The main three topics of the workshop will be discussed along the three-day according to the following schedule:
· Day 1: Amplitudes structure and algebraic periods
The perturbative computation of scattering amplitudes requires the evaluation of highly complex Feynman integrals, whose number grows rapidly with the loop order. Remarkably, many of these integrals exhibit cancellations and simplifications a posteriori, pointing to deep redundancies and the presence of hidden structures. Anticipating these structures in advance is crucial: it allows the reduction of the otherwise overwhelming number of integrals, thereby making multi-loop amplitude calculations tractable. Recent advances in algebraic geometry are playing an increasingly central role in this endeavor. They reveal that Feynman integrals are intimately connected to the underlying geometry, often manifesting as algebraic periods, i.e. integrals of algebraic differential forms over topologically defined cycles. This perspective not only provides conceptual clarity but also opens new avenues for systematically organizing and computing multi-loop amplitudes.
· Day 2: Intersection Theory and String Amplitudes
The study of Feynman integrals through the lens of intersection theory—viewing them as products between suitable homology classes (integration contours) and cohomology classes (differential forms)—offers a unifying framework for their analysis. This approach captures both the linear relations (such as integration-by-parts identities) and the quadratic relations (such as double-copy structures) that arise among integrals. In doing so, it provides a powerful method for systematically reducing them to the so-called master integrals, the minimal set of integrals spanning the entire family. An exceptionally successful testing ground for this formalism is provided by string amplitudes. Their computation beyond tree level—even in the bosonic case—remains highly non-trivial. String amplitudes naturally incorporate geometric and modular structures, making them an ideal laboratory for applying intersection-theoretic methods and exploring the interplay between algebraic geometry, topology, and physics. In this way, they not only offer new computational strategies but also serve as a bridge between perturbative quantum field theory and deeper mathematical frameworks. The motivation for studying them extends far beyond string theory itself: they also uncover hidden properties of its low-energy limit, namely Quantum Field Theory, thereby enriching our understanding of both frameworks.
· Day 3: Gravitational waves and Differential equations
A natural application of the ideas outlined above takes place in the gravitational setting, where modern scattering amplitude techniques are applied to compute gravitational-wave observables. The problem of computing the dynamics and gravitational waveforms for compact binary systems reduces to the evaluation of complex families of multi-loop Feynman integrals, further revealing deep dualities between gauge theories and gravity. The analysis of the structure underlying these amplitude computations is revealing fundamental mathematical properties and providing concrete advancements in the precision of gravitational wave observables.
A particularly powerful framework for analyzing the integrals appearing in these perturbative calculations is provided by the theory of D-modules and Gel’fand–Kapranov–Zelevinsky (GKZ) systems. Instead of direct integration, this algebraic framework derives the differential equations governing the integrals, making their singularity structure and monodromy manifest. By systematically classifying the solution spaces of these equations, these methods enable high-precision perturbative computations, bridging abstract algebraic geometry with gravitational phenomenology.
Day-to-day the workshop blends broad perspective review talks (60'), pedagogical bridge lectures on advanced topics (45' ), and six slots for PhD students and early-career researchers talks (25'+5') to present new results and receive detailed feedback. Each day concludes with a special seminar by a senior expert.
