Mathematical aspects of quantum field theory

In this lecture we provide an introduction to recent mathematical approaches to quantum field theory (QFT). After a short review of classical field theory, we shall focus on the locally covariant approach to algebraic QFT. The axiomatic formalism will be outlined and discussed at the level of examples. In the second part of the lecture we shall focus on the important class of gauge theories. Their description requires us to introduce suitable techniques from homotopical algebra, in particular homotopy sheaf theory, in order to formalize descent in (quantum) gauge theories. Some background material for this course can be found in: [1] C. J. Fewster and R. Verch, "Algebraic quantum field theory in curved spacetimes," arXiv:1504.00586 [math-ph]. [2] W. G. Dwyer and J. Spalinski, "Homotopy theories and model categories," in: Handbook of algebraic topology, 73-126, North-Holland, Amsterdam (1995). [3] S. Hollander, "A homotopy theory for stacks," Israel J. Math. 163, 93 (2008). [4] M. Benini, A. Schenkel and R. J. Szabo, "Homotopy colimits and global observables in Abelian gauge theory," Lett. Math. Phys. 105, no. 9, 1193 (2015).