Biomathematics Group


General Information

Below you can find the lectures and seminars offered by our group in the winter semester 2019/20. Afterwards you can find the lectures that were offered during the summer semester 2019.

Further, you can find possible topics for bachelor and master theses. These are only suggestions and according to prior agreement also other topics are possible. Please contact us if you are interested in writing a thesis or in the reading course.

Important Links:

  • KIS: dates of the lectures and exercise classes
  • URM: enrollment into the exercise classes
  • OpenOLAT: course material and further information (you will get the access codes in the first lecture)

Lectures for mathematics students during the winter semester 2019/20

Our group offers the following lectures for mathematics students during the winter semester 2019/20:

Nonlinear PDE

Contents

  • Short review of functional analysis background
  • Linear parabolic PDEs
  • Nonlinear elliptic and parabolic boundary value problems: fixed point arguments, monotone operators.

Literature

  • R.A. Adams, Sobolev Spaces,
  • H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,
  • L.C. Evans, Partial Differential Equations,
  • G.M. Lieberman, Second Order Parabolic Partial Differential Equations,
  • M. Ruzicka, Nichtlineare Funktionalanalysis,
  • R. Showalter, Monotone operators in Banach space and nonlinear partial differential equations,
  • E. Zeidler, Nonlinear Functional Analysis and its Applications II/B: Nonlinear Monotone Operators,
  • C. Surulescu, Mathematical Analysis of PDEs, lecture notes at the TU Kaiserslautern.

Extent

4 SWS / 60 h lecture

+ 2 SWS / 30 h exercises

 

 

Requirements

  • Functional analysis
  • helpful but not necessary: Sobolev spaces

Dates

Lectures:

Monday, 13:45 - 15:15 (31-302 IBZ)

Tuesday, 11:45 - 13:15 (31-302 IBZ)

Exercises:

Thursday, 13:45-15:15 (31-302 IBZ)

Material

Seminars and proseminars during the winter semester 2019/20

Our group offers the following additional courses during the winter semester 2019/20:

Proseminar: Differenzengleichungen und Anwendungen in der Biologie

Inhaltliche Voraussetzungen

  • Grundlagen der Mathematik I und II;
  • hilfreich, aber nicht notwendig: Matlab-Kenntnisse.

Termin

Die Vorbesprechung für das Proseminar findet am Freitag, den 19. Juli 2019, um 11:30 im Raum 31-251 statt.

Materialien

Seminar: Mathematical Models in Life Sciences

Topics

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    Dates

    tba

    Material

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    Lectures for mathematics students during summer term 2019

    Our group has offered following lectures for mathematics students during summer term 2019.

    Biomathematics

    Contents

    Single species models in continuous time:

    • Population growth models: Malthus, habitat variability, Verhulst, contest and scramble, generalist predation.
    • Elementary bifurcations and catastrophes (example: insect outbreak), harvesting.
    • Spatial spread of a single population, traveling waves, and similarity solutions.

    Multispecies models in continuous time:

    • Lotka-Volterra models (predator-prey, competition, symbiosis).
    • Enzyme kinetics, inner and outer solutions, asymptotic methods.
    • Turing pattern formation, animal coat patterns.
    • Invariant sets.
    • Spatial spread of populations: reaction-diffusion equations, traveling waves, taxis, models for tumor growth and invasion.

    Kinetic transport equations: multiscale modeling, upscaling, method of moments.

    • Examples: Brain tumors and their spread in anisotropic tissues,  animal movement on path networks.

    Literature

    • L.C. Evans,  Partial Differential Equations,  AMS,  2010 (second edition).
    • J.D. Murray,  Mathematical Biology I, II,  Springer,  2004.  
    • S.H. Strogatz,  Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering,  Westview Press,  2001.
    • C. Surulescu, Mathematical Biology, lecture notes at the TU Kaiserslautern, 2018.

    Literature about stability and ODE theory:

    • Hartman, P.: Ordinary Diff erential Equations, SIAM, 2002 (second edition).
    • Meiss, J.D.: Di fferential Dynamical Systems, SIAM, 2007.
    • Perko, L.: Diff erential Equations and Dynamical Systems, Springer, 1991.

    Extent

    4 SWS / 60 h lecture

    + 2 SWS / 30 h exercises

     

     

    Requirements

    • Einführung: Gewöhnliche Differentialgleichungen,
    • Numerics of ODE,
    • PDE: An Introduction.

    Dates

    Lectures:

    Tuesday, 08:15 - 09:45 (31-302 IBZ)

    Thursday, 08:15 - 09:45 (31-302 IBZ)

    Exercises:

    Thursday, 11:45-13:15 (31-302 IBZ)

    The exercise class starts on 25 April.

    Material

    Exam

    The oral exams will take place on

    Wednesday, 24 July and

    Wednesday, 21 August.

    Bachelor and master theses and reading course

    Bachelor thesis / Master thesis / Reading Course

    Topics

    • Multiscale modeling of brain tumors: from subcellular dynamics to tumor space-time evolution
    • SDE(stochastic differential equations)-driven modeling of tumor growth with phenotypic heterogeneites.
    • Multiphase modeling of glioma pseudopalisading
    • Reaction-diffusion models for microvascular hyperplasia and glioma pseudopalisading
    • Acidity-driven progression of GBM (glioblastoma multiforme) and therapy approaches
    • Modeling mesenchymal cell invasion and differentiation in a fibrous tissue: steps towards meniscus regeneration
    • Mathematical modeling of buruli ulcer

    Further topics are possible.

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