# Condition: the geometry of numerical algorithms

**Lecturer:**Paul Breiding**Date:**Tuesday, 09:00 - 10:00**Room:**G3 10**Language:**English**Target audience:**MSc students, PhD students, Postdocs**Keywords:**Condition numbers, complexity of numerical algorithms, loss of precision, average analysis, smoothed analysis**Literature:***Condition: the geometry of numerical algorithms*by Bürgisser and Cucker (Springer, 2013)*The condition number of join decompositions*by Breiding and Vannieuwenhoven (SIAM J. Matrix Anal. and Appl., 39(1), 287–309)**Prerequisites:**Participants should have a good understanding of linear algebra and a basic knowledge about probability theory and differential geometry. Expertise in linear programming, algebraic geometry or multilinear algebra is helpful but not required.

## Abstract:

Numerical data rarely is given exact, but usually comes with errors due to noise, measurement errors or errors caused by prior calculations. Consequently, algorithms which take such data as input can't produce correct answers.

In this course we will learn about the concept of *condition numbers* and how it helps to understand how much the solution of a computational problem is changed, if the input data is perturbed. We will also discuss how condition numbers are related to the complexity of numerical algorithms. The concepts of average and smoothed analysis will be introduced.

We will consider the condition numbers of the following problems: linear equation solving, polynomial equation solving and computing tensor decompositions. If time permits, we will also cover condition numbers of problems in linear programming.

## 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