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Random Real Algebraic Geometry

  • Lecture: 2h, 5 ECTS
  • Eligible as BMS Advanced Course in Area 2
  • Anrechenbar als Modul „Fortgeschrittene Themen der Algebra”

Schedule and Organization

The lecture starts on May 2 and takes place in person.

Course Hours
1200 - 1330
MA 144
Prof. Dr. Peter Bürgisser


Algebraic geometry provides insights about the solutions of polynomial equations in generic situations. Well known examples are:

  • A general polynomial of degree d has exactly d zeros.
  • There are exactly two lines meeting two general lines in 3-space.
  • A smooth cubic surface contains exactly 27 lines.

But this only holds when counting complex zeros. How about real solutions? When assuming that the given objects are random, then we may ask about their typical behaviour. A first rough measure may be the expected number of real solutions. When the solutions set is infinite, we may ask about its typical topology, measured by Euler characteristic, number of connected components or Betti numbers etc.

Recently, a theory has emerged that provides first answers to such questions. It relies on a mix of algebra with techniques from probability and integral geometry. Random matrix theory is relevant as well.

There are nice recent lecture notes by Paul Breiding and Antonio Lerario on this topic, I intend to follow them partially.

People in my group are expected to participate in this course: it provides the background for some exciting research we are currently undertaking.


  • random polynomials (Kac, Kostlan)
  • invariant probability distributions
  • tools from differential geometry
  • coarea formula
  • integral geometry formula
  • random fields (Kac-Rice formula)
  • Euler characteristic
  • Betti numbers


  • Paul Breiding and Antonio Lerario. Lectures on Random Algebraic Geometry. https://pbrdng.github.io/rag.pdf
  • Peter Bürgisser and Antonio Lerario. Probabilistic Schubert Calculus. Crelles Journal 760, 2020.
  • Alan Edelman and Eric Kostlan. How many zeros of a random polynomial are real. Bull. AMS 32, 1995.
  • Marc Kac. On the average number of real roots of a random algebraic equation. Bull. AMS 49, 1943.
  • Antonio Lerario. Lectures on Metric Algebraic Geometry.
  • Peter Sarnak and Igor Wigman. Topology and nesting of the zero set components of monochromatic random waves. Comm. Pure App. Math. 2018.

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