Topologie

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

Schedule

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.

Description

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

Topics

• 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

Bibliography

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

Link to Moodle

Topologie

Link to RWTH Online

Topologie

Seminar zu Dynamischen Systemen

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

Schedule

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

Topics

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

Bibliography

• 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.

Link to RWTH Online

Seminar zu Dynamischen Systemen

Seminar on Symplectic Geometry

Prof. Dr. Umberto Hryniewicz, Dr. Matthias Meiwes

Schedule

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

Description

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.

Link to RWTH Online

Seminar on Symplectic Geometry

Symplectic Geometry

Prof. Dr. Umberto Hryniewicz, Dr. Tobias Soethe

Schedule

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.

Goal

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.

Topics

• 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.

Bibliography

• 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.

Link to Moodle

Symplectic Geometry

Link to RWTH Online

Coming soon...

Differentialgeometrie I

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

Schedule

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).

Description

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

Topics

Preliminary syllabus:

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

Bibliography

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.

Link to Moodle

Differentialgeometrie I

Link to RWTH Online

Differentialgeometrie I

Dynamische Systeme

Prof. Dr. Umberto Hryniewicz, Dr. Matthias Meiwes

Schedule

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).

Content

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.

Bibliography

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.

Link to Moodle

Dynamische Systeme

Link to RWTH Online

Dynamische Systeme

Seminar zur Topologie

Prof. Dr. Umberto Hryniewicz, Dr. Tobias Soethe

Schedule

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

Topics

• 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.

Bibliography

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.

Link to RWTH Online

Seminar zur Topologie

Variationsrechnung I

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

Schedule

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).

Description

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.

Topics

Preliminary syllabus:

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

Bibliography

• 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.

Link to Moodle

Variationsrechnung I

Link to RWTH Online

Variationsrechnung I

Topologie

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

Schedule

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)

Description

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

Topics

• 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

Bibliography

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

Link to Moodle

Topologie

Link to RWTH Online

Topologie

Symplektische Geometrie

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

Schedule

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)

Goal

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.

Topics

• 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.

Bibliography

• 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.

Link to Moodle

Symplektische Geometrie

Link to RWTH Online

Symplecktische Geometrie