Reaction-diffusion processes underpin countless biological phenomena. In these systems, fluctuations play an important role, particularly when the number of interacting units is small. In chemical reactions, for example, mean-field approaches such as the law of mass action neglect fluctuations and thus fail to capture absorbing states. In this talk, I will introduce Doi-Peliti field theory (DPFT), a microscopic framework that uses ladder operators to represent particle dynamics in second-quantized form. Its path-integral formulation is exact at the level of the action and allows observables to be calculated systematically in a perturbative expansion.

I will show how DPFT can be used to model biological systems, including oscillating neuronal avalanches [1] and disease spreading on general graphs [2]. I will also present recent progress in microscopic theories of active matter, describing the emergence of effective attraction between interacting run-and-tumble particles [3,4] and the role of delay in reconfigurable chiral microswimmers [5].

[1] J. Pausch, R. Garcia-Millan, and G. Pruessner, Time-dependent branching processes: a model of oscillating neuronal avalanches, Sci. Rep. 10, 13678 (2020). 
[2] I. Bordeu, S. Amarteifio, R. Garcia-Millan, B. Walter, N. Wei, and G. Pruessner, Volume explored by a branching random walk on general graphs. Sci. Rep. 9, 15590 (2019).
[3] R. Garcia-Millan, L. Cocconi, Z. Zhang, M. Bothe, L. Chen, Z. Zhen, G. Pruessner, Effective attraction by repulsion, arXiv:2605.01421 (2026). 
[4] R. Garcia-Millan, Z. Zhang, L. Cocconi, M. Bothe, L. Chen, Z. Zhen, G. Pruessner, Microscopic theory of soft run- and-tumble particles, arXiv:2605.01422 (2026).
[5] J. Peltier, C. Britton, L. Alvarez, R. Garcia-Millan, in preparation (2026).

I will describe a circle-valued Morse theory for simplicial complexes. The central objects of study are partial matchings which admit certain zigzag cycles; these cyclic matchings lift canonically to acyclic matchings on the infinite cyclic cover of the underlying simplicial complex. From the lifted acyclic matchings, we obtain a finitely generated Morse chain complex defined over the Novikov ring, which consists of power series in one variable with finite negative support. We then establish a quasi-isomorphism between this Morse-Novikov complex and the simplicial chain complex of the cyclic cover, duly completed over the Novikov ring. As a pleasant consequence, we can define new computable invariants to detect (obstructions to) the fiberedness of tame knots.