In many situations in Differential or Algebraic Geometry, one forms moduli spaces $\cal M$ of geometric objects, such that $\cal M$ is a manifold, or something close to a manifold (a derived manifold, Kuranishi space, …). Then we can ask whether $\cal M$ is orientable, and if so, whether there is a natural choice of orientation.

  This is important in the definition of enumerative invariants: we arrange that the moduli space $\cal M$ is a compact oriented manifold (or derived manifold), so it has a fundamental class in homology, and the invariants are the integrals of natural cohomology classes over this fundamental class.

  For example, if $X$ is a compact oriented Riemannian 4-manifold, we can form moduli spaces $\cal M$ of instanton connections on some principal $G$-bundle $P$ over $X$, and the Donaldson invariants of $X$ are integrals over $\cal M$.

  In the paper arXiv:2503.20456, Markus Upmeier and I develop a theory of "bordism categories”, which are a new tool for studying orientability and canonical orientations of moduli spaces. It uses a lot of Algebraic Topology, and computation of bordism groups of classifying spaces. We apply it to study orientability and canonical orientations of moduli spaces of $G_2$ instantons and associative 3-folds on $G_2$ manifolds, and of Spin(7) instantons and Cayley 4-folds on Spin(7) manifolds, and of coherent sheaves on Calabi-Yau 4-folds. These have applications to enumerative invariants, in particular, to Donaldson-Thomas type invariants of Calabi-Yau 4-folds.

   All this is joint work with Markus Upmeier.

Feynman integrals are the essential building blocks of observables in perturbative Quantum Field Theories. As precision experiments in high-energy physics are becoming more common, understanding the structure of higher loop integrals has become very important from a phenomenology point of view. On the mathematical physics side, such investigations have led to profound connections to geometry. In particular, there is a correspondence between Feynman integrals and algebraic varieties and knowing what geometry a given Feynman integral corresponds to offers invaluable lessons in solving it. In this talk, I will start with a pedagogical review of modern methods to solve higher loop integrals. Then, with a simple example, I will show how one can infer the geometry associated with an integral and discuss some of the implications of this connection.