Periodic Reporting for period 1 - HiProLoop (Scattering Amplitudes for Higgs Production at High-Order as touchstone for Automated Multiloop Feynman Calculus) Reporting period: 2018-09-01 to 2020-08-31 Summary of the context and overall objectives of the project My project is concerned with particle physics, and more specifically the phenomenology of particle scattering and the underlying mathematical structures. Particle physics of this sort, is of relevance to experiments such as those taking place at the Large Hadron Collider at CERN, for which the crowning achievement of the LHC was the 2012 discovery of the Higgs particle. To get even more out of these experiments, high accuracy precision calculations of the scattering cross sections of the processes observed at colliders are crucial in order to diminish the theoretical uncertainties to levels below that of the ever decreasing experimental uncertainty, and be able to see signs of potential new physics that may be hiding behind the overwhelming background of known physical effects. Shortly after my arrival in Padova I started working on a new direction that had been opened up by my collaborator in Padova, professor Pierpaolo Mastrolia. This project was concerned with the use of the mathematical field of intersection theory in connection with particle physics and Feynman integrals. This became my main project during my time in Padova, having resulted in 3 papers so far, one of which got published in the prominent journal Physical Review Letters (PRL). Another project on which I spent a lot of effort during the two years, concerns a specific aspect of the physics of the Higgs boson (its production together with a hadronic jet). That project has given rise to two publications during my time in Padova. Another project I was working on was concerned with the electro-weak sector of the Standard model with regards to the simultaneous production of a Z-boson and a hadronic jet, in which the main task was computing the relevant Feynman integrals in a specific approximation. This has resulted in one publication containing the integrals, and the scattering amplitude and the cross section are forthcoming. In total I had six publications during my time in Padova.The conference I proposed in my application got organized under the name of MathemAmplitudes, and it was held December 18-20 2019 in Padova. I consider MathemAmplitudes a success, as it for the first time gathered together physicists and mathematicians to discuss the new directions opened up with regards to the relation between Feynman integrals and the mathematical field of intersection theory. Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far 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. Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far) The intersection-based approach to the derivation of integral relations is so beyond the state of the art that it would have been unthinkable three years ago. At its core this approach utilizes the underlying vector space structure of Feynman integrals, as opposed to the brute force approach of the traditional method. The intersection approach itself is completely general, and what prevents it from being applied to any arbitrary case are merely computational issues. The adoption of new mathematical methods into physics characteristic for the intersection project, has time and time again shown itself to be the way to genuine paradigm shifts in physics. If the intersection based approach to Feynman integrals will prove itself to be at that level is for time to tell, but I do believe it has the potential to genuinely change the approach of the community to computations of Feynman integral relations. Yet no matter what, these developments are so prominent that they are going to be defining for me in the eyes for the research community. The semi-numerical method used for the solution of the integrals in the Higgs project must also be said to be beyond the state of the art.All and all my time in Padova has been a huge success. Besides the research discussed above, the six publications and the numerous conference presentations, I have gained two grants, the Carlsberg Foundation Reintegration Fellowship, and the Interactions Cofund, which together allow me to return to my native Denmark to continue my career. I have also got experience with conference organizing through the MathemAmplitudes conference, including the creation of the YouTube channel containing the talks. I do believe that the Marie Sklodowska-Curie Individual Fellowship has give me great opportunities that I could not have got without, and that the research done during the past two years will set the stage for the rest of my career.