Data Loci course

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Next lectures
30.03.2026, 10:00 (G3 10 (Lecture hall))
09.04.2026, 10:00 (G3 10 (Lecture hall))
13.04.2026, 10:00 (G3 10 (Lecture hall))
20.04.2026, 10:00 (G3 10 (Lecture hall))
27.04.2026, 10:00 (G3 10 (Lecture hall))
04.05.2026, 10:00 (G3 10 (Lecture hall))
11.05.2026, 10:00 (G3 10 (Lecture hall))
18.05.2026, 10:00 (G3 10 (Lecture hall))
21.05.2026, 10:00 (G3 10 (Lecture hall))

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Abstract

Many optimization problems can be represented by systems of parametric polynomial equations. The choice of parameters influences the geometry of the solutions associated with these polynomial systems. A sufficiently "generic'' or "random'' selection of parameters often results in solution sets that exhibit expected properties, such as the expected dimension or cardinality. In contrast, non-generic parameters give rise to algebraic subsets known as Data Loci. Each data locus is formed by imposing additional conditions on the solution sets.

In this course, we will provide an overview of the most relevant types of data loci in algebraic optimization, with a particular emphasis on data loci in distance optimization - a related field referred to as Metric Algebraic Geometry - and on the computation of Nash equilibria in Game Theory.

Each lecture will contain several examples and computations in Macaulay2 and Julia.

Tentative schedule

1) March 30 (10:00-12:00, G310): The intersection theory you need

2) April 09 (10:00-12:00, G3 10): Polar classes and projective duality

3) April 13 (10:00-12:00, G310): The first data loci: evolutes and ED discriminants

4) April 20 (10:00-12:00, G310): Offsets and ED polynomials

5) April 27 (10:00-12:00, G310): Relative polar classes and conditional data loci

6) May 4 (10:00-12:00, G310): Kalman varieties (CC equations meet RR equations)

7) May 11 (10:00-12:00, G310): Higher-order polar classes and higher-order distance loci

8) May 18 (10:00-12:00, G310): Data loci in Game Theory 1: Nash discriminants and resultants

9) May 21 (10:00-12:00, G310): Data loci in Game Theory 2: Nash Loci