Offer a Seminar on Topics of Linear Algebra and Mathematical Analysis for IST 1st cycle students to nurture talent

In 2015/16, in the 3rd year of lecturing the 1st year semester courses Linear Algebra (Winter semester) and Differential and Integral Calculus II (Spring Semester) to the IST Mathematics, Physics and Biomedical Engineering undergraduate programs, after returning to IST following being President of the Knowledge Society Agency, Luis Magalhães created a voluntary Seminar on Topics of Linear Algebra for his students on Linear Algebra, with one 3 hours/week session, toB> give interested students the opportunity t/B>o: since he found that the best students were needing this type of learning opportunity outside the necessarily very constarined environment of large classes. This would make students to deeply learn the topics they would present to other students and to become acquainted with several other topics by attending the other students presentations.

To avoid that students would have to invest time in searching and selecting well good resources for their presentations, they were provided with lecture notes specially written to be used by the Seminar students to prepare the presentations.

This initiative, which was strictly voluntary and did not provide official credits to students, had excellent results with many students reaching a Mathematics maturity very unusual at such an early stage of studies, and the Seminar was continued year after year, only interrupted for 1 year by the COVID pandemic.

Some students even developped great enthusiasm for the Seminar and got actively involved in informing other students about its value. One of these students (Carolina Figueiredo, then 1st year student of the IST Physics undergraduate program, and later completing the 1st year IST Applied Mathematics Master Program before going to study for a PhD in Physics at Princeton U., where her research was highly successful leading her to earn the 2026 Vera Rubin New Frontiers Prize by the Breakthrough Prize Foundation, in the first edition of this Prize) asked that the Seminar be also offered in the Spring Semester while students were taking the Differential and Integral Calculus II course. This was inconvenient to Luis Magalhães since his work load in the Spring Semester was more demanding, and he resisted to offering the Seminar then, but Carolina insistence was such that he told her that he would offer the Seminar if she would come up with at least 6 interested students. She came with 7, the Seminar had to be offered and, in fact, was especially successful. Due to this, the Seminar was exceptionally offered also in the Spring Semester, but only once.

As Luis Magalhães had to stop lecturing official courses at his Jubilação in his 70th anniversary at the end of 2021, since he was not lecture courses in the Winter Semester of 2021/22, he decided to continue offering the Seminar expanding it with major changes: Thus, beginning in 2021/22, the Seminar took the name Seminar on Topics of Linear Algebra and Mathematical Abalysis (abbreviated Seminar TALAM) and began being offered in the new format every academic year.

Since in that academic year IST adopted an ill-informed and disastrous curricular change: some students attending the Seminar requested lessons on Complex Analysis.

This led to extra Seminar sessions of 3 hours/week for such lessons, lectured by Manuel Ricou and Luis Magalháes, which have been offered every year in the Spring Semester since 2021/22 following the 8 initial chapters of the book Magalhães, Luis T., Complex Analysis and Dynamics in One Variable with Applications, Springer, Cham, Switzerland, 2025, which before 2025 was made available o the students as lecture notes. Some of the students who attended these lectures, having noticed the content of the other chapters of this book, requested lessons on the Geometric Theory of One Variable Complex Functions. Such lessons began to be offered in extra Seminar sessions of 3 hours/week in the Winter Semester, with basis in chapters 9 to 12 of that book, lectured by Luis Magalhães and Manuel Ricou. The students who attended these lessons further requested lessons on Complex Dynamics, the final chapter of the book, and again this was offered in extra Seminar sessions of 3 hours/week. This was the first time ever that a detailed and deep course on the important subject Complex Dynamics, for which many top mathematicians contributed, with particular incidence since 1985 and including an impressive number of Fields Medalists, was offered in Portugal. This demonstrates the power that voluntary learning opportunities can unleash!

Since 2023, Pedro Gonçalves Henriques also helped in the Seminar supervision and attended the Seminar sessions.

Examples of topics chosen by students participating in this Seminar in the past are:
  1. Values of n for which it is possible to define the cross product of pairs of vectors in Rn and Hurwitz's 1,2,4,8 Theorem
  2. Perron-Frobenius Theorem and Markov Chains
  3. Cardinality
  4. Discrete and continuous linear dynamics (with difference and differential equations)
  5. Cartesian equations of conics, point-body motion under central force and Kepler's laws
  6. Mechanical oscillations
  7. Rigid body motion
  8. Linear codes
  9. Graphs
  10. Derivative of several real variable functions and Optimization of quadratic Functions
  11. Linear superposition in Probability (and probability distributions)
  12. Divergence, Curl, Differential forms and Stokes' Theorem
  13. Hilbert spaces
  14. Tensors
  15. Quaternions
  16. Algebraic Numbers
  17. Linear Algebra in Topology
  18. Distributions
  19. Computational Complexity: Matrix Product
  20. From Cauchy's Rule to the Calculus of Transcendental Functions
  21. Riemann Integrability and Continuity Almost Everywhere
  22. The Dirichlet, Cantor and Volterra Sets
  23. Abel's Theorem for Power Series
  24. Wallis Product and Stirling Approximation for n!
  25. The Van der Waerden function (continuous and non-differentiable at all points)
  26. Axiomatics of the reals, compactness, Heine-Borel theorem, cardinality of the reals
  27. Differentiation and integration of series
  28. Curves length
  29. Lebesgue measure and integral
  30. Fourier analysis and applications
  31. Frames
  32. Maxwell's equations of electromagnetism
  33. Differential geometry of curves and surfaces
  34. Banach-Tarski paradox
  35. Spectral theory of linear operators
  36. Matroids
  37. Exponential of real or complex square matrix, product of exponentials of non-commutative matrices, and Golden-Thompson inequality,

    and examples of other choices offered are:

  38. Lebesgue differentiation theorem
  39. Wavelets
  40. Compressed sensing
  41. Linear control systems
  42. Diffusion, propagation and equilibrium processes
  43. Eigenvalues and cyclic spaces, minimal polynomial and rational canonical form of a matrix
  44. Moore-Penrose pseudoinverse of a matrix
  45. Courant and Wielandt minimax principles
  46. Matrix representation of orthogonal projections
  47. Localization problem of Distance Geometry
  48. One-dimensional elastic equilibrium systems
  49. Isotropic functions by the orthogonal group
  50. Invariant polynomials
  51. Weyl and Schur inequalities and Schatten norms
  52. Eigenvalues and main diagonal of a Hermitian matrix (Schur-Horn Theorem)
  53. Kadison-Singer Problem
  54. Tietäväinen and van Lint theorem characterizing perfect codes
  55. Bounded duration or bounded bandwidth signals.

The set of lecture notes used in this Seminar is:
Luis Magalhães thanks Pedro Gonçalves Henriques and many students who pointed out flaws in his lecture notes over more than one decade when preliminary versions in Portuguese were used by more than one hundred students to prepare individual projects on topics they presented in the Seminar, especially the following students of the IST undergraduate degree programs: