 7, pp. 1603 – 1645Contributions to Journals: Articles
On orientations for gauge-theoretic moduli spacesAdvances in Mathematics, vol. 362, 106957Contributions to Journals: Articles
Chern's contribution to the Hopf problem: An exposition based on Bryant's paperDifferential Geometry and its Applications, vol. 57, pp. 138-146Contributions to Journals: Articles