Biomathematics Group


Allgemeine Information

Below you can find possible topics for bachelor and master theses or a reading course. 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.

Further, you can find the lectures and seminars that are offered during the winter semester 2021/22.

Important Links:

  • KIS: Dates of 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 wintersemester 2022/23

Our group offeres the following lectures for mathematics students during the winte rsemester 2022/23:

Einführung in die Funktionalanalysis

Contents

  • Beispiele für Banachräume und Hilberträume
  • Kompaktheit, Heine-Borel, Arzelà-Ascoli
  • beschränkte lineare Operatoren, adjungierte Operatoren, Neuman-Reihe
  • Orthogonalität, Hilbertraum-Basis, Riesz-Darstellung, Lax-Milgram, selbstadjungierte Operatoren, Spektraltheorie

Literature

  • H.W. Alt: Lineare Funktionalanalysis. Springer, Berlin, 2006
  • H. Heuser: Funktionalanalysis. Teubner, Wiesbaden, 2006.

Extent

2 SWS lectures

1 SWS exercises

Requirements

Grundlagen der Mathematik I + II

Dates

Lectures:

Mittwoch, 11:45 - 13:15, Raum 48-208

Exercises:

Montag, 10:00 - 11:30, Raum 46-268 (14-tgl.)

Donnerstag 10:00 - 11:30, Raum 46-267 (14-tgl.)

 

Material

Introduction to the Theory of Sobolev Spaces

Contents

  • Weak derivatives and partial integration, mollifiers, properties.
  • Hölder spaces, boundaries and regularity.
  • Sobolev spaces, approximations by smooth functions.
  • Extensions and traces.
  • Sobolev inequalities and embeddings.
  • Poincare's inequality.
  • Some applications to elliptic PDEs.

We recommend this lecture for everyone who wants to work with partial differential equations

Literature

  • R.A. Adams: Sobolev Spaces, Academic Press, 1975.
  • H. Brezis: Functional Analysis, Sobolev Spaces and Partial Di erential Equations, Springer, 2011.
  • P. Ciarlet: Linear and Nonlinear Functional Analysis with Applications, SIAM, 2013.
  • L.C. Evans: Partial Di erential Equations, AMS 2010.
  • G. Leoni: A rst course in Sobolev spaces, AMS 2009.

Contact Time

2 SWS lecture + 1 SWS exercise classes.

The exercise classes will take place every second week. Turnus and dates to be announced after lecture start.

Requirements

Functional Analysis, Measure and Integration Theory.

Dates

Lectures:

Tuesday, 11:45 - 13:15, building 31-302 IBZ

Exercises:

Monday, 10:00 - 11:30, building 46-268

Thursday, 10:00 - 11:30, building 46-267

every 2 weeks

Material

Partial Differential Equations: An Introduction

Contents

Partial Differential Equations play an important role in natural sciences and engineering. Stationary processes can be modelled by differential equations that involve more than one spatial variable, e.g. equations for membrans or electrostatic and gravitational potentials. The equations can also include time derivatives to describe transient processes like growth processes, wave propagation, heat transfer or fluid flow. In this introductory lecture the three most important types of second order PDEs are presented: elliptic, parabolic and hyperbolic equations. Explicit solution techniques and the qualitative baviour of solutions is discussed. Special knowledge of results from functional analysis is not required.

Literature

  • Lecture notes
  • L.C. Evans: Partial differential equations. AMS 1998;
  • F. John: Partial differential equations. Springer-Verlag 1986.

Contact time

2 SWS lecture + 1 SWS exercise classes.

The exercise classes will take place every second week. Turnus and dates to be announced after lecture start.

Requirements

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

Dates

Lectures:

Tuesday, 08:15 - 09:45, building 46-267

Thursday, 08:15 - 09:45, building 46-267

Exercises:

Tuesday, 15:30-17:00, building 44-380

The lecture and exercise class take place in the second half of the semester.

Material

Lectures for mathematics students during summer semester 2022

Our group offered the following lectures for mathematics students during the summer semester 2022:

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.

Extent

4 SWS lectures

2 SWS exercises

Requirements

Bachelor or equivalent ODE

Dates

Lectures:

Tuesday, 12:00 - 13:30

Thursday, 12:00 - 13:30

Exercises:

Thursday, 14:00 - 15:30

The lectures and the exercises will take place in 31-302 IBZ.

Material

Einführung: Gewöhnliche Differentialgleichungen

Contents

  • Differentialgleichungen erster Ordnung: Autonome Differentialgleichungen erster Ordnung, Variation der Konstanten, explizit lösbare Fälle, Anfangswertprobleme,
  • Existenz und Eindeutigkeit: Funktionalanalytische Grundlagen, Banachscher Fixpunktsatz, Satz von Picard-Lindelöf, Fortsetzbarkeit von Lösungen, Existenzsatz von Peano,
  • Qualitatives Verhalten: Lemma von Gronwall, Stetige Abhängigleit von den Daten, Ober- und Unterfunktionen,
  • Lineare Differentialgleichungen: Homogene lineare Systeme, Matrix-Exponentialfunktion, Variation der Konstanten, Differentialgleichungen n-ter Ordnung,
  • Stabilität: Dynamische Systeme, Phasenraum, Hamiltonsche Systeme, Asymptotisches Verhalten, Stabilitätstheorie nach Lyapunov.

Extent

2 SWS lectures

1 SWS exercises

Requirements

Dates

Lectures:

Friday, 12:00 - 13:30

Exercises:

Monday, 10:00 - 11:30

Wednesday, 12:00 - 13:30

Wednesday, 16:00 - 17:30

every 2 weeks

Material

Lectures for mathematics students during winter semester 2021/22

Our group offered the following lectures for mathematics students during the  winter semester 2021/22:

Partial Differential Equations: An Introduction

Contents

Partial Differential Equations play an important role in natural sciences and engineering. Stationary processes can be modelled by differential equations that involve more than one spatial variable, e.g. equations for membrans or electrostatic and gravitational potentials. The equations can also include time derivatives to describe transient processes like growth processes, wave propagation, heat transfer or fluid flow. In this introductory lecture the three most important types of second order PDEs are presented: elliptic, parabolic and hyperbolic equations. Explicit solution techniques and the qualitative baviour of solutions is discussed. Special knowledge of results from functional analysis is not required.

Literature

  • Lecture notes
  • L.C. Evans: Partial differential equations. AMS 1998;
  • F. John: Partial differential equations. Springer-Verlag 1986.

Extent

2 SWS lecture

1 SWS exercises

 

Requirements

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

Dates

Lectures:

Tuesday, 08:00 - 09:30 (OpenOLAT)

Thursday, 08:00 - 09:30 (OpenOLAT)

Exercises:

Tuesday, 16:00-17:30 (44-380)

The lecture and exercise class take place in the second half of the semester.

Material

Lectures for mathematics students of other disciplines during winter semester 2021/22

Our group offered the following lectures for mathematics students of other disciplines during the winter semester 2021/22 :

Höhere Mathematik III

Inhalte

Vektoranalysis: Mehrdimensionale Integralrechnung, insbesondere:

  • Parametrisierung von Kurven und Flächen im Rn,
  • Berechnung von Oberflächen- und (skalaren und vektoriellen) Kurvenintegralen im Rn,
  • Tangentialräume und Differential, 
  • Klassische Operatoren auf Vektorfeldern: div, rot, grad
  • Integralsätze von Gauß und Stokes, Green’sche Formeln, Anwendungen im 3-dimensionalen Euklidischen Raum

Differentialgleichungen: Grundlegende Konzepte zur Behandlung gewöhnlicher und partieller Differentialgleichungen:

1a. Gewöhnliche Differentialgleichungen: 

  • Differentialgleichungen erster Ordnung: Existenz und Eindeutigkeit, Autonome Differentialgleichungen erster Ordnung, Separationsansatz, Variation der Konstanten, explizit lösbare Fälle, Anfangswertprobleme
  • Lineare Differentialgleichungen:  Homogene lineare Systeme, Matrix-Exponentialfunktion, Variation der Konstanten, Differentialgleichungen n-ter Ordnung

1b. Partielle Differentialgleichungen:

  • Klassifikation und Wohlgestelltheit von partiellen Differentialgleichungen 2. Ordnung
  • Wellengleichung, Poissongleichung, Fouriertransformation
  • Lösungsmethoden: Separationsansatz, Fouriertransformation

1c. Numerische Lösung von Differentialgleichungen:

  • Einzelschrittverfahren (implizit/explizit)
  • Runge-Kutta-Verfahren
  • Schrittweitensteuerung

 

Kontaktzeit

4 SWS / 60 h Vorlesung

2 SWS / 30 h Hörsaalübung

2 SWS / 30 h Präsenzübung

Inhaltliche Voraussetzungen

Höhere Mathematik I und II

Termine

Die Vorlesung wird als zeitunabhängige Aufzeichnungen zur Verfügung gestellt.

Fragestunde 1. Semesterhälfte: Do, 08:00 - 09:30 (42-115)

Präsenzübung und Hörsaalübung: KIS

Anmeldung

Materialien

Abschlussarbeiten und 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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