#### Alladi Ramakrishnan Hall

#### Feynman Integral: From Integration-By-Parts reduction to Intersection Theory

#### Manoj Kumar Mandal

##### Universita' degli Studi di Padova, Italy

*Scattering amplitudes encode crucial information about collision phenomena in our universe, from the smallest to the largest scales. Evaluation of multi-loop Feynman integrals is an integral part of the determination of these scattering amplitudes and related quantities. The Feynman integrals obey linear relations, which are exploited by employing the standard Integration-By-parts identities to simplify the evaluation of scattering amplitudes: they can be used both for decomposing the scattering amplitudes in terms of a basis of functions referred to as master integrals (MIs) and for the evaluation of the latter using the differential equation. I will show that they are better understood using the Intersection Numbers, which act as scalar products between the vector spaces of the Feynman Integral. Application to few Feynman integrals at one- and two-loops will be shown, thereby sketching the first step towards potential applications to generic multi-loop integrals.*

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