Special Year Research Seminar

Ideas from tame geometry have recently begun to find their way into quantum field theory and string theory, suggesting that consistent effective theories and their observables may admit definable descriptions of finite complexity. In this talk I will discuss recent joint work exploring this perspective and its connections with arithmetic geometry, Hodge theory, and o-minimality. A guiding question is whether finiteness in physics can be understood as a manifestation of tameness. I will discuss finiteness questions for spaces of theories and distinguished loci in moduli space, as well as structural constraints on amplitudes and observables. Perturbatively, amplitudes are naturally expressed in terms of period integrals, while beyond perturbation theory even simple examples seem to force broader tame frameworks involving non-analytic, multisummable phenomena. I will emphasize the role of complexity, especially in the sharper quantitative form suggested by sharp o-minimality, and the challenge of identifying the right tame structures for physics. Throughout, I will indicate some open questions and directions, both on the physics side and in the mathematics that these developments bring into focus.