These projects constitute the main line of research of my postdoc (project proposal).
Recently [1], a framework, similar to the probabilistic approach used for Bosonic fields, was presented to describe Fermionic systems. We aim to extend and apply this framework to the construction of interacting Fermionic quantum field theories in at least three directions. The first is the generalization to the Fermionic case of the variational methods for (Bosonic) quantum field theory introduced by E. Nelson in finite dimension. The second direction consists in applying the mentioned probabilistic method in the construction of the Hamiltonian of Fermionic fields on the space generated by the Fermionic fields at time zero. The third application is in the direction of non-relativistic systems.
The goal is twofold. The first research line is to generalize to the time-dependent case the results obtained in the stationary setting. A second research line of the present project concerns the identification of an effective and useful stochastic scheme associated to the Gross-Pitaevskii model of a BE condensate.
Goal: contribute the extend the database of available theorems written in ``computer understandable'' language. More concretely I am interested in contributing to the Lean mathlib library by proving in Lean facts from probability. The goal would be to obtain a fairly complete set of fundamental results in probability transcribed in Lean.
Goal: write a set of notes and examples as an introduction to deep learning from a mathematical point of view. The work in progress can be tracked on Github.