Dr Markus Upmeier

Dr Markus Upmeier
Dr Markus Upmeier

Dr Markus Upmeier

Lecturer

About

Department of Mathematics, University of Aberdeen, Elphinstone Rd, Aberdeen AB24 3UE, U.K.

 

Office: Fraser Noble Building, Room 157

Biography

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.

CV

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.

Teaching

Non-course Teaching Responsibilities

  • Two MX4023 4th year projects on "Lie groups and representation theory"
Publications

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  • Canonical orientations for moduli spaces of G_2-instantons with gauge group SU(m) or U(m)

    Joyce, D., Upmeier, M.
    Journal of Differential Geometry
    Contributions to Journals: Articles
  • A categorified excision principle for elliptic symbol families

    Upmeier, M.
    Quarterly Journal of Mathematics, vol. 72, no. 3, pp. 1099–1132
    Contributions to Journals: Articles
  • On spin structures and orientations for gauge-theoretic moduli spaces

    Joyce, D., Upmeier, M.
    Advances in Mathematics, vol. 381, 107630
    Contributions to Journals: Articles
  • Orientation data for moduli spaces of coherent sheaves over Calabi-Yau 3-folds

    Upmeier, M., Joyce, D.
    Advances in Mathematics, vol. 381, 107627
    Contributions to Journals: Articles
  • Connections on central extensions, lifting gerbes, and finite-dimensional obstruction vanishing

    Biswas, I., Upmeier, M.
    Contemporary Mathematics, vol. 766
    Contributions to Journals: Articles
  • Closed almost-Kähler 4-manifolds of constant non-negative Hermitian holomorphic sectional curvature are Kähler

    Upmeier, M., Lejmi, M.
    Tohoku Mathematical Journal, vol. 72, no. 4
    Contributions to Journals: Articles
  • Integrability theorems and conformally constant Chern scalar curvature metrics in almost Hermitian geometry

    Upmeier, M., Lejmi, M.
    Communications in Analysis and Geometry, vol. 28
    Contributions to Journals: Articles
  • On orientations for gauge-theoretic moduli spaces

    Upmeier, M., Joyce, D., Tanaka, Y.
    Advances in Mathematics, vol. 362
    Contributions to Journals: Articles
  • Chern's contribution to the Hopf problem: An exposition based on Bryant's paper

    Upmeier, M., Tralle, A.
    Differential Geometry and its Applications, vol. 57, pp. 138-146
    Contributions to Journals: Articles
  • The canonical 2-gerbe of a holomorphic vector bundle

    Upmeier, M.
    Theory and Applications of Categories, vol. 32, no. 30, pp. 1028-1049
    Contributions to Journals: Articles