The first major piece of work I undertook during my stay at the university of Padova, was on the use of the univariate form of the intersection number to derive relations between Feynman integrals on the maximal cut. My contributions were concerned with the use of specific parametrizations and with the use of regulators for some integrals. In the end the research group and I did around 30 examples of the use of univariate intersection theory to Feynman integrals.
The next major project was on the development and use of a multivariate intersection theory, which is needed in order to handle generic Feynman integrals. The multivariate formula was found and implemented, and tested successfully for a few paradigmatic Feynman integrals at one and two loops. This marked the first time a generic integral relation has been found without the use of the traditional methods, and it resulted in the PRL article mentioned above.
These developments led to the idea of the MathemAmplitudes conference, which covered these topics. Aside from the purely scientific content, this event helped put Padova on the map in the minds of the attending scientists, as a place where cutting edge research is still taking place.
In parallel with these developments I was working on the Higgs project. A breakthrough happened with the development of a semi-numeric technique to solve the differential equations used to identify the unknown Feynman integrals. With this we became able to compute all the remaining Feynman integrals. These remaining integrals fell into two integral families and we dedicated a paper to each. Yet my main work over the two years in this direction has been on the computation of the scattering amplitude, which is not yet completed. On a different project, that regarding the Z-boson, the development was parallel. We finished the computation of the master integrals, where my contribution was the definition of the families and the integration of the differential equation system.
During the Covid 19 outbreak and the associated lockdown, progress had to take a different form, due to the absence of face-to-face interaction between me and the other members of my research group. During this time the group in Padova and I published a paper containing a more detailed explanation of the multivariate case, along with more examples and a thorough discussion of three different reduction methods. Also some so-far unpublished work on closed-form expressions for the intersection number was developed by my collaborators and I during this time, and I should also mention some work I did myself on light-by-light scattering with a focus on symbols and special functions.
