# Introduction to Random Algebraic Geometry

**Lecturer:**Antonio Lerario**Date:**Tuesday, 09:00 - 10:00, from February 6 to March 27**Room:**G3 10**Language:**English**Target audience:**MSc students, PhD students, Postdocs**Keywords:**Real Algebraic Geometry, Random Matrix Theory, Integral Geometry**Prerequisites:**Basic knowledge from Diﬀerential Geometry, Probability and Algebraic Geometry

## Abstract:

This course will deal with the basic problem of understanding the structure (e.g. the geometry and topology) of the set of solutions of real polynomial equations with *random* coeﬃcients. The simplest case of interest is the count of the number of real zeroes of a random univariate polynomial $$p(x) = a_{0} +· · ·+a_{d}x^{d}\: , where\: a_{0}, . . . , a_{d}$$ are random Gaussian variables – this problem is “classical” and was pioneered by Kac in the 40s. More generally algebraic geometers might be interested, for example, in the number of components of a* random* real plane curve of degree d, or in the *expected *number of real solutions of more advanced counting problems (e.g. enumerative problems). This is a very fresh and modern approach to real algebraic geometry: when the outcome is highly sensitive to the choice of the parameters (in general, there is no notion of “generic” in the real world), the attention is shifted to the “typical” situation. I will present the basic techniques for attacking this type of questions, trying to emphasize the connections of classical algebraic geometry with convex geometry, random matrix theory and random ﬁelds.

## Regular Lectures (Winter 2018/2019)

**Condition: the geometry of numerical algorithms**- Paul Breiding
- Tuesday, 09:00 - 10:00, MPI MiS G3 10, till January 30

**Information, Physics and Computation**- Pau Vilimelis Aceituno
- Wednesday, 13:00 - 14:30, MPI MiS A3 02

**Introduction to Random Algebraic Geometry**- Antonio Lerario
- Tuesday, 09:00 - 10:00, MPI MiS G3 10, from February 6 to March 27

**Random dynamical systems and stochastic porous media equations with nonlinear noise**- Benjamin Gess
- Monday, 16:15 - 17:45, MPI MiS A3 01

**Artiﬁcial Neural Networks and Machine Learning: Theoretical Foundations**- Nihat Ay
- Thursday, 11:15 - 12:45, MPI MiS A3 01
- First Lecture: November, 15th

**Geometric and topological methods for the analysis of data in biology, neurobiology and other fields**- Jürgen Jost
- Friday, 13:30 - 15:00, MPI MiS A3 01
- First Lecture: October, 26th

**Important abstract mathematical concepts (algebraic structures, category theory, (pre)sheaves and schemes)**- Jürgen Jost
- Monday, 13:30 - 15:00, MPI MiS A3 01
- First Lecture: November, 12th

**Singular stochastic PDEs**- Nicolas Perkowski
- Wednesday, 11:15 - 12:45, MPI MiS A3 01

**Seminar on inverse spectral theory**- Angkana Rüland
- Thursdays, 9:15 - 10:45, Leipzig University, SG 3-14

**Introduction into regularity structures**- Felix Otto
- Wednesday, 09:10 - 11:00, MPI MiS A3 01

**Toric Varieties**- Mateusz Michalek
- Friday, 11:00 - 12:30, MPI MiS G3 10

**Morse Theory and Closed Geodesics**- Hans-Bert Rademacher
- Wednesday, 13:15 - 14:45, Leipzig University, SG 3-10

**An introduction to Information Geometry and Mathematical Population Genetics**- Tat Dat Tran
- Friday, 15:15 - 16:45, MPI MiS A3 02

**Reading Seminar: Representation theory - The basics**- Joscha Diehl
- first meeting: Tuesday 16.10.2018 at 11:00h, F3 Section

**Reading Seminar: Geometric Measure Theory**- Renan Assimos Martins, Kostas Zemas
- Monday, 15:00 - 17:00, MPI MiS A3 02

**Geometric and Algebraic Combinatorics**- Bernd Sturmfels
- October, 29 – November 2 and November 16

**IMPRS-Ringvorlesung**- Nihat Ay, Christiane Görgen, Guido Montúfar, André Uschmajew, Stefan Hollands
- Tuesday, 10:00 - 12:00 (tba), MPI MiS G3 10