Research

Interests

Broadly speaking, my research interests lie in the field of differential geometry and its interplay with mathematical physics.

I work with higher geometry methods in multisymplectic geometry and, most of all, I am interested in structures “up to homotopies” inspired by geometry and mechanics.

Physically inspired mathematics
Research Program

My research project focuses on the notions of L algebra of observables and homotopy comomentum maps. These are the higher analogues of smooth observables and the ordinary comoment map in the context of n-plectic geometry. My key results are related to the explicit construction of non-trivial instances of these objects with possible application in different areas of mathematics (e.g. knot theory) and physics (e.g. hydrodynamics).

Nowadays, I am investigating the notion of gauge transformations and (symplectic) reduction in the context of multisymplectic geometry.

A deep motivation for pursuing this line of research can be tracked in my interest in understanding the axiomatic and geometric foundation of classical field theories (i.e. mechanical systems with infinite and continuous degrees of freedom). In this perspective, I am always interested in expanding my knowledge both in the pure direction (homotopical algebra, higher categories) than in the applied direction (symplectic integrators, numerical simulations).

Typical academic picture
Keywords
  • Multi-symplectic geometry and L-Algebras
  • Higher structures and Homotopy structures
  • Axiomatization and mathematical foundations of classical and quantum field theories
  • Geometric Mechanics of Field Systems
  • Geometric and Algebraic Quantization
Publications
Sep 2022Observables of multisymplectic manifolds and higher Courant algebroids.
Miti A. M. and Zambon M. ; arXiv:2209.05836.
[bibtex] [arxiv]
Jun 2022Reduction of L∞-algebras of observables on multisymplectic manifolds.
Blacker C. , Miti A. M. and Ryvkin L. ; arXiv:2206.03137.
[bibtex] [arxiv]
Feb 2021A Hydrodynamical Homotopy Co-momentum Map And A Multisymplectic Interpretation Of Higher-order Linking Numbers.
Miti A. M. and Spera M. ; Journal of the Australian Mathematical Society.
[bibtex] [arxiv]
Dec 2020Multisymplectic actions of compact Lie groups on spheres.
Miti A. M. and Ryvkin L. ; Journal of Symplectic Geometry.
[bibtex] [arxiv]
Aug 2020Derivation of the HOMFLYPT knot polynomial via helicity and geometric quantization.
Miti A. M. and Spera M. ; Bollettino dell'Unione Matematica Italiana.
[bibtex] [arxiv]
Theses
Apr 2021Homotopy comomentum maps in multisymplectic geometry.
Phd thesis @ Università Cattolica del Sacro Cuore & KU Leuven;
Advisors: Mauro Spera, Marco Zambon.
[bibtex] [file]
Nov 2015Algebraic quantization of Jacobi fields and geometrica approach to Peierels brackets.
Master Thesis @ Università Degli Studi di Milano;
Advisors: Claudio Dappiaggi, Livio Pizzocchero.
[bibtex] [file]
Nov 2010Teoria dei gruppi di Lie e applicazione alla meccanica del corpo rigido.
Bachelor thesis @ Università degli Studi di Milano Bicocca;
Advisors: Franco Magri.
[bibtex] [file]
Presentations
May 2024Multisymplectic approach to LFT and the Stress-Energy tensor.
Talk @ Geometric conservation laws in physics, Università di Torino.  [Slides]
Mar 2024Multisymplectic observables and higher Courant algebroids.
Talk @ Higher structures in Caprarola, INFN, Caprarola.  [Slides]
Jul 2023The multisymplectic structure of Lagrangian field theories.
Talk @ Higher structures working group, UniGoettingen, Gottingen.  [Slides]
May 2023Symmetries and reduction of multisymplectic manifolds.
Talk @ Joint seminar in mathematical physics, UniPv, Pavia.  [Slides]
Feb 2023On the Lie infinity algebra associated to higher Courant algebroids.
Talk @ Higher structures working group, ICJ, Lyon.  [Slides]
Jan 2023First steps in geometric quantum mechanics.
Talk @ PhD Seminar, UCSC, Brescia.  [Slides]
Dec 2022Multisymplectic observables and higher Courant algebroids.
Talk @ Séminaire Physique mathématique, ICJ, Lyon.  [Slides]
Aug 2022Symmetries and reduction of multisymplectic manifolds.
Talk @ Higher Structures and Field Theory, Esi, Vienna.  [Slides]
Jun 2022L∞-structures via the Nijenhuis-Richardson algebra.
Talk @ Good Morning Sfars, online.  [Slides]
Apr 2022Symmetries and reduction of multisymplectic manifolds.
Talk @ MPI-Oberseminar, MPIM, Bonn.  [Slides]
Mar 2022Morphisms and Yoneda lemma for stacks.
Talk @ Reading seminars on differentiable stacks, MPIM, Bonn.  [Slides]
Dec 2021Gauge transformations of multisymplectic manifolds and L-infinity observables.
Talk @ Higher geometry seminar, MPIM, Bonn.  [Slides]
Jul 2021Gauge transformations of multisymplectic manifolds and L-infinity observables.
Talk @ Young Researchers' Virtual Multisymplectic Geometry Conference, EIMI, Saint Petersburg.  [Slides]
Apr 2021What is: Geometric mechanics?.
Talk @ PhD colloquium, KU Leuven.  [Slides]
Feb 2021Phd Preliminary Defence.
Talk @ -, KU Leuven.  [Slides]
Sep 2020Homotopy co-moment maps for compact actions on spheres.
Talk @ Workshop on Multisymplectic Geometry, KU Leuven.  [Slides]
May 2020Gauge transformations of multisymplectic manifolds and L_infinity observables.
Talk @ Poisson seminar, KU Leuven.  [Slides]
Dec 2019MultiSymplectic manifolds and homotopy co-momentum maps.
Poster @ Supergeometry conference, Luxembourg.  [Slides]
Mar 2019Multisymplectic Geometry and Knots.
Talk @ UniSa, Fisciano.  [Slides]
Dec 2018Homotopy co-momentum Map in Hydrodynamics.
Talk @ 13th GMC Young researcher's workshop, Coimbra.  [Slides]
May 2018On some (multi)symplectic aspects of link invariants.
Talk @ -, KU Leuven.  [Slides]
Nov 2017An invitation to Geometric Mechanics.
Talk @ UCSC, Brescia.  [Slides]
Jul 2017Multi-symplectic Geometry and Covariant Phase Space.
Poster @ 11th GMC summer school, ICMAT, Madrid.  [Slides]
Miscs
Feb 2019MultiSymplectic Geometry and Classical Field systems. (WIP)
On the multisymplectic framework of classical field theories.   [Notes]
Feb 2018Duffing system in Python.
Solving Duffing Equation by means of Runge-Kutta method.   [Code]
Oct 2017Cartan calculus on smooth manifolds.
A compact cheatsheet with the basic computation rules.   [Notes]
May 2017Covariant Phase Space Via Multysimplectic Geometry.
On the covariant phase space in the multisymplectic framework.   [Notes]