Biomathematics Group


General 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 2020/21.

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:

Introduction to the Theory of Sobolev Spaces

Contents

  • Convergence theorems, Lp spaces and properties.
  • 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

The lecture and exercise classes will take place online. The lecture and the solutions of the exercise sheets will be uploaded as notes and/ or time-independent recordings. Additionally, we offer a weekly question time.

Materials

Lectures for students from other departments during the winter semester 2020/21

Our group offers the following lectures students from other departments during the winter semester 2020/21:

Höhere Mathematik I

Inhalte

  • Grundlegende Konzepte und Rechentechniken: Mengentheorie, Reelle und komplexe Zahlen (speziell kartesische Koordinaten und Polarkoordinaten, Wurzeln komplexer Zahlen), Lösung von Gleichungen und Ungleichungen
  • Funktionen einer Variablen:Grundlegende Konzepte und elementare Funktionen, Stetigkeit, Symmetrie, Monotonie, Umkehrfunktionen, rationale Funktionen, Asymptoten, Folgen und Reihen (Grenzwertbegriff, Rechenregeln), Potenzreihen (Konvergenzverhalten und Rechnen mit Potenzreihen), Exponentialfunktion und Logarithmus, trigonometrische Funktionen
  • Differenziation (eindimensional): Definition von Grenzwerten und Bedeutung der Ableitung, Rechentechniken, implizite Ableitung, Mittelwertsatz, Extremwerte, Regel von de l’Hospital, Taylor-Entwicklung, Darstellung von Funktionen durch Taylorreihen, Anwendungen (Fehlerabschätzung und Approximation)
  • Integration (eindimensional): Definites/Indefinites Integral (Stammfunktion, Riemann-Summe, Hauptsatz der Differential- und Integralrechnung, Mittelwertsatz), Integrationstechniken (Substitution, partielle Integration) Integration von Potenzreihen und rationalen Funktionen, Ideen der numerischen Integration, uneigentliche Integrale, verschiedene Anwendungen

Literatur

  • Günter Bärwolff, Höhere Mathematik, Spektrum Akademischer Verlag (2005), L INF 25
  • Thomas Rießinger, Mathematik für Ingenieure, Springer (2005), ARB 057/170
  • Thomas Rießinger, Übungsaufgaben zur Mathematik f. Ing., Springer (2004), MAS 024/021
  • Neunzert, Eschmann, Blickensdörfer, Schelkes: Analysis 1, L mat 1296.

Kontaktzeit

4 SWS Vorlesung

2 SWS Hörsaalübung

2 SWS Präsenzübung

Termine

Vorlesung: zeitunabhängige Aufzeichnung

Hörsaalübung: zeitunabhängige Aufzeichnung

Anmeldung

URM (bis spätestens Freitag, 30.10, 12:00 Uhr)

Materialien

Seminars and proseminars during the winter semester 2019/20

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

Seminar: Mathematical Models in Life Sciences

Topics

tba

    Extent

    2 SWS seminar

    Requirements

    tba

    Dates

    The first meeting will be at the beginning of the winter semester and will be online.

    If possible the seminar will take place as block course before the christmas holidays.

    Material

    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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