• Elements of dynamical systems: phase space, equilibrium and stability, bifurcations and limit cycles, homoclines and heteroclines. 
  Application of the theory to examples in chemical kinetics and population dynamics. Effective potential and ionic currents: cellular excitation. 
  The model by Hodgkin and Huxley and the model by FitzHugh and Nagumo. [KS, chapter 5] 
• Reaction--diffusion equations. Qualitative analysis: traveling fronts, traveling pulses, spiral waves. Application of the theory to the propagation of electical signals 
  and to the depolarization of living cells. [KS, chapter 6]
• Stability in continuous media. Elements of Fourier analysis, linear stability analysis. Application of the theory to reaction-diffusion systems of equations: 
  Turing instability and pattern formation. Dynamical patterns. [P, chapter 6]  [S, chapter 5.2] 
• Elements of homogenization theory: two-scales expansion, reaction-diffusion equations with periodic and non-periodic microstructure. 
  Homogenization of the flow in porous media. [H, chapter 5] 
• Mixture theory. Volume fraction, saturation constraint, partial stress tensor, boundary conditions. Porosity and permeability, Darcy equation.  Application 
  of the theory to the growth of cell populations and to the diffusion-degradation of drugs in living tissues. [H2] Growth of a tumor spheroid. [P, chapter 10]  
• Inverse problems in biomedicine. Radon transform. Ill-posed problems and regularization. The Tichonov functional, the adjoint operator. Differentiability 
  according to Frechet and Gateaux derivative. Application to medical images. [K] 
• Elements of continuum mechanics. [L, chapters 1-2] Fluid-structure interaction, systems of coordinates and interface conditions. The ALE method. 
  Application of the theory to the fluid dynamics of large vessels.   
• The material symmetry group. Isotropic, transverse isotropic and orthotropic materials: strain invariants and strain energies. The role of fibres 
  in biomechanics: collagen and cardiac fibres. Cardiac mechanics: the anatomy and physiolgy of the heart, the PV-loop, fibres shortening and torsion, 
  active stress and active strain. [L, chapter 5]  
• Viscoelastic materials. Linear lumped-elements models, dynamic rheometry. Application to biological tissues. [J]

Testi
[KS] J. Keener and J. Sneyd, ``Mathematical Physiology'', Springer.
[P] B. Perthame, ``Growth, reaction, movement and diffusion from biology''. http://www.ann.jussieu.fr/~perthame/
[H] M. Holmes, ``Introduction to Perturbation Methods'', Springer, 1995.
[H2] M. Holmes, ``Mixture Theories for the Mechanics of Biological Tissues'', RPI Web Book, 1995.
[J] D.D. Joseph, ``Fluid dynamics of viscoelastic liquids'', Springer
[S] S.Salsa, F.Vegni, A.Zaretti, P.Zunino, ``Invito alle Equazioni a Derivate Parziali'', Springer Italia, 2009.
[L] I-Shih Liu, A Continuum Mechanics Primer On Constitutive Theories of Materials, Lecture Notes (2006). http://www.im.ufrj.br/~liu/ 
[K] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer (2011)

Esame orale su tutto il programma. E' *fortemente* consigliato aver seguito con profitto gli esami di Meccanica Razionale e dei Continui (obbligatorio per la laurea triennale) e il corso di Equazioni Differenziali alle Derivate Parziali EDP1 (facoltativo per alcuni indirizzi della laurea triennale).