Prof. Dr. Umberto Hryniewicz, Dr. Mathias Meiwes

**The classes happen:** Mondays, 10:30 - 12:00, SG23 (1810|023) and Tuesdays, 12:00 - 14:30, SG13 (1810|013)

**The exercise sessions happen:** Wednesday, 10:30 - 12:00, SG12 (1810|012)

Execise sheet 1. pdf

Execise sheet 2 and 3. pdf

Execise sheet 4. pdf

Execise sheet 5. pdf

Execise sheet 6. pdf

Execise sheet 7. pdf

Execise sheet 8. pdf

You can find the supplementary material for the course here. pdf (Updated October 1 2020)

- DO CARMO, M. -
*Differential Geometry of Curves and Surfaces*

Englewood Cliffs, Prentice-Hall, 1976 - DO CARMO, M. -
*Riemannian geometry*

Birkhäuser Boston, 1992 - WARNER, F. -
*Foundations of differentiable manifolds and Lie groups*

Springer-Verlag, New York-Berlin, 1983

Prof. Dr. Umberto Hryniewicz, Dr. Louis Merlin

**The seminars happen:** Tuesdays, 16:00 - 18:00, Soziologiegebäude (1830|001), Eilfschornsteinstr. 7

- ARNOLD, V.I. -
*Mathematical Methods of Classical Mechanics*

Springer-Verlag, New York-Berlin, 1989. - BREZIS, H. -
*Functional Analysis, Sobolev Spaces and Parial Differential Equations*

Springer-Verlag, New York-Berlin 2011 - MILNOR, J. -
*Morse theory*

Annals of Mathematical Studies, Princeton University Press, 1973

- WARNER, Frank K. -
*Foundations of Differentiable Manifolds and Lie Groups*

Springer-Verlag, New York-Berlin 1983

Prof. Dr. Umberto Hryniewicz, Dr. Louis Merlin

The first meeting takes place Tuesday April 21st, 14.00 via Zoom.

- ARNOLD, V.I. -
*Mathematical Methods of Classical Mechanics*

Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1989. - MOSER, J. and ZEHNDER, E. -
*Notes on Dynamical Systems.*

Courant Lecture Notes in Mathematics 12, AMS, Providence, 2005

Prof. Dr. Umberto Hryniewicz, Leonardo Masci, M. Sc.

Lectures: Monday, from 10:30 to 12:00 and Tuesday, from 12:30 to 14:00.

Exercise class: Monday, from 8:30 to 10:00.

The links to the Zoom meetings can be found in the Moodle page of the course.

In this course we will show how certain classes of differential equations arise from the search of critical points of functionals on spaces of functions. We will then apply tools coming from functional analysis to find solutions to these equations.

Preliminary syllabus:

- Euler-Lagrange equations.
- Examples: Brachistochrona, Lagrangians for conservative force fields, Lagrangian with magnetic terms.
- Symmetries and Noether's theorem.
- Holonomic constraints, geodesics on manifolds.
- Hamiltonian formalism and the action functional.
- Dirichlet's problem and its variants.
- Review on functional analysis and Sobolev spaces.
- Direct methods in the Calculus of Variations.

- ARNOLD, V.I. -
*Mathematical Mehtods of Classical Mechanics*

Graduate Texts in Mathematics**60**, Springer-Verlag, New York/Berlin, 1989. - BRÉZIS, H. -
*Functional Analysis, Sobolev Spaces and Partial Differential Equations*

Universitext, Springer-Verlag, New York 2011.

Prof. Dr. Umberto Hryniewicz, Dr. Louis Merlin

The meetings take place on Monday, from 14:00 to 15:30, via Zoom.

In this seminar we will introduce Riemann surfaces and work towards their classification. We will use tools from topology, analysis and hyperbolic geometry.

- Riemann surfaces.
- Fundamental group, covering spaces and topological classification of surfaces.
- Harmonic functions, Perron's method and Dirichlet's problem.
- Classification into elliptic, parabolic or hyperbolic Riemann surfaces.
- Uniformization.
- Hyperbolic surfaces.
- Arithmetic groups and compact hyperbolic surfaces.
- Basic Teichmüller theory.

Basics on Riemann surfaces:

- DONALSON, S. -
*Riemann Surfaces*, Chapter 1 to 5

Oxford Graduate Texts in Mathematics**22**, Oxford University Press, Oxford, 2011.

Uniformisation:

- FARKAS, H.M. and KRA, I. -
*Riemann Surfaces*, Sections IV.1 to IV.6

Second Graduate Texts in Mathematics**71**, Springer-Verlag, New York 1992.

Hyperbolic Riemann Surfaces and Arithmetic Groups:

- KATOK, S. -
*Fuchsian Groups*

Chicago Lectures in Mathematics, University of Chicago Press, Chicago 1992.

Basic Teichmüller Theory:

- BUSER, P. -
*Geometry and Spectra of Compact Riemann Surfaces,*Chapter 6,

Modern Birkhäuser Classics, Birkhäuser Boston, Boston 2010.

Prof. Dr. Umberto Hryniewicz, Dr. Matthias Meiwes

The meetings take place on Wednesday, from 10:30 to 12:00, via Zoom.

Symplectic manifolds are the phase spaces of Hamiltonian systems. The goal of this seminar is to cover basic facts on symplectic geometry and Hamiltonian systems. Topics range from simple examples from classical mechanics, to more advanced topics like, for instance, the analysis of the Action functional and the geometry of the group of Hamiltonian diffeomorphisms.

- Symplectic linear algebra.
- Maslov index.
- Hamiltonian systems and calculus of variations.
- Basic facts about symplectic manifolds.
- Symplectic diffeomorphisms, Flux homomorphism, Hamiltonian diffeomorphisms, Calabi homomorphism.
- Symplectic capacities and non-squeezing.
- Basics on Hofer geometry.
- Basic facts about contact manifolds.

Symplectic Geometry:

- MCDUFF, D. and SALAMON, D. -
*Introduction to Symplectic Topology*, Chapter 2 and 3

Oxford Graduate Texts in Mathematics**27**, Oxford University Press, Oxford, 2017.

Hamiltonian dynamics and Symplectic Capacities:

- HOFER, H. and ZEHNDER, E. -
*Symplectic Invariants and Hamiltonian Dynamics*Chapter 1 and 2

Modern Birkhäuser Classics, Birkhäuser Boston, Boston 2011.

Seminar on Symplectic Geometry

Prof. Dr. Umberto Hryniewicz, Leonardo Masci, M. Sc.

Lectures: Monday, from 10:30 to 12:00 and Tuesday, from 10:30 to 12:00.

Exercise class: Wednesday, from 10:30 to 12:00.

The links to the Zoom meetings can be found in the Moodle page of the course.

This course will serve as an introduction to the language of differentiable manifolds, with an eye towards applications in geometry, topology and dynamical systems.

Preliminary syllabus:

- Differentiable manifolds, examples, tangent bundle, differentiable maps.
- Vector fields and Lie brackets.
- Immersions, embeddings and submanifolds.
- Regular values and Sard's Theorem.
- Vector bundles and connections.
- Riemannian metrics, Levi-Civita connection, geodesics.
- Differential forms, integration, Stokes theorem, de Rham cohomology.
- Introduction to degree theory, Poincaré-Hopf theorem. Gauss-Bonnet Theorem.

- LEE, J.M. -
*Introduction to Smooth Manifolds*

Graduate Texts in Mathematics**218**, Springer, New York, 2013. - JÄNICH, K. -
*Topologie*

Springer-Lehrbuch, Springer, Berlin, 2005. - DO CARMO, M. -
*Riemannian Geometry*

Birkhäuser Boston, Boston, 1992 - WARNER, F. W. -
*Foundations of Differentiable Manifolds and Lie Groups*

Graduate Texts in Mathematics**94**, Springer-Verlag, New York-Berlin, 1983. - MILNOR, J.W. -
*Topology from the Differentiable Viewpoint*

Princeton Landmarks in Mathematics, Princeton University Press, Princeton, 1997.

**Further reading**

- LANG, S. -
*Fundamentals of Differential Geometry*

Graduate Texts in Mathematics**191**, Springer-Verlag, New York, 1999. - BOTT, R. and TU, L.W. -
*Differential Forms in Algebraic Topology*

Graduate Texts in Mathematics**82**, Springer-Verlag, New York-Berlin, 1982.

Prof. Dr. Umberto Hryniewicz, Dr. Louis Merlin

Our seminar will be focused on Fourier series, and on its applications to problems in abstract Mathematics. The choice of applications is biased, with emphasis on differential equations coming from Mathematical Physics.

- Fourier Series and their convergence properties.
- Poincaré's rotation number, and dynamical systems on the circle.
- Basic Ergodic Theory and applications to Number Theory.
- Fourier series for the action functional from Classical Mechanics.
- Fourier series for the wave and heat equations. Applications to the physics of sound and music.

You can find a preemptive schedule of the talks in the following document. [PDF]

- ARNOL'D, V.I. -
*Geometrical Methods in the Theory of Ordinary Differential Equations,*

Grundlehren der Mathematischen Wissenschaften**250**, Springer-Verlag, New York, 1988. - ARNOL'D, V.I. -
*Mathematical Methods of Classical Mechanics,*

Graduate Texts in Mathematics**60**, Springer-Verlag, New York, 1989. - BENSON, D. -
*Music: a Mathematical Offering,*

Cambrige University Press, Cambrige, 2007. - COURANT, R. -
*Differential and Integral Calculus, Vol I,*

Wiley Classics Library, John Wiley and Sons Inc., New York, 1988. - COURANT, R. and HILBERT, D. -
*Methods of Mathematical Physiscs, Vol I,*

Interscience Publishers Inc., New York, 1953. - EVANS, L. C. -
*Partial Differential Equations,*

Graduate Studies in Mathematics**19**, American Mathematical Society, Providence, 2010. - HOFER, H. and ZEHNDER, E. -
*Symplectic Invariants and Hamiltonian Dynamics,*

Modern Birkhäuser Classics, Springer-Verlag, Basel, 2011.

Introductory Seminar on Analysis

Prof. Dr. Umberto Hryniewicz, Leonardo Masci, M. Sc.

Lectures: Monday, from 16:30 to 18:00 and Wednesday, from 10:15 to 11:00.

Exercise class: Tuesday, from 16:15 to 17:00.

The links to the Zoom meetings can be found in the Moodle page of the course.

The aim of the course is to introduce the student to the theory of point-set topology and topological manifolds.

- Topologies and constructions with topological spaces.
- Filters, nets, convergence.
- Continuous functions, initial and final topologies
- Compactness and completeness, metrization theorem.
- Separation axioms.
- Topological manifolds, their dimension and metrizability.
- Connectedness and path connectedness

We will follow Professor Hartmut Führ's script for the Topology course. Additional material will be posted on the Moodle.

Prof. Dr. Umberto Hryniewicz, Leonardo Masci M. Sc.

The meetings take place on Fridays, from 10:30 to 12:00, via Zoom.

- Basic notions on dynamical systems: recurrence, ergodicity, entropy.
- Basic notions on hyperbolic dynamics.
- Morse theory and gradient flows.
- Basic notions on Hamiltonian dynamics.

- ZEHNDER, E. -
*Lectures on Dynamical Systems. Hamiltonian vector fields and symplectic capacities,*

EMS Textbooks in Mathematics, European Mathematical Society, Zürich, 2010. - ARNOL'D, V.I. -
*Geometrical Methods in the Theory of Ordinary Differential Equations,*

Grundlehren der Mathematischen Wissenschaften**250**, Springer-Verlag, New York, 1988. - KATOK, A. and HASSELBLATT, B. -
*Introduction to the modern theory of Dynamical Systems,*

Encyclopedia of Mathematics and its Applications**54**, Cambridge University Press, Cambridge, 1995.

Prof. Dr. Umberto Hryniewicz, Dr. Matthias Meiwes

The meetings take place on Tuesday, from 14:00 to 15:30, via Zoom.

The main theme of this seminar is the interaction between dynamics and geometry. Some selected topics: dynamics of Hamiltonian systems, Hamiltonian Floer homology, Contact homology, Clarke duality for the action functional, Geodesic flows and Morse theory.

Seminar on Symplectic Geometry

Prof. Dr. Umberto Hryniewicz, Dr. Tobias Soethe

Lectures: Monday and Wednesday, from 14:00 to 15:30.

Exercise class: Thursday, from 12:30 to 14:00.

The links to the Zoom meetings and the notes from the lectures can be found in the Moodle page of the course.

The student will develop a basic understanding of symplectic manifolds seen as phase spaces of Hamiltonian systems, and their connections to Classical Mechanics and Riemannian Geometry. Moreover, the student will be exposed to the basic variational properties of the action functional, and some of its applications to non-squeezing phenomena.

- Symplectic vector spaces and the symplectic linear group.
- Symplectic manifolds, Darboux's theorem, sympletic and Hamiltonian vector fields, symplectic and Hamiltonian diffeomorphisms, Lagrangian neighborhood theorem.
- Contact manifolds, contact vector fields, Darboux's theorem for contact forms, Legendrian neighborhood theorem, Gray's stability.
- Flux homomorphism.
- The action functional, symplectic capacities and Gromov's non-squeezing.

- MCDUFF, D. and SALAMON, D. -
*Introduction to Symplectic Topology,*

Oxford Graduate Texts in Mathematics, Oxford University Press, Oxford, 2017. - HOFER, H. and ZEHNDER, E. -
*Symplectic Invariants and Hamiltonian Dynamics,*

Modern Birkhäuser Classics, Springer-Verlag, Basel, 2011. - ZEHNDER, E. -
*Lectures on Dynamical Systems. Hamiltonian Vector Fields and Symplectic Capacities,*

EMS Textbooks in Mathematics, European Mathematical Society, Zurich, 2010.

*Coming soon...*

Prof. Dr. Umberto Hryniewicz, Leonardo Masci, M. Sc.

The lectures are scheduled on Mondays and Tuesdays, from 10:30 to 12:00 in room AS (1050|U101). The exercise sessions are scheduled on Tuesdays, from 14:30 to 16:00 in room SG 513 (1810|513).

This course will serve as an introduction to the language of differentiable manifolds, with an eye towards applications in geometry, topology and dynamical systems.

Preliminary syllabus:

- Differentiable manifolds, examples, tangent bundle, differentiable maps.
- Vector fields and Lie brackets.
- Immersions, embeddings and submanifolds.
- Regular values and Sard's Theorem.
- Vector bundles and connections.
- Riemannian metrics, Levi-Civita connection, geodesics.
- Differential forms, integration, Stokes theorem, de Rham cohomology.
- Introduction to degree theory, Poincaré-Hopf theorem. Gauss-Bonnet Theorem.

The course will follow lecture notes written by the lecturer. They will be available on the Moodle page of the course.

**Further reading**

- LEE, J.M. -
*Introduction to Smooth Manifolds*

Graduate Texts in Mathematics**218**, Springer, New York, 2013. - JÄNICH, K. -
*Topologie*

Springer-Lehrbuch, Springer, Berlin, 2005. - DO CARMO, M. -
*Riemannian Geometry*

Birkhäuser Boston, Boston, 1992 - WARNER, F. W. -
*Foundations of Differentiable Manifolds and Lie Groups*

Graduate Texts in Mathematics**94**, Springer-Verlag, New York-Berlin, 1983. - MILNOR, J.W. -
*Topology from the Differentiable Viewpoint*

Princeton Landmarks in Mathematics, Princeton University Press, Princeton, 1997. - LANG, S. -
*Fundamentals of Differential Geometry*

Graduate Texts in Mathematics**191**, Springer-Verlag, New York, 1999. - BOTT, R. and TU, L.W. -
*Differential Forms in Algebraic Topology*

Graduate Texts in Mathematics**82**, Springer-Verlag, New York-Berlin, 1982.

Prof. Dr. Umberto Hryniewicz, Dr. Matthias Meiwes

The lectures are scheduled on Mondays, from 16:30 to 18:00 in room SG 23 (1810|023) and Wednesdays, from 10:30 to 12:00 in room SG 512 (1810|512). The exercise sessions are scheduled on Fridays, from 12:30 to 14:00 in room SG 413 (1810|413).

Preemptive syllabus:

- Recurrence, ergodicity and entropy.
- Morse theory and gradient flows.
- Basic notions on Hamiltonian systems, examples coming from mathematical physics.
- Basic notions on hyperbolic dynamics.

Main reference:

- ZEHNDER, E. -
*Lectures on dynamical systems. Hamiltonian vector fields and capacities.*

EMS Textbooks in Mathematics. European Mathematical Society (EMS), Zürich, 2010.

Further reading:

- HASSELBLATT, B. and KATOK, A. -
*Introduction to the modern theory of dynamical systems*

Encyclopedia of Mathematics and its Applications,**54**. Cambridge University Press, Cambridge 1995. - ARNOL'D, V.I. -
*Geometrical Methods in the Theory of Ordinary Differential Equations, Second edition*

Grundlehren der Mathematischen Wissenschaften,**250**. Springer-Verlag, New York 1998.

Prof. Dr. Umberto Hryniewicz, Dr. Tobias Soethe

Meetings are scheduled on Fridays, from 10:30 to 12:00 in room SG 203 (1810|203).

- Fundamental group and covering spaces.
- Classification of surfaces.
- Homology and cohomology theories.
- Applications: invariance of domain, Brouwer degree, Brouwer fixed point theorem, Lefschetz fixed point theory.
- Higher homotopy groups and their applications.

Main bibliography:

- JÄNICH, K -
*Topologie*

Springer-Verlag, Berlin, 2005 - VICK, J. W. -
*Homology theory. An introduction to algebraic topology*

Graduate Texts in Mathematics,**145**Springer-Verlag, New York, 1994.

Auxiliary bibliography:

- HATCHER, A. -
*Algebraic Topology*

Available online at the author's personal website. - SPANIER, E. H. -
*Algebraic Topology*

McGraw-Hill Book Co., New York-Toronto, Ont.-London 1996.

Prof. Dr. Umberto Hryniewicz, Dr. David Bechara Senior

Lectures take place on Tuesdays, from 14:30 to 16:00, in room V (1010|213), and on Wednesdays, from 12:30 to 14:00, in room III (1010|107).

Exercise sessions take place on Mondays, from 16:30 to 18:00, in room AS (1050|U101).

In this course we will show how certain classes of differential equations arise from the search of critical points of functionals on spaces of functions. We will then apply tools coming from functional analysis to find solutions to these equations.

Preliminary syllabus:

- Euler-Lagrange equations.
- Examples: Brachistochrona, Lagrangians for conservative force fields, Lagrangian with magnetic terms.
- Holonomic constraints, geodesics on manifolds.
- Hamiltonian formalism and the action functional.
- Dirichlet's problem and its variants.

- ARNOLD, V.I. -
*Mathematical Mehtods of Classical Mechanics*

Graduate Texts in Mathematics**60**, Springer-Verlag, New York/Berlin, 1989. - BRÉZIS, H. -
*Functional Analysis, Sobolev Spaces and Partial Differential Equations*

Universitext, Springer-Verlag, New York 2011.

Prof. Dr. Umberto Hryniewicz, Dr. Daniel Rudolf, Leonardo Masci M. Sc.

The lectures take place on Mondays, from 10:30 to 12:00 in room SG 513 (1810|513), and on Wednesdays, from 10:15 to 11:00 in room AS (1050|U101).

The exercise sessions take place on Tuesdays, from 16:15 to 17:00, in room klPhys (1090|334)

The aim of the course is to introduce the student to the theory of point-set topology and topological manifolds.

- Topologies and constructions with topological spaces.
- Filters, nets, convergence.
- Continuous functions, initial and final topologies
- Compactness and completeness, metrization theorem.
- Separation axioms.
- Topological manifolds, their dimension and metrizability.
- Connectedness and path connectedness

We will follow Professor Hartmut Führ's script for the Topology course. Additional material will be posted on the Moodle.

Prof. Dr. Umberto Hryniewicz, Dr. David Bechara Senior, Leonardo Masci M. Sc.

Lectures take place on Wednesdays, from 14:30 to 16:00, and Fridays, from 12:30 to 14:00, in room SG 413 (1810|413).

Exercise sessions take place on Fridays, from 14:30 to 16:00, in room SG 12 (1810|012)

The student will develop a basic understanding of symplectic manifolds seen as phase spaces of Hamiltonian systems, and their connections to Classical Mechanics and Riemannian Geometry. Moreover, the student will be exposed to the basic variational properties of the action functional, and some of its applications to non-squeezing phenomena.

- Symplectic vector spaces and the symplectic linear group.
- Symplectic manifolds, Darboux's theorem, sympletic and Hamiltonian vector fields, symplectic and Hamiltonian diffeomorphisms, Lagrangian neighborhood theorem.
- Contact manifolds, contact vector fields, Darboux's theorem for contact forms, Legendrian neighborhood theorem, Gray's stability.
- Flux homomorphism.
- The action functional, symplectic capacities and Gromov's non-squeezing.

- MCDUFF, D. and SALAMON, D. -
*Introduction to Symplectic Topology,*

Oxford Graduate Texts in Mathematics, Oxford University Press, Oxford, 2017. - HOFER, H. and ZEHNDER, E. -
*Symplectic Invariants and Hamiltonian Dynamics,*

Modern Birkhäuser Classics, Springer-Verlag, Basel, 2011. - ZEHNDER, E. -
*Lectures on Dynamical Systems. Hamiltonian Vector Fields and Symplectic Capacities,*

EMS Textbooks in Mathematics, European Mathematical Society, Zurich, 2010. - CANNAS DA SILVA, A. -
*Lectures on Symplectic Geometry*

Lecture Notes in Mathematics, Springer-Verlag, Berlin, Heidelberg, 2008.