*International Young Researchers Workshop on Geometry, Mechanics, and Control*is a yearly event in which early career researchers from each of the three areas can share their work and initiate new collaborations. The workshop consists of three minicourses and ten contributed talks. The 15th edition was meant to take place in the University of Utrecht (The Netherlands). However, due to the constraints imposed by Corona, we held the meeting online instead:

**When:**November 30th – December 4th 2020. The sessions start at 2:30pm (CET). You can find the precise schedule below.**Where:**Zoom. Videos and slides of the event can be found below, together with the abstracts and titles.**How:**Registration is now closed.**Other questions?**You can contact us at 15thiyrwgmc[at]gmail.com.

**Minicourses**

**Title:** *Infinite-dimensional Geometry: theory and applications*

**Speaker:** Alice Barbara Tumpach (Université de Lille, France)

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

** Slides: **

First lecture

Second lecture

Third lecture

**Title:** *C ^{0} Symplectic Geometry*

**Speaker:** Lev Buhovski (Tel Aviv University, Israel)

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.

** Slides: **

First lecture

Second lecture

Third lecture

**Title:** *The Pontryagin maximum principle*

**Speaker:** María Soledad Aronna (Escola de Matemática Aplicada, Brazil)

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

** Slides: **

Complete set of notes

**Contributed talks**

November 30, 2020

**Title:** *Reduction by local symmetries in field theories*

**Speaker:** Álvaro Rodríguez Abella (Instituto de Ciencias Matemáticas - Universidad Complutense de Madrid)

Symmetries have a major role in the study of Mechanical systems and Field theories. If a Lagrangian (or Lagrangian density, in the case of Field theories) is invariant under the action of a group of symmetries, we can define the so-called reduced Lagrangian (density) on the quotient of the corresponding phase space by this action. The variational principle is transferred to this quotient space, providing a new set of equations for the reduced Lagrangian. This procedure is known as Reduction and has been thoroughly treated in the literature. However, a wide variety of problems involve local symmetries, which are not given by group actions, but by fiberwise actions of certain Lie group bundles. In other words, the group of symmetries varies as it does the point of the base space (which is the spacetime, typically). Gauge symmetries are the main instance of this situation.

The aim of this talk is to determine the reduction procedure for a first order Lagrangian density which has local symmetries, obtaining the reduced equations.

November 30, 2020

**Title:** *On Topological Equivalence in Linear Quadratic Optimal Control*

**Speaker:** Wouter Jongeneel (EPFL)

For the closed-loop time-one maps resulting from the family of Linear Quadratic Optimal Control problems, we will describe the quotient space under topological conjugacy. As it turns out, these equivalence classes can be characterized as being disjoint path-connected sets in the Symplectic group. Understanding this equivalence has applications in the discretization of continuous flows and provides new insights in ``structural tuning'' of controlled behaviour.

December 1, 2020

**Title:** *Stratification of the transverse momentum map*

**Speaker:** Maarten Mol (Universiteit Utrecht)

Let $G$ be a Lie group. The momentum map $J: (S,omega) to mathfrak{g}^*$ of a Hamiltonian $G$-action descends to a map from the orbit space $S/G$ into the orbit space $mathfrak{g}^*/G$ of the coadjoint action. We call this the transverse momentum map. When $G$ is compact, both of these orbit spaces are naturally stratified by orbit types. However, the transverse momentum map need not be a morphism of stratified spaces.

The aim of this talk is to describe a finer stratification of the orbit space $S/G$, with respect to which the transverse momentum map does become a morphism of stratified spaces into $mathfrak{g}^*/G$ (where the latter is still stratified by orbit types). After its introduction, we will discuss the compatibility between this stratification, the reduced Poisson bracket and the symplectic reduced spaces.

December 1, 2020

**Title:** *The evolution vector field on contact manifolds and thermodynamics*

**Speaker:** Manuel Lainz (Instituto de Ciencias Matemáticas)

On a contact manifold $(M,eta)$, given a Hamiltonian function $H$ one can naturally define the Hamiltonian vector field $X_H$. The evolution vector field $E_H = X_H + H R$, where $R$ is the Reeb vector field can also be constructed from the Hamiltonian function and the contact structure alone.

This vector field coincides with the Hamiltonian vector field on the zero set of the Hamiltonian, but differs from it on the rest of $M$. The evolution vector field is always tangent to the kernel of the contact form. The thermodynamic interpretation of this fact is that its integral curves fulfill the first law of thermodynamics. In addition, with a simple assumption on the Hamiltonian, the second law of thermodynamics is fulfilled.

On this talk, we will explain the geometric and dynamical properties of this vector field and its applications to the description of isolated thermodynamic systems.

[1] Simoes Alexandre Anahory, de León Manuel, Valcázar Manuel Lainz and de Diego David Martín. 2020, emph{Contact geometry for simple thermodynamical systems with friction}. Proc. R. Soc. A. 476:20200244. http://doi.org/10.1098/rspa.2020.0244

December 2, 2020

**Title:** *Stochastic processes on surfaces in three-dimensional contact sub-Riemannian manifolds*

**Speaker:** Karen Habermann (University of Warwick)

We are concerned with stochastic processes on surfaces in three-dimensional contact sub-Riemannian manifolds. Employing the Riemannian approximations to the sub-Riemannian manifold which make use of the Reeb vector field, we obtain a second order partial differential operator on the surface arising as the limit of Laplace-Beltrami operators. The stochastic process associated with the limiting operator moves along the characteristic foliation induced on the surface by the contact distribution. We show that for this stochastic process elliptic characteristic points are inaccessible, while hyperbolic characteristic points are accessible from the separatrices. We illustrate the results with examples and we identify canonical surfaces in the Heisenberg group, and in SU(2) and SL(2,R) equipped with the standard sub-Riemannian contact structures as model cases for this setting. Our techniques further allow us to derive an expression for an intrinsic Gaussian curvature of a surface in a general three-dimensional contact sub-Riemannian manifold.

December 2, 2020

**Title:** *Covariant brackets in particle dynamics and first order Hamiltonian field theories*

**Speaker:** Luca Schiavone (Federico II University of Naples - Universidad Carlos III de Madrid)

Poisson brackets are a central tool within the problem of quantization where it is customary to start with a classical algebra of observables equipped with the Poisson bracket on the phase space and build an algebra of quantum observables in which a Lie product substitute the Poisson bracket in a suitable sense.

On the other hand, many physical theories appear to be manifestly covariant with respect to some (Lie) group. Accordingly, it would be relevant to retain such a covariance within the quantization in order to obtain a quantum theory manifestly covariant with respect to the same group as its classical counterpart. In this respect, it is necessary that the ``classical'' bracket is covariant with respect to the same group. This is not the case, for instance, for the Poisson bracket of Hamiltonian dynamics and its generalization to field theories when the group is the Poincaré one. This is because it is defined on an algebra of observables at a fixed time and, thus, a space-time splitting which brake the covariance, is used from the very beginning. We propose a way to define a bracket directly on the space of solutions of the dynamical system. Consequently, such a bracket is covariant with respect to the relevant group by construction both in particle dynamics and in field theory. We also show the relation with the standard Poisson bracket of Hamiltonian dynamics. The mathematical setting we will use is the multisymplectic formulation of first order Hamiltonian field theories.

December 3, 2020

**Title:** *Geometrical splitting methods for contact Hamiltonian systems*

**Speaker:** Federico Zadra (Bernoulli Insitute, University of Groningen)

Contact Hamiltonian systems attracted a lot of interest during the last years due to their implications in Mechanics, Thermodynamics, Fields Theory, and many other topics in Physics. Even if there is a rich geometrical framework behind, many Hamiltonian systems do not admit an exact solution. In the talk, contact Hamiltonian splitting methods will be introduced: they allow to have a numerical method compatible with the geometrical structure. We will consider different examples, such as: Lane–Emden, Spin-Orbit and Lienard Systems, which permit to analyze better its features and limits.

December 3, 2020

**Title:** *Hopf-Rinow theorem of sub-finslerian geometry*

**Speaker:** Layth M. Alabdulsada (University of Debrecen)

The sub-Finslerian geometry means that the metric $F$ is defined only on a given subbundle of the tangent bundle, called a horizontal bundle. In the paper, a version of the Hopf-Rinow theorem is proved in the case of subFinslerian manifolds, which relates the properties of completeness, geodesically completeness, and compactness. The sub-Finsler bundle, the exponential map and the Legendre transformation are deeply involved in this investigation.

December 4, 2020

**Title:** *Two charged particles on a sphere*

**Speaker:** Nataliya Balabanova (University of Manchester)

The Hamiltonian approach to the motion of two charged non-relativistic particles in the plane or in three-dimensional space in the presence of a magnetic field has been studied for the cases with the symmetry group $text{SE}(2)$. Our take on the similar problem is considering a setup with $text{SO}(3)$-symmetry: to this end, we place the particles on a sphere in $mathbb{R}^3$ and assume the magnetic field to be spherically symmetric and of uniform strength. We will discuss the Hamiltonian formalism of this problem, establish existence of relative equilibria and discuss in detail the case of two identical particles, drawing the comparisons between our problem and that of gravitational interaction between two bodies on a sphere.

This talk is the result of joint work with Dr James Montaldi.

December 4, 2020

**Title:** *The topology of Bott integrable fluids*

**Speaker:** Robert Cardona (Universitat Politècnica de Catalunya)

The Euler equations describe the dynamics of an inviscid and incompressible fluid on a Riemannian manifold. In the context of stationary solutions, Arnold's structure theorem marked the birth of the modern field of Topological Hydrodynamics. This theorem provides an almost complete description of the rigid behavior of a solution whose Bernoulli function is analytic or Morse-Bott. However, very few examples of such fluids exist in the literature.

We prove a realization and topological classification theorem for non-vanishing Eulerisable flows (steady solutions for some metric) with a Morse-Bott Bernoulli function. The proof combines the geometry of the equation for a varying metric with the theory of integrable systems. If we drop the non-vanishing assumption, we investigate how the topology of the ambient manifold can be an obstruction to the existence of any Bott integrable fluid for any metric, answering a question raised by Daniel Peralta-Salas in the Morse-Bott case.

#### ** Schedule: **

Note that the schedule is in CET (the local time in Utrecht).
**Some extra information about the Workshop**

**discussion sessions**at the end of each day. These will provide an opportunity for attendees to talk with the speakers about the content of the talks/lectures. The way this will be implemented is through the “breakout rooms” functionality in Zoom: We will create 4 distinct rooms, one for each speaker, and attendees will be assigned randomly to one of them. After that, attendees will be free to move in-between rooms.

- Make sure your interactions with others during the Workshop are civil. We want to create an inclusive atmosphere in which everyone feels welcome to contribute.
- You are welcome to ask questions to the minicourse lecturers. It is best if you type your question in chat and let the chairperson read it to the speaker. Alternatively, you can use the “raise your hand” functionality in Zoom, and wait for the chairperson to give you the turn.
- For the short talks, wait until the end to ask questions to the speaker. We will not have much time inbetween talks, so recall that you can further interact with the speaker in the discussion session at the end of the day.
- The format of each discussion session will be decided by the corresponding speaker. Generally speaking, you can expect a more informal setting than during the talks.
- If you have any questions, please contact us in the organisation!

**Online discussion board of the Workshop**

**short videos**(at most 10-15 minutes, preferably shorter) in which they explain some of their work. You may then upload it to Youtube (or some other streaming service), and share the link in a new post in the Group.

#### ** Scientific Committee: **

- María Barbero (Universidad Politécnica de Madrid, Spain)
- Cédric M. Campos (Universidad Rey Juan Carlos, Spain)
- Madeleine Jotz Lean (Georg-August Universität Göttingen, Germany)
- François Gay-Balmaz (CNRS, École Normale Supérieure, France)
- Bahman Gharesifard (Queen’s University, Canada)
- Ramiro Lafuente (The University of Queensland, Australia)
- Juan Margalef (Penn State’s University, USA)
- Rodrigo T. Sato Martín de Almagro (Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany)

** Organising Committee: **

- Álvaro del Pino (Utrecht University, The Netherlands)