Niels Benedikter

Sistemi Hamiltoniani (Laboratorio), Milan 2020/2021

This course deals with Hamiltonian mechanics and their perturbation theory. Recordings for my part of the theory lectures (corresponding to Chapter 2 "Canonical Transformations" of the lecture notes) can be found on Ariel (link below).

For materials and recording of the associated programming course, please see the official Ariel page. Every section on Ariel contains a subsection "Lezioni" and "Laboratorio"; for the programming course go to "Laboratorio". Here is the overview of the topics I discussed up to now:

  • Lab 1: Introduction to Linux, Installing Mizar, Introduction to C
  • Lab 2: Working with arrays, memcpy, and a first idea of numerics: solving a variational problem
  • Lab 3: Mizar and the forward Euler method
  • Lab 4: Symplectic integrators: the Leapfrog method
  • Lab 5: analysis of harmonic oscillators with linear coupling, Poincare sections
  • Lab 6: Henon-Heiles model, algorithm for Poincare sections [Henon1982], analytic solution for linearly coupled HOs
  • Lab 7: Splitting methods from Leapfrog to SABA3
  • Lab 8: SABA3 with corrector and "smarter" splitting; introduction to construction of first integrals
  • Lab 9: conclusion numerics, algebraic computation with polynomials in one variable
  • Lab 10: algebraic computation with polynomials in multiple variables, Poisson bracket in complex representation
  • Lab 11: iterative construction of first integrals for the Hénon-Heiles model
  • Lab 12: analysis of algebraically constructed truncated first integrals along numerical trajectories
  • Lab 13: plotting level sets of truncated first integrals
  • Lab 14: comparison of level sets to Poincare sections
Matematica del continuo (Sicurezza dei sistemi e delle reti informatiche), Milan 2020/2021

Insegnerò solo il secondo semestre, da aprile 2021. La prima parte (fino a fine marzo) è tenuta da Prof. Maggis. Per informazioni guardate la pagina Ariel del corso.

I teach only the second semester, from April 2021. The first part (until end of march) is taught by Prof. Maggis. For more information, please look at the Ariel page of the course. Essercizi aggiuntivi e lezioni.

Matematica del continuo (Sicurezza dei sistemi e delle reti informatiche), Milan 2019/2020

L'insegnamento fornisce gli strumenti base dell'Analisi Matematica, sia dal punto di vista teorico che pratico, indispensabili per poter seguire con profitto un corso universitario di carattere scientifico. Le conoscenze proposte sono propedeutiche ad altri corsi base del CdS.

Descrizione sulla pagina dell'Università: Matematica del continuo. Tutte le informazioni su Ariel. Per ora (per il corona virus) le lezioni sono su Zoom in orario normale (martedì 10:30-13:30, mercoledì 8:30-11:30).

Riassunto di quasi tutto: riassunto per stampare. Attenzione: non contiene le lezioni di Lorenzo Luperi Baglini (in particolare il capitolo sul integrale definito), quelli si trovano solo su Ariel. Invece, tutti i riassunti dell'elenco seguente sono contenuti nel file mdc2up.pdf.

Riassunti (anche su Ariel):

Esercizi (anche su Ariel):

Teaching assistant, IST Austria 2018

Teaching assistant for the course "Stability of Matter in Quantum Mechanics" with Prof. Robert Seiringer.

Research into the stability of matter has been one of the most successful chapters in mathematical physics, and is a prime example of how modern mathematics can be applied to problems in physics. This course provides a self-contained description of research on the stability of matter problem. It introduces the necessary aspects of functional analysis as well as the quantum mechanical background. The topics covered include Lieb-Thirring and other inequalities in spectral theory, inequalities in electrostatics, stability of large Coulomb systems, and gravitational stability of stars.

Advanced Mathematical Physics, Copenhagen 2017

Lecture Notes:

Lectures Niels:

Lectures Jérémy:

Assignments (to be handed in at the beginning of the Friday seminar!):

Summer School on Current Topics in Mathematic Physics at the University of Zurich, Switzerland, July 17 - July 21:

Seminar: All talks have to be about 40 minutes long (not more!). Schedule is subject to changes depending on our progression in the lecture. The summary is due on Monday after the seminar. Please remember to provide a list of references. Contact Jérémy or me at least two weeks before your talk for a briefing. Topics and summaries:

Criteria for passing the course:
Reach approximately 50% of the points averaged over the four assignments.
Give a seminar talk and produce a summary of your talk for the other participants.

Literature: The lecture notes should be self-contained for most of the course. If you are looking for additional reading, here are some recommendations.

  • Gerald Teschl: Mathematical Methods in Quantum Mechanics. Online version

    Contains essentially all topics of the lecture, but in a different arrangement. Sometimes details are missing.

  • Stephen J. Gustafson, Israel Michael Sigal: Mathematical Concepts of Quantum Mechanics. Online version

    Nice selection of material, but sometimes sketchy. Good overview.

  • Michael Reed, Barry Simon: Methods of Modern Mathematical Physics, Volumes I-IV.

    Recommended as a reference work, not as a text book. Concise but rather dense.

  • Elliott H. Lieb, Michael Loss: Analysis. Errata

    A functional analysis book aimed at applications in quantum mechanics. Contains many useful explicit estimates.

If you want to read up on quantum mechanics from the physicist point of view (which is not the focus of this course):

  • Gordon Baym: Lectures on Quantum Mechanics
  • Leslie E. Ballentine: Quantum Mechanics, A Modern Development
  • Steven Weinberg: Lectures on Quantum Mechanics
Advanced Mathematical Physics, Copenhagen 2016

Thank you all for following the course! New materials for the 2017 course will appear above.

Teaching assistant, Bonn 2011-2014

I assisted Prof. Benjamin Schlein at the University of Bonn for the courses

and for a Summer School at the University of Heidelberg