Seminar topics for PhD students - Isabel S. Labouriau
- last update 10/IX/2019
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Here are some topics in Geometry, Analysis and Dynamical Systems that are background reading for
some of the mathematics I am doing at the moment.
They also make interesting seminar topics in their own right.
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Geometry
- Symmetric patterns, crystals and quasicrystals
One of the many forms of interactions of algebra and geometry,
this was originally developed for application to real crystals, but has other applications.
Particularly interesting are the aperiodic tilings of the plane, that may arise as
projections of periodic patterns in higher dimension.
Another interesting topic is the introduction of colours.
References
M. Senechal, "Quasicrystals and Geometry", Cambridge University Press, 1995.
R.L.E.Schwarzenberger, "Colour symmetry", Bull. London Math. Soc, 16 (1984), 209-240.
I.S. Labouriau and E.M.O. Pinho,
"Periodic Functions, Lattices and Their Projections",
preprint CMUP n. 2018-11 September 2018 - arXiv: 1809.07298 [math.DS]
I.S. Labouriau and E. M. O. Pinho,
"On the projection of functions invariant under the action of a crystallographic group",
Journal of Pure and Applied Algebra 218 (2014) 37-51
M.G.M. Gomes, I.S. Labouriau and E.M. Pinho,
"Spatial Hidden Symmetries in Pattern Formation" --- in
M. Golubitsiky, D. Luss, S.H. Strogatz (eds.),"Pattern Formation in Continuous and Coupled Systems", Springer-Verlag, 2001
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Analysis
- Eigenvalues of linearized partial differential equations under symmetric boundary conditions
Part of the art of solving p.d.e.'s lies in choosing appropriate boundary conditions.
When the boundary conditions are symmetric, the eigenvalues of a linear p.d.e.
inherit some of these symmetries.
This may, in turn, be used to provide information on
a nonlinear p.d.e. under the same boundary conditions.
References
P.G. Drazin, "Introduction to Hydrodynamic Stability", Cambridge University Press, 2002.
S.B.S.D. Castro, I.S. Labouriau, J.F. Oliveira,
"Projections of patterns and mode interactions",
Dynamical Systems an International Journal 33-4 (2018) 547-564
M.G.M. Gomes, I.S. Labouriau and E.M. Pinho,
"Spatial Hidden Symmetries in Pattern Formation" --- in
M. Golubitsiky, D. Luss, S.H. Strogatz (eds.),"Pattern Formation in Continuous and Coupled Systems", Springer-Verlag, 2001
Chapter 5 of M. Golubitsky and I. Stewart, "The Symmetry Perspective", Birkhauser, 2002.
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Dynamical Systems
- Geometry of continuous-time dynamical systems (ordinary differential equations)
The best way of learning about this subject is to work out an example in low dimension.
Choose a family of o.d.e.'s depending on parameters (I can help) and try to understand its geometry.
Use numerics, if you like.
Examples come in all flavours: from applied to pure, from wild to symmetric.
Some of my favourite appear in the general references below.
In particular I am interested in:
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Generic dynamics associated to a projected pattern
Now that we know the symmetries of projected patterns, we are interested in finding how its dynamics compares to those that do not come from a a projection. This may start with a p.d.e. with symmetric boundary conditions and its reduction to an o.d.e. or may be approached directly studying o.d.e.s with prescribed symmetries.
See
M.G.M. Gomes, I.S. Labouriau and E.M. Pinho,
"Spatial Hidden Symmetries in Pattern Formation"
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Partial stability issues in differential equations near networks
See
I. Melbourne. "An example of a non-asymptotically stable attractor" Nonlinearity, 4, 835-844 (1991)
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Slow-fast dynamics and applications to nerve impulse
See
H. W. Broer, T. J. Kaper and M. Krupa, "Geometric desingularization of a cusp singularity in slow-fast systems with applications to Zeeman's examples", preprint (2008)
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Transition to chaos in non-autonomous differential equations
See
I. S. Labouriau, E. Sovrano,
"Chaos in periodically forced reversible vector fields", preprint
CMUP n. 2019-01 January 2019
- arXiv:1901.09009v2 [math.DS]
I.S. Labouriau, A.A.P. Rodrigues, "Dynamics near a periodically forced robust heteroclinic cycle", preprint CMUP n. 2018-10 September 2018
- arXiv: 1809.04006v2 [math.DS]
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General references
On dynamics with symmetry
M. Golubitsky and I. Stewart, "The Symmetry Perspective", Birkhauser, 2002.
- this book was awarded a
prize
M. Field, "Lectures on Bifurcation, Dynamics and Symmetry", Longman, 1996.
Models in biology
two classics:
J.D. Murray, "Mathematical Biology", Springer, 2002/2003.
A.T. Winfree, "The Geometry of Biological Time", Springer, 2001
On dynamics in general
J. Guckenheimer and P. Holmes,
"Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields", Springer, 1983
a real classic:
S. Lefschetz and J. La Salle, "Stability by Liapunov direct method", Academic Press, 1961
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For fun
I.N. Stewart, "Letters to a Young Mathematician", Basic Books, 2006
or in portuguese translation "Cartas a uma jovem matemática", Relógio D'água, 2006
Contains a chapter on "How to choose a PhD supervisor".