Mathematics for Data Science | Marcel Oliver’s Website

Winter Semester 2025/2026

Summary

This class constitutes the mathematical core of Master in Data Science program. It gives a rigorous, structured review of Linear Algebra and Analysis with particular emphasis on applications in Data Science, Computation, and Modeling and a view toward working in high-dimensional or infinite-dimensional vector spaces. The following topics will be covered:

Review of linear algebra, including inner product spaces, least squares, diagonalization and singular value decomposition
Calculus in Banach spaces (metric space topology, Fréchet-derivative, general Taylor theorem, fundamental theorem of calculus)
Measures and integration
Basic harmonic analysis (Fourier transform, sampling, wavelets)
If time allows: Review of elementary probability theory

Quick Links

Main Textbook

J. Humpherys, T.J. Jarvis, and E.J. Evans, Foundations of Applied Mathematics, Vol. 1, Mathematical Analysis, Society for Industrial and Applied Mathematics, 2017

Additional Reading

E.W. Cheney, Analysis for Applied Mathematics, Springer, 2001
D. Calvetti and E. Somersalo, Mathematics of Data Science — a computational approach to clustering and classification, Society for Industrial and Applied Mathematics, 2021

Grading

The grade for this class is determined by a final written exam.
Successful submission of weekly exercises will add a grade bonus of 1/3 of a grade step to your final grade.
There will be a mock exam 3-4 weeks before the final exam. A passing grade on the mock exam will add another grade bonus of 1/3 of a grade step to your final grade.
Note that the pass/fail decision is not affected by the bonus, and the top grade can be achieved without the bonus.

Topics (tentative, subject to change!)

Mon, October 13, 2025: Review of linear algebra: vector spaces, subspaces, linear independence (HJE, pp. 3-13)
Thu, October 16, 2025: Review of linear algebra (ctd.): replacement and basis exchange theorems, dimension (HJE, selected topics from pp. 14-27)
Mon, October 20, 2025: Linear transformations: examples, kernel, range (HJE, pp. 31-36)
Thu, October 23, 2025: Linear transformations (ctd.): matrix representations, change of basis, linear systems (HJE, pp. 37-65)
Mon, October 27, 2025: Determinants (HJE, pp. 65-78)
Thu, October 30, 2025: Inner product spaces, orthonormal sets, orthogonal projections, QR (HJE, pp. 87-105)
Mon, November 03, 2025: Normed linear spaces, operator norm, norm inequalities (HJE, pp. 110-120)
Thu, November 06, 2025: Adjoints, least squares (HJE, pp. 120-130)
Mon, November 10, 2025: Eigenvalues and eigenvectors, invariant subspaces, diagonalization (HJE, pp. 139-150)
Thu, November 13, 2025: Schur’s lemma, spectral theorem, normal and Hermitian matrices (HJE, pp. 150-159)
Mon, November 17, 2025: Singular value decomposition (HJE, pp. 159-171)
Thu, November 20, 2025: Metric spaces, open sets, continuous functions and limits (HJE, pp. 179-190)
Mon, November 24, 2025: Closed sets, sequences, convergence (HJE, pp. 190-195)
Thu, November 27, 2025: Completeness and uniform continuity (HJE, pp. 195-201)
Mon, December 01, 2025: Compactness (HJE, pp. 201-208)
Thu, December 04, 2025: Banach spaces, uniform convergence, continuous linear extensions (HJE, pp. 210-217)
Mon, December 08, 2025: Topologically equivalent metrics, homeomorphisms, connectedness (HJE, pp. 219-227)
Thu, December 11, 2025: Banach-valued integration (HJE, pp. 227-233)
Mon, December 15, 2025: Differentiation: directional derivatives, partial derivatives, Fréchet derivative in finite dimensions (HJE, pp. 241-251)
Thu, December 18, 2025: General Fréchet derivative, properties of derivatives (HJE, pp. 252-259)
Mon, December 22, 2025: Mean value theorem and fundamental theorem of calculus (HJE, pp. 260-265)
Thu, January 08, 2026: Taylor’s theorem (HJE, pp. 266-272)
Mon, January 12, 2026: Contraction mapping principle, uniform contraction mapping principle, Newton’s method (HJE, pp. 277-284)
Thu, January 15, 2026: Newton’s method (HJE, pp. 286-293)
Mon, January 19, 2026: Implicit and inverse function theorem (HJE, pp. 293-300)
Thu, January 22, 2026: Multivariable integration (HJE, pp. 319-330)
Mon, January 26, 2026: Measurability, monotone convergence (HJE, pp. 331-340)
Thu, January 29, 2026: Fatou’s lemma, dominated convergence, differentiation under the integral (HJE, pp. 340-349)
Mon, February 02, 2026: Change of variables (HJE, pp. 349-356)
Thu, February 05, 2026: TBA