Dr Markus Upmeier

Biography

I am a Lecturer in Mathematics at the University of Aberdeen. I joined Aberdeen in 2021, after positions at the University of Oxford and as Akademischer Rat a.Z. at the University of Augsburg.

My research lies in global analysis, topology, and geometry, focusing on index-theoretic structures on moduli spaces.

At Aberdeen, I am Director of Postgraduate Teaching in Mathematics and Programme Coordinator for MSc Financial Technology. I currently supervise PhD and MSc students, and am involved in postgraduate education and outreach.

I have organised scientific events including the Newton Institute Satellite Programme Topology, Representation Theory and Higher Structures, the ICMS workshop Geometric Moduli Spaces, and a forthcoming Oberwolfach workshop. I have been awarded external funding for research, including a DFG Heisenberg Fellowship.

With Dominic Joyce, I am preparing an OUP monograph based on the preprint Bordism categories and orientations of moduli spaces. It gives a systematic account of orientation problems in moduli theory, and aims to make the subject accessible to a broad mathematical audience.

Selected Publications

My research has two connected strands. The first concerns global analysis and index theory, especially determinant line bundles and Fredholm problems arising from moduli spaces. The second concerns homological structures on moduli spaces.

Global analysis and index theory

Bordism invariance of orientations and real APS index theory,
Advances in Mathematics 461, 2025.

Canonical orientations for moduli spaces of G₂-instantons (with D. Joyce),
Journal of Differential Geometry 124, 2023.

Orientation data for moduli spaces of coherent sheaves over Calabi–Yau 3-folds (with D. Joyce),
Advances in Mathematics 381, 2021.

Homological structures on moduli spaces

Homological Lie brackets on moduli spaces and pushforward operations in twisted K-theory,
Journal of Topology 18, 2025.

Vertex F-algebras and their associated Lie algebra,
SIGMA 22, 2026.

Research Overview

My research lies in global analysis, topology, and geometry, with a particular focus on applications of index-theoretic methods to moduli spaces.

Many moduli spaces arising in gauge theory, special holonomy, and algebraic geometry are governed by elliptic operators. While their numerical index gives the expected dimension, further refined structures - determinant line bundles, orientations, spin structures, spectral flow, and Fredholm boundary data - are essential for constructing deformation-invariant enumerative invariants.

Current Research

A central aim of my work is to develop systematic tools for constructing and comparing these refined structures. This includes orientation and spin problems for gauge-theoretic moduli spaces, moduli spaces of coherent sheaves on Calabi-Yau manifolds, and moduli spaces arising in G2- and Spin(7)-geometry.

With Dominic Joyce, I am preparing an OUP monograph based on our arXiv preprint Bordism categories and orientations of moduli spaces which aims to give a systematic and accessible account of orientation problems in moduli theory for a broad mathematical audience.

I am also developing a Fredholm theory for manifolds and domains with boundary and corners.

A related, more topological strand of my research studies homological structures on moduli spaces, including operations in generalized cohomology and their connections with vertex-algebraic structures.

Teaching Responsibilities

I teach across pure and applied mathematics. My recent teaching includes Real Analysis, Complex Analysis, Financial Mathematics, and Mathematics for Sciences.

At Aberdeen, I am Director of Postgraduate Teaching in Mathematics and Programme Coordinator for the MSc Financial Technology. These roles involve curriculum planning, student support, programme development, and coordination of postgraduate teaching. I currently supervise PhD and MSc students, as well as undergraduate projects.

I also contribute to mathematical outreach, including school visits and events such as Maths Challenge Scotland.

I am an Associate Fellow of the Higher Education Academy.