Integrable Systems Course

This is the webpage for Math 595 Integrable Systems which I am teaching at UIUC in Spring 2023. At the moment this page is "in progress", below I will add a list of relevant books and links to some relevant papers. Later, I may add notes for the course.

Some useful textbooks, in no particular order:

• Lectures on Symplectic Geometry, Ana Cannas da Silva

• Integrable Hamiltonian Systems, Bolsinov & Fomenko

• Symplectic techniques in physics, Guillemin & Sternberg

• Introduction to symplectic topology, McDuff & Salamon

• Symplectic theory of completely integrable Hamiltonian systems, Pelayo & Vu Ngoc

  • • • https://arxiv.org/abs/1306.0115 

      • This is a survey article is a nice overview of the theory with lots of references

• Lectures on Lagrangian torus fibrations, Jonny Evan

  • • • https://arxiv.org/abs/2110.08643 

      • These notes are very useful! They cover many topics related to our course

Some suggested topics for presentations in the course

There is no particular order here, but some of the topics that I think would be best for presentations are in bold. 

Here are the ideas:

• Construction of Duistermaat's Chern class

  • • • https://onlinelibrary.wiley.com/doi/10.1002/cpa.3160330602 (we talked about this paper in class but I skipped the part about the Chern class)

• Atiyah-Guileman-Sternberg convexity theorem (there is a proof in Ana Canas da Silva's book)

• Periodic Hamiltonian flows on four dimensional manifolds, Karshon

  • • • https://arxiv.org/abs/dg-ga/9510004 

• From semi-toric systems to Hamiltonian S^1-spaces, Hohloch-Sabatini-Sepe

  • • • https://arxiv.org/abs/1305.7040 

• On semi-global invariants for focus-focus singularities, Vu Ngoc

  • • • https://arxiv.org/abs/math/0208245

• Vianna's papers on exotic Lagraigian tori

  • • • https://arxiv.org/abs/1305.7512

      • https://arxiv.org/abs/1409.2850 

• Four dimensions from two in symplectic topology, Symington

  • • • https://arxiv.org/abs/math/0210033 

• Almost toric symplectic four manifolds, Symington

  • • • https://arxiv.org/abs/math/0312165 

• Another good topic would be discussing how almost toric fibrations can be used to come up with bounds for certain symplectic capacities. There are several papers about this with overlapping authors, such as:

  • • • https://arxiv.org/abs/2004.13062 

      • Section 2.5.1 of https://arxiv.org/pdf/2210.06415.pdf lists some papers about this

• Quantization of integrable systems could be a good topic, there are many papers about this such as this one: https://arxiv.org/abs/1111.5985 

• Several papers about parabolic singular points (classification and circle actions)

  • • • https://arxiv.org/abs/1802.09910

      • https://arxiv.org/abs/2106.04838 

      • https://arxiv.org/abs/2110.13758 

• Zung's papers on the topology of integrable systems (surprisingly, the second paper actually may be easier to read first)

  • • • https://arxiv.org/abs/math/0106013

      • https://arxiv.org/abs/math/0010181 

Some other papers related to the course

These papers are also related to the course and would possibly make good presentations. Taking a look at these could also lead you to another topic that you are interested in. As before, they are in no particular order.

• Semitoric families, Le Floch-Palmer

  • • • https://arxiv.org/abs/1810.06915 

• Extending compact Hamiltonian S^1-spaces to integrable systems with mild degeneracies in dimension four, Hohloch-Palmer

  • • • https://arxiv.org/abs/2105.00523 

• Semitoric systems of non-simple type, Palmer, Pelayo, Tang

  • • • https://arxiv.org/abs/1909.03501 (the arxiv version is out of date, but I can give you a better version if you want)

• Vu Ngoc's conjecture on focus-focus singular fibers with multiple pinched points, Pelayo and Tang

  • • • https://arxiv.org/abs/1803.00998 

• A family of compact semitoric systems with two focus-focus singularities, Hohloch and Palmer

  • • • https://arxiv.org/abs/1710.05746 

• Minimal models of compact symplectic semitoric manifolds, Kane Palmer and Pelayo

  • • • https://arxiv.org/abs/1610.05423 

• Symplectic G-capacities and integrable systems, Figalli Palmer and Pelayo

  • • • https://arxiv.org/abs/1511.04499