Instructors: Herbert Edelsbrunner, Carl-Philipp Heisenberg

Teaching Assistants: Vanessa Barone, Daniel Capek, Florian Pausinger

This is a semester-long course revolving around the general topics of Shapes and Patterns. In biology, shape is intimately connected to function, and the quantification of shape and its variation is at the heart of biological organization. In neuroscience, the dynamic pattern of brain activity is overlaid on an intricate arrangement of neuronal cells. In physics, the sometimes deterministic, random, chaotic behavior of systems has a profound impact on our understanding of how the world works. The course material follows mathematical categories, focusing on Symmetric, Smooth, and Fractal Shapes at different times during the semester. The translation between vocabularies in different disciplines will be emphasized. Throughout the course, we will encourage interdisciplinary approaches through the formation of mixed teams of students.

**Projects**

The backbone of the course structure is mathematical, and the meat are topics from the sciences, balancing theory and applications within each topic. To encourage interactions across the disciplines, we will organize the students in mixed groups of about four members. Each group will work on a class project throughout the semester. Possible projects include background search, computer

simulation, mathematical investigation, etc. Here is a preliminary list of possible topics:

1- **Diffusion in different environments.** What are relevant scenarios inbiology? What is known about this question in mathematics and physics? Do simulations under varying conditions. ** ***Karla, Bor, Enikö, Kreso.*

2- **Cellular aggregates and shape changes.** What is known in mathematics and computer science about these local changes? What are the biological scenarios where they are relevant? * Pepa, Julio, Cheta, Catarina.*

3- **Synchronization and relevance in somitogenesis**. What is the fundamental mechanism that leads to synchroniation? What is the specific realization that plays out in the development of somites? *Andreas, Shayan, Shamsi, Jason.*

4- **Measuring cells.** Study conventional measures such as length, area, volume to cells. How about different types of curvature? What is the biological relevance? *Minji, Ran, Corinna, Simon.*

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**Exams and Grade**

There will be homework questions asked during the lectures and solutions will be collected three times during the semester. There will be a written final exam at the end of the course. The grade assignment will depend on the

- class participation: 10%

- class project: 35%

- homework: 20%

- final exam: 35%

Date | Topic (Math) | Topic (Nature) | Notes | Assignments |
---|---|---|---|---|

Tue Oct 07 | Introduction and Organization | |||

I. SYMMETRIC SHAPES | ||||

Thu Oct 09 | Evolution of symmetric forms | Lecture 01 | ||

Tue Oct 14 | Symmetry groups | Lecture 02 | ||

Thu Oct 16 | Chirality | Lecture 03 | ||

Tue Oct 21 | Lattices | Lecture 04 | ||

Thu Oct 23 | Cell sorting in development | Lecture 05 | ||

Tue Oct 28 | Voronoi tessellations | Lecture 06 | ||

Thu Oct 30 | Knots and links | Lecture 07 | Homework I | |

Tue Nov 04 | Project proposals | |||

II. SMOOTH SHAPES | ||||

Tue Nov 11 | Surface tension | Lecture 08 | ||

Thu Nov 13 | Shape formation in plants | Lecture 09 | ||

Tue Nov 18 | Curves and surfaces | Lecture 10 | ||

Thu Nov 20 | Morse theory | Lecture 11 | ||

Tue Nov 25 | Lateral inhibition | Lecture 12 | ||

Thu Nov 27 | Singularities | Lecture 13 | Homework II | |

Tue Dec 02 | Project progress report | Homework IIb | ||

Thu Dec 04 | Project progress report | |||

III. FRACTAL SHAPES | ||||

Tue Dec 09 | Geometric probability | Lecture 14 | ||

Thu Dec 11 | Intrinsic volume | Lecture 15 | ||

Tue Dec 16 | Single cell migration | |||

Thu Dec 18 | Collective migration | Lecture 17 | ||

Thu Jan 08 | Fractal dimension | Lecture 18 | Homework III | |

Tue Jan 13 | Gene networks | |||

Thu Jan 15 | Patterns in the brain | Lecture 20 | ||

Fri Jan 23 | Project final reports | |||

Tue Jan 27 | Preparation for final exam | |||

Thu Jan 29 | Final exam | |||

Hermann Weyl. *Symmetry.* Princeton University Press, Princeton, New Jersey, 1952. *(For background on symmetry; mathematics, biology, and architeture.)*

Herbert Edelsbrunner. *Geometry and Topology for Mesh Generation.* Cambridge University Press, Cambridge, England, 2001. *(For background on Voronoi diagrams and Delaunay triangulations; mathematics and algorithms.)*

D'Arcy Wentworth Thompson. * On Growth and Form.* Cambridge University Press, Cambridge, England, 1961. * (Classic work on biological form. Interesting ideas but slow reading.)*

René Thom. * Structural Stability and Morphogenesis.* Benjamin, Reading Massachusetts, 1975. * (A broad approach to morphology based on smooth functions; mathematics and philosophy but rather controversial.)*

Benoit B. Mandelbrot. *The Fractal Geometry of Nature.* Freeman and Company, New York, New York, 1983.* (A standard text on fractals.)*

Colin C. Adams. *The Knot Book.* Freeman and Company, New York, New York, 1994. *(A textbook on the mathematical theory of knots; very pleasant reading.)*

Yukio Matsumoto. *An Introduction to Morse Theory*. American Mathematical Society, Providence, Rhode Island, 2000. *(An introductory text for Morse theory.)*

J. W. Bruce and P. J. Giblin. *Curves and Singularities*. Cambridge University Press. *(An introduction to curves but also surfaces; pleasant reading.)*

**Left right a/symmetry**

Blum, M., Weber, T., Beyer, T., and Vick, P. (2009). Evolution of leftward flow.Semin Cell Dev Biol 20, 464–471.

Lee, J.D., and Anderson, K.V. (2008). Morphogenesis of the node and notochord: the cellular basis for the establishment and maintenance of left-right asymmetry in the mouse. Dev Dyn 237, 3464–3476.

**Cell sorting **

Krens SF1, Heisenberg CP. Cell sorting in development. Curr Top Dev Biol. 2011;95:189-213. (Review on cell sorting theories and cell sorting events in development)

Lecuit T1, Lenne PF. Cell surface mechanics and the control of cell shape, tissue patterns and morphogenesis. Nat Rev Mol Cell Biol. 2007 Aug;8(8):633-44. (on the idea that the function of cell-cell adhesion is increasing contact size)

Krieg M1, Arboleda-Estudillo Y, Puech PH, Käfer J, Graner F, Müller DJ, Heisenberg CP. Tensile forces govern germ-layer organization in zebrafish. Nat Cell Biol. 2008 Apr;10(4):429-36. (cited in the lecture)

Maître JL1, Berthoumieux H, Krens SF, Salbreux G, Jülicher F, Paluch E, Heisenberg CP. Adhesion functions in cell sorting by mechanically coupling the cortices of adhering cells. Science. 2012 Oct 12;338(6104):253-6. (cited in the lecture)

Maître JL1, Heisenberg CP. Three functions of cadherins in cell adhesion. Curr Biol. 2013 Jul 22;23(14):R626-33. (cited in the lecture)

**Gene Networks**

Molecular Genetics of Bacteria - Jeremy W Dale, Simon F Park - 5th edition (can be borrowed from the library or from Calin Guet)

Recitation slides can be found here: http://pub.ist.ac.at/~barone/

Additional references are mentioned during class or can be retrieved from the lectures pdfs published on-line.