The **Scientific board and the Organizing Commitee** Welcome You to the **FIFTH edition of the DomoSchool, the International Alpine School in Mathematics and Physics** in Domodossola. **Domoschool 2024 Edition** will consists of** FOUR courses for a total of 20 hours **and it has been organised **with the support of FONDAZIONE CRT, Comune Domosdossola, INDAM - GNSAGA, I.N.F.N.**

The Domoschool 2024 edition will focus on modern mathematical methods for calculating Feynman integrals and scattering amplitudes.

The last century has witnessed an explosion of scientific developments. Among the great successes of theoretical physics, the theory of General Relativity, Quantum Mechanics and the Standard Model of elementary particles stand out. In all these areas, experimental verification is crucial to support the validity of a theory or refute it. Contemporary technological advances have led to the possibility of making very high-precision measurements, in large accelerators (CERN, Stanford, etc.) for the Standard Model, LIGO-Virgo and large space missions for General Relativity and almost every laboratory for classical or quantum physics.

One of the main challenges of modern physics is therefore to use theoretical models to make extremely accurate predictions to compare with experimental results. The crucial point is that, despite being very different from each other, these research areas (and not only) seem to have a common underpinning mathematical structure which emerges through the so-called Feynman integrals, in honour of theoretical physicist and Nobel Prize winner Richard Feynman, who first invented them.

This school aims to provide an introduction to the complex interplay between advanced mathematical methods and their fundamental role in theoretical physics, which can be explored through Intersection Theory, specifically addressing key concepts such as twisted co-homology and intersection numbers, that have been recently discovered to rule the algebraic properties of Feynman integrals as well as Euler-Mellin integrals and generalised hypergeometric functions.

These methods are of great importance, given the complete transversality of the applications, but being quite recent, the school offers the opportunity to young physicists and mathematicians to experience the “unreasonable effectiveness of mathematics'', observed by Eugene Wigner, namely the mysterious phenomenon in which areas of pure mathematics, originally constructed without regard to application, are suddenly discovered to be exactly what is required to describe the structure of the physical world – as Michael Berry said.

The courses will provide an introduction to the following topics:

-basic notions of Riemann surfaces, homology, and bilinear Riemann relations;

-intersection theory for twisted homology and cohomology;

-linear algebra for Feynman integrals, and finite fields arithmetic;

-Euler-Mellin Integrals, maximum likelihood, and Landau determinants.

These methods are of great importance, given the complete **transversality of the applications**, but being quite recent and using methods that are unusual for physicists, they are still not very available, especially to the community of young physicists and mathematicians.

**Topics:**

**basic notions of Riemann surfaces, homology, up to the bilinear Riemann relations;****intersection theory for twisted homology and cohomology;****decomposition algorithms in Master Integrals, based on methods over finite fields;****Euler-Mellin Integrals, maximum likelihood, and Landau determinants.**

The aim of the school is to introduce PhD students, postdocs and young researchers, both in mathematics and in physics, to this topic. Interaction with and among professors will be also encouraged.

**Courses will be conducted at PhD level by world leading experts.**

Moreover, students will have the opportunity to give **20 minutes talks** presenting their research on themes related to the main topics of the courses.

Informal discussions will be strongly encouraged, and specific time slots (short talks) will be reserved for this important part of the school.

*We are looking forward to welcome you in Domodossola!*

For any questions or information do not hesitate to contact us, filling in the following form