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    Nonlinear Algebra HT2026

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    Nonlinear algebra

    The toolbox of linear algebra is often not sufficient to meet the nonlinear world. Polynomials offer a natural generalization. The course offers participants hands-on investigations (via both proofs and coding) of wide classes of algebraic optimization problems and algebraic inverse problems, coming from various fields of engineering (robotics, computer vision, deep learning, etc).

    The course covers: Overview of the necessary background in algebra/algebraic geometry. Solving polynomial systems with symbolic and numeric methods, optimization over semi-algebraic sets, tensor geometry, metric algebraic geometry. Applications of the theory to solve problems within engineering, machine learning, data science, AI.

    Teaching

    Teachers: Kathlén Kohn, kathlen[at]kth[dot]se
    TA:  Kim Kiehn, kiehn[at]kth[dot]se

    Course literature

    • Mateusz Michałek & Bernd Sturmfels, Invitation to Nonlinear Algebra, American Mathematical Society [2021]
    • Paul Breiding, Kathlén Kohn & Bernd Sturmfels, Metric Algebraic Geometry, Springer Nature [2023]

    • Igor. R. Shafarevich, Basic Algebraic Geometry Vol 1: Varieties in Projective Space. 3rd ed. Springer [2013]

    Schedule:  starting on September 2: 9:00-12:00 on Wednesdays

    see details here
    The class always starts at 9:00 sharp!

    For detailed planning (lecture/exercise) see sections below

    Examination

    Examiner:  Sofia Tirabassi

    Examination Form:
    Written exam (in-person, on campus): 70%,
    Labs (in-person, on campus): 30%. 
    Participants can obtain up to 10% bonus points for active participation in class.

    Preliminary grading scale:

    A B C D E
    94% 83% 72% 61% 50%

    To obtain passing grade E, participants have to take part in at least 1 Lab.

    Labs:
    There will be 2 labs, each giving 15 points. Each lab will consist of an individual part (like a kontrollskrivning) and a team project. Students are allowed to bring one handwritten, two-sided A4 sheet of notes to the lab.
    The labs are taking place during 9:00-12:00 at the following days:
    Lab 1: We, September 30, 2026
    Lab 2: We, November 4, 2026
    In case a participant cannot take part in one of the 2 labs or is not satisfied with the amount of points during one of the 2 labs, there will be a bonus lab. The final amount of points obtained from the labs will be calculated as follows:
    Points(Lab1)+Points(Lab2)+Points(BonusLab)-min{Points(Lab1),Points(Lab2),Points(BonusLab)}.
    Time and date of the bonus lab will be determined at a later point.

    Resources
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    Week 1

    Topics: polynomial equation systems, ideals, varieties, projective space, first examples of maps between varieties: low-rank matrix completion / LU factorization  / linear convolutional networks

    Literature: Invitation to Nonlinear Algebra, Chapters 1.1 & 2.1

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    Week 2

    Topics: projective space, algebraic varieties, Zariski topology, semi-algebraic sets, constructible sets, maps between varieties and their images

    Literature: Invitation to Nonlinear Algebra, Chapters 2.1, 2.2, 4.3

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    Week 3

    Topics: irreducibility, dimension, fibers, fiber-dimension theorem, first principles of real vs. complex algebraic geometry, dimension computations via Jacobians, tangent spaces

    Literature: Invitation to Nonlinear Algebra: Chapter 2.1;
    Shafarevich: Chapter 1.6.3 and Lemma 2.4 in Chapter 2.6.2 (careful: this book uses slightly different definitions, which makes the statements seem different - at first sight - from the statements in my slides)

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    Week 4

    Topics: polynomial homotopy continuation, monodromy, discriminants, critical loci of maps, regular points of polynomial equation systems, tensors and their rank

    Literature: Metric Algebraic Geometry: Chapter 3.2 & 3.3

    Software: HomotopyContinuation.jl

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    Week 5

    Lab 1

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    Week 6

    Topics: Gröbner bases (monomial order, elimination theorem, counting solutions), companion matrix, fundamental theorem of algebra, Bézout's theorem, total degree start system, algebraic inverse problems and their degrees

    Literature: Metric Algebraic Geometry: Chapter 3

    Software: Macaulay2

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    Week 7

    RANSAC, application example: the structure-from-motion problem in computer vision

    Literature: Metric Algebraic Geometry: Chapter 13.1 and 13.3
    Section 5.2 in Snapshot of Algebraic Vision

    Software: HomotopyContinuation.jl

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    Week 8

    algebraic optimization, Lagrange multiplier theorem for varieties, Euclidean distance degree, Eckart-Young theorem

    Literature: Metric Algebraic Geometry: Chapter 2.1 and 2.2

    Software: Macaulay2, HomotopyContinuation.jl

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    Week 9

    Singularities as minima, Voronoi cells, exposedness, relative Euclidean boundaries and algebraic boundaries, medial axis

    Literature: Metric Algebraic Geometry: Chapter 8.1, 8.2, 8.4, (7.1)

    Software: Macaulay2

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    Week 10

    Lab 2

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    Day 11

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    Day 12

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    Day 13

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    Day 14

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    Day 15

    Review and outlook