Research

Research interests

• Noncommutative geometry of algebras of pseudodifferential operators, index theory, K-theory

• Local solvability and regularity of the tangential Cauchy-Riemann complex, the corresponding Kohn Laplacian and generalizations to locally integrable structures

• Microlocal Analysis: lower bounds for pseudodifferential operators and systems

• Classical Harmonic Analysis on Euclidean space, with applications to nonlinear PDEs: restriction theorem for the Fourier transform, Strichartz estimates for dispersive equations, Fourier integral operators

• Time-frequency analysis: boundedness of operators on modulation and Wiener amalgam spaces; sparse representation of Fourier integral operators via Gabor frames; Cohen distributions and interferences

• Nonlinear PDEs: holomorphic extension and exponential decay of solutions of PDE's; in particular, homoclinics for semilinear equations and travelling waves. Propagation of singularities for NLS.

• Mathematical aspects of Quantum Mechanics and Quantum Field Theory, in particular quantization rules and Feynman path integrals: rigorous definition via the Trotter formula, convergence in several function spaces (L^p, Kato-Sobolev spaces) for low regular potentials.

• Representations of semisimple groups, asymptotics of matrix coefficients.

• Optimization/variational problems in time-frequency analysis and quantum mechanics: uncertainty principles,  Wehrl entropy, etc., even in quantitative form.

• Rigorous analysis of convolutional neural networks (in particular scattering network) from an analytic point of view (stability, energy propagation, etc.)

• Mathematical aspects of quantum information 

Publications

 Recent papers (download available)

Books

• • F. Nicola, S. Ivan Trapasso, Wave Packet Analysis of Feynman Path Integrals, Lecture Notes in Mathematics. volume 2305, 2022

  • F. Nicola, L. Rodino, Global Pseudo-differential Calculus on Euclidean Spaces, Birkhauser, Basel, 2010

  • F. Nicola, Analisi Matematica I - Appunti delle lezioni, CLUT, Torino, 2013.

  • P. Boggiatto, E. Cordero, H. Feichtinger, M. de Gosson, F. Nicola, A. Oliaro, A. Tabacco (ed.), Landscapes of Time-frequency Analysis, Springer's series ANHA: Applied and Numerical Harmonic Analysis (editor in chief J.J. Benedetto), ISBN 978-3-030-05209-6, Springer 2019

  • P. Boggiatto, T. Bruno, E. Cordero, H. G. Feichtinger, F. Nicola, A. Oliaro, A. Tabacco and M. Vallarino (ed) Landscapes of time-frequency analysis—ATFA 2019. Papers from the 2nd international conference Aspects of Time-Frequency Analysis held at Politecnico di Torino, Turin, June 25–27, 2019. Applied and Numerical Harminic Analysis Birkhäuser/Springer, Cham, [2020]

Ph.D. students

• Alessandra Cauli (2015-2018)

• S. Ivan Trapasso (2017-2020)

• Federico Bastianoni (2019-2022)

• Federico Riccardi (2023-2026)