This minicourse is an introduction to Differential Geometry, with highlights on the infinite-dimensional case. It will be divided into 3 sections:
– Basic notions of manifolds and fiber bundles modelled on Hilbert, Banach or Fréchet spaces. Examples used in Geometry, Shape Analysis, or Gauge Theory.
– Inverse Function Theorems: the Banach version and the Nash-Moser version. Some applications to submanifolds.
– Some pathologies concerning Riemannian, complex, symplectic and Poisson structures in the infinite-dimensional setting.

During the lecture, the notions introduced will be illustrated with examples related to projective spaces, grassmannians, diffeomorphisms groups, spaces of sections, spaces of curves, and others.

References:

– R.S. Hamilton. The inverse function Theorem of Nash and Moser . Bulletin (New Series) of the American Mathematical Society, Volume 7, Number 1, 1982.
– W. Klingenberg. Riemannian Geometry . Walter de Gruyter, New York, 1982.
– A. Kriegl and P. W. Michor. The convenient setting of Global Analysis . Mathematical Surveys and Monographs, Volume 53.
– S. Lang. Fundamentals of Differential Geometry . Graduate Texts in Mathematics, Springer-Verlag, 1999.
– S. Lang. Differential and Riemannian Manifolds . Graduate Texts in Mathematics, Springer-Verlag, 1995.

Videos: First lecture, Second lecture, Third lecture

Throughout the development of mathematical methods in symplectic geometry and Hamiltonian dynamics, interest has arisen in studying continuous counterparts of the objects from these fields, as well as behaviour of the objects under uniform limits. One example is the study of Hamiltonian homeomorphisms in dimension two, related to the Arnold conjecture [1,2,19,20,23,33]. Another such example was an attempt to understand whether for a diffeomorphism, the property of being symplectic, survives under uniform limits. Eventually, this C^0 rigidity property of symplectic diffeomorphisms was confirmed in the celebrated Eliashberg-Gromov theorem [10,14].

Still, only relatively recently, attempts were made for providing a more systematic approach under the name of C^0 symplectic geometry. The foundational paper [31] made a major step in this direction, introducing central notions of the field, studying their properties, and indicating important directions for a further study. One of motivations for the work [31] was the celebrated Fathi question asking whether the group of area-preserving homeomorphisms of a 2-dimensional disc is simple. The further research inspired by the work [31] investigated topics such as uniqueness properties of Hamiltonians generating continuous/topological Hamiltonian flows, C^0 continuity of spectral invariants, C^0 rigidity versus flexibility of submanifolds, C^0 Arnold conjecture, C^0 contact geometry [3,4,5,6,7,9,15,16,17,18, 21,22,24,25,26,27, 28,29,30,32,35, 36,37,38,39,40]. In my lectures I will give an overview of some of these topics.

The Fathi question was very recently answered in [9]. The solution of the question in [9] uses symplectic topological tools of Floer theory, and the ideas involved in it are largely inspired by previous ideas of Oh as well as by earlier works on C^0 symplectic geometry which appeared after [31].
Another important subject within C^0 symplectic geometry is the study of the functional-theoretic properties of the Poisson bracket operator, named Function theory on symplectic manifolds [34]. Here, the space of functions on the corresponding symplectic manifold is typically equipped with the uniform (C^0) norm. Function theory on symplectic manifolds was initiated in the works [8,11,12,13]. If time will permit, I will try to give a brief overview of that subject as well.

References:

[1] G. D. Birkhoff, Proof of Poincaré’s geometric theorem, Trans. Amer. Math. Soc., 14 (1913) 14-22.
[2] G. D. Birkhoff, Surface transformations and their dynamical applications, Acta Math., 43 (1922), 1-119.
[3] L. Buhovsky, V. Humilière, and S. Seyfaddini, A C^0 counterexample to the Arnold conjecture, Invent. Math., 213(2):759-809, 2018
[4] L. Buhovsky, V. Humilière, and S. Seyfaddini, The action spectrum and C^0 symplectic topology, arXiv:1808.09790, 2018.
[5] L. Buhovsky, V. Humilière, and S. Seyfaddini, An Arnold-type principle for non-smooth objects, https://arxiv.org/abs/1909.07081
[6] L. Buhovsky and E. Opshtein, Some quantitative results in C^0 symplectic geometry, Invent. Math., 205(1):1-56, 2016.
[7] L. Buhovsky and S. Seyfaddini, Uniqueness of generating Hamiltonians for topological Hamiltonian flows, J. Symplectic Geom., 11(1):37-52, 2013.
[8] F. Cardin and C. Viterbo, Commuting Hamiltonians and Hamilton-Jacobi multi-time equations, Duke Math. J. 144 (2008), 235-284.
[9] D. Cristofaro-Gardiner, V. Humilière, and S. Seyfaddini, Proof of the simplicity conjecture, https://arxiv.org/abs/2001.01792
[10] Ya. M. Eliashberg, A theorem on the structure of wave fronts and its application in symplectic topology, Funktsional. Anal. i Prilozhen., 21(3):65-72, 96, 1987.
[11] M. Entov and L. Polterovich, Quasi-states and symplectic intersections, Comment. Math. Helv., 81(1):75-99, 2006.
[12] M. Entov and L. Polterovich, C^0-rigidity of Poisson brackets, in Proceedings of the Joint Summer Research Conference on Symplectic Topology and Measure-Preserving Dynamical Systems (eds. A. Fathi, Y.-G. Oh and C. Viterbo), 25-32, Contemporary Mathematics 512, AMS, Providence RI, 2010.
[13] M. Entov, L. Polterovich, and F. Zapolsky, Quasi-morphisms and the Poisson bracket, Pure Appl. Math. Q., 3(4, Special Issue: In honor of Grigory Margulis. Part 1):1037-1055, 2007.
[14] M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math., 82(2):307-347, 1985.
[15] V. Humilière, R. Leclercq, and S. Seyfaddini, Coisotropic rigidity and C^0-symplectic geometry, Duke Math. J., 164(4):767-799, 2015.
[16] V. Humilière, R. Leclercq, and S. Seyfaddini, Reduction of symplectic homeomorphisms, Ann. Sci. Ec. Norm. Supér. (4), 49(3):633-668, 2016.
[17] V. Humilière, R. Leclercq, and S. Seyfaddini, New energy-capacity-type inequalities and uniqueness of continuous Hamiltonians, Comment. Math. Helv., 90 (2015), 1-21.
[18] Y. Kawamoto, On C^0-continuity of the spectral norm for symplectically non-aspherical manifolds, https://arxiv.org/abs/1905.07809.
[19] P. Le Calvez, Une version feuilletée équivariante du théorème de translation de Brouwer, Publ. Math. Inst. Hautes Études Sci., (102):1-98, 2005.
[20] P. Le Calvez, Periodic orbits of Hamiltonian homeomorphisms of surfaces, Duke Math. J., 133(1):125-184, 2006.
[21] F. Le Roux, Simplicity of Homeo(D^2, ∂D^2, Area) and fragmentation of symplectic diffeomorphisms, J. Symplectic Geom., 8(1):73-93, 2010.
[22] F. Le Roux, S. Seyfaddini, and C. Viterbo, Barcodes and area-preserving homeomorphisms, arXiv:1810.03139, 2018.
[23] S. Matsumoto, Arnold conjecture for surface homeomorphisms, In Proceedings of the French-Japanese Conference “Hyperspace Topologies and Applications” (La Bussière, 1997), volume 104, pages 191-214, 2000.
[24] S. Müller, The group of Hamiltonian homeomorphisms in the $L^infty$-norm, J. Korean Math. Soc. 45 (2008), no. 6, 1769-1784.
[25] S. Müller, C^0-characterization of symplectic and contact embeddings and Lagrangian rigidity, International Journal of Mathematics Vol. 30, No. 09, 1950035 (2019).
[26] S. Müller, P. Spaeth, Topological contact dynamics II: topological automorphisms, contact homeomorphisms, and non-smooth contact dynamical systems, Trans. Amer. Math. Soc. 366 (2014), no. 9, 5009-5041.
[27] S. Müller, P. Spaeth, Gromov’s alternative, Eliashberg’s shape invariant, and C^0-rigidity of contact diffeomorphisms, Internat. J. Math. 25 (2014), no. 14, 1450124.
[28] S. Müller, P. Spaeth, Topological contact dynamics I: symplectization and applications of the energy-capacity inequality, Adv. Geom. 15 (2015), no. 3, 349-380.
[29] S. Müller, P. Spaeth, Topological contact dynamics III: uniqueness of the topological Hamiltonian and C^0-rigidity of the geodesic flow, J. Symplectic Geom. 14 (2016), no. 1, 1-29.
[30] Y-G. Oh, The group of Hamiltonian homeomorphisms and continuous Hamiltonian flows, In Symplectic topology and measure preserving dynamical systems, volume 512 of Contemp. Math., pages 149-177. Amer. Math. Soc., Providence, RI, 2010.
[31] Y-G. Oh and S. Müller, The group of Hamiltonian homeomorphisms and C^0-symplectic topology, J. Symplectic Geom., 5(2):167-219, 2007.
[32] E. Opshtein, C^0-rigidity of characteristics in symplectic geometry, Ann. Sci. Éc. Norm. Supér. (4), 42(5):857-864, 2009.
[33] H. Poincaré, Sur un théorème de géométrie, Rend. Circ. Mat. Palermo, 33 (1912), 375-407.
[34] L. Polterovich, D. Rosen, Function theory on symplectic manifolds, American Mathematical Society, 2014.
[35] S. Seyfaddini, C^0-limits of Hamiltonian paths and the Oh-Schwarz spectral invariants, Int. Math. Res. Not. IMRN, (21):4920-4960, 2013.
[36] S. Seyfaddini, The displaced disks problem via symplectic topology, C. R. Math. Acad. Sci. Paris, 351(21-22):841-843, 2013.
[37] E. Shelukhin, Viterbo conjecture for Zoll symmetric spaces, https://arxiv.org/abs/1811.05552
[38] E. Shelukhin, Symplectic cohomology and a conjecture of Viterbo, https://arxiv.org/abs/1904.06798
[39] M. Usher, Local rigidity, contact homeomorphisms, and conformal factors, https://arxiv.org/abs/2001.08729
[40] C. Viterbo, On the uniqueness of generating Hamiltonian for continuous limits of Hamiltonians flows, Int. Math. Res. Not., pages Art. ID 34028, 9, 2006.

In this minicourse we will discuss the Pontryagin Maximum Principle, which is the standard first order optimality condition applied in Optimal Control. We will revisit the main features of its proof [3,2]. In particular, we will focus on the construction of control variations, which are the main ingredient of Optimal Control Theory [4].

If there is enough time, we will see an adaptation of Pontryagin’s result to Impulsive Optimal Control [1], in which variations involve Lie brackets in an essential way.

References:

[1] M.S. Aronna, M. Motta, F. Rampazzo. A Higher-order Maximum Principle for Impulsive Optimal Control Problems. SIAM Journal on Control and Optimization 58.2 (2020), 814-844.
[2] A. Bressan, B. Piccoli. Introduction to the mathematical theory of control . Springfield: American institute of mathematical sciences Vol. 1, 2007.
[3] H. Schättler, U. Ledzewicz. Geometric optimal control: theory, methods and examples. Springer Science & Business Media Vol. 38, 2012.
[4] H.J. Sussmann. A Strong Version of the Lojasiewicz Maximum Principle. Optimal Control of Differential Equations. Chap. 19, 293-310, 2020.

Videos: First lecture, Second lecture, Third lecture