Research group in Nonlinear Algebra

Bernd Sturmfels

Contact: Email
+49 (0) 341 - 9959 - 750

Inselstr. 22
04103 Leipzig

Administrative Assistant:
Saskia Gutzschebauch
Email, Phone/Fax:
+49 (0) 341 - 9959
- 752
- 658


Summer 2019

Enumerative Geometry

  • Lecturer: Francesco Galuppi
  • Date: Wednesday 09:30 - 11:00 and Thursday 13:30 - 15:00
  • Room: MPI MiS, G3 10
  • Language: English
  • Target audience: MSc students, PhD students, Postdocs
  • Keywords: Algebraic geometry, 27 lines, 3264 conics, intersection theory
  • Prerequisites: An undergraduate course on algebraic geometry. Enthusiasm about doing exercises and about teamwork are very welcome.


We all know that there is exactly 1 conic through 5 general points in the plane, but as soon as we ask how many conics are tangent to 5 given smooth conics the answer is somewhat less obvious. This kind of enumerative questions has been considered since the ancient times.
Enumerative geometry is a great playground and provides plenty of concrete problems to work with, but is sometimes left out of a basic course on algebraic geometry. This reading group aims to fill the gap. We will study how to precisely formulate enumerative questions and how to use intersection theory to find the answers.
While doing that, we will incidentally learn a lot of geometry and bump into key concepts such as Chow rings, Schubert calculus, Grassmanians, Hilbert schemes and Chern classes.

Invitation to Nonlinear Algebra


Participants will work through the 13 chapters of the book manuscript with that title. The course will be a mix of lectures by participants and study sessions, with 2 hours devoted to each chapter. A tentative schedule is as follows:

1. Polynomial Rings (June 4, 10 - 12, changed room: A3 01)
2. Varieties (June 4, 14 - 16, changed room: A3 01)
3. Solving and Decomposing (June 5, 10 - 12)
4. Mapping and Projecting (June 5, 14 - 16)
5. Linear Spaces and Grassmannians (June 6, 10 - 12)
6. Nullstellensaetz (June 6, 14 - 16)
7. Tropical Algebra (June 11, 10 - 12, changed room: A3 01)
8. Toric Varieties (June 11, 14 - 16, changed room: A3 01)
9. Tensors (June 12, 14 - 16)
10. Representation Theory (June 13, 10 - 12)
11. Invariant Theory (June 13, 14 - 16)
12. Semidefinite Programming (June 14, 10 - 12)
13. Combinatorics (June 14, 14 - 16)

Each chapter will be assigned to one or two leading presenters. Participants from nearby universities (Leipzig, Berlin, Magdeburg,...) are most welcome. Travel support is available for assigned leaders.

Hodge Theory and Periods of Varieties

  • Lecturer: Emre Sertöz
  • Date: Block course: May 6 - 10, time tba
  • Room: MPI MiS, G3 10
  • Language: English
  • Target audience: MSc students, PhD students, Postdocs
  • Keywords: Curves, surfaces, periods, cohomology
  • Prerequisites: Linear algebra, calculus, visualization skills
  • Remarks: This is a block course, consisting of two blocks with each block lasting a week. There will be 4 hours of lectures per day with an additional 2 hours for working on exercises in groups. The second block will be late in summer.


Starting from 19th century, elliptic and hyperelliptic integrals captivated the imagination of almost every mathematician. The initial observations provided the intuition for much of the development of modern complex geometry. At our present day, one starts learning the subject from these modern abstractions. However, in this lecture, we will work with concrete examples to see for ourselves what the ancients have seen in order to develop our intuition. This is intended so that we can anticipate what the abstract technical framework must look like, without doing anything technically demanding. We will end the first block by looking ahead and studying projective hypersurfaces, their Hodge decomposition and their periods.

For a daily breakdown of the contents, as well as the plan for the second block, see at Emre Sertöz homepage.

23.11.2020, 13:55