A mathematician who has worked on orthogonal polynomials.

Institutional biography: dlmf.nist.gov/about/bio/THKoornwinder

A mathematician known for work in number theory.

Biography: mathshistory.st-andrews.ac.uk/Biographies/Kummer/

A mathematician whose work includes number theory and elliptic integrals.

Biography: mathshistory.st-andrews.ac.uk/Biographies/Legendre/

A mathematician and mathematics educator who taught at the Community College of Philadelphia.

Academic memorial: faculty.utrgv.edu/eleftherios.gkioulekas/OGS/santos.html

A mathematician whose research includes harmonic analysis, operator theory and functional analysis.^[ref]

Homepage: heil.math.gatech.edu/

A mathematician known for work in group theory, Galois theory and real analysis, including bounded variation.^[ref]

A mathematician who worked on Fourier series and group theory.^[ref]

Al-Khwarizmi wrote on algebra and treated quadratic equations using both algebraic procedures and geometry.

John Joseph O'Connor is a mathematician and a creator of the MacTutor history archive.^[ref]

François Viète worked on algebraic notation and relations between polynomial roots and coefficients.^[ref]

Ilya Piatetski-Shapiro was a mathematician whose research linked analytic number theory, representation theory and algebraic geometry. His early work included the distribution of prime numbers in special sequences and uniqueness questions for trigonometric series. His later work on automorphic forms and representations of Lie groups helped connect arithmetic questions with harmonic analysis.^[ref] These connections are powerful because the same object can carry both arithmetic information and a rich set of symmetries: methods for studying its transformations can then reveal properties of its arithmetic coefficients.

Identity and discipline: en-exact-sciences.tau.ac.il/math/in-memoriam

Israel Gohberg was a mathematician working in functional analysis, operator theory and matrix methods. His research and collaborations linked abstract properties of linear operators with concrete integral equations and matrices. He also founded the journal Integral Equations and Operator Theory, providing a meeting point for these subjects.^[ref] The relationship is useful in applications: an integral equation can encode a boundary-value problem, while a finite-dimensional approximation produces a linear system. Understanding invertible linear maps, the spectrum and perturbations helps explain when such approximations are meaningful and when small errors can have large consequences.

Identity and discipline: en-exact-sciences.tau.ac.il/math/in-memoriam

David Milman was a mathematician who helped develop the geometry of Banach spaces and the use of extreme points in functional analysis. He studied with Mark Krein in Odessa and later worked at Tel Aviv University. Their Krein-Milman theorem describes a compact convex set through the closed convex hull of its extreme points.^[ref] He also proved that a uniformly convex Banach space is a reflexive Banach space. This connection makes a geometric property of the norm control a duality property of the entire space.

Memoir: www.math.tau.ac.il/images/stories/documents/david_milman.pdf

Mark Krein was a mathematician whose research joined functional analysis, operator theory, harmonic analysis and mathematical physics. He worked on the moment problem, integral equations and the spectral theory of linear operators, building an influential research school despite severe institutional discrimination.^[ref] The Krein-Milman theorem, proved with David Milman, gives a geometric expression of this work: a nonempty compact convex set in a locally convex Hausdorff space is the closed convex hull of its extreme points. In finite dimensions, the corresponding picture is a convex polygon reconstructed from its vertices.

Nicholas J. Williams studies representation theory for finite-dimensional associative algebras and its connections to combinatorics. His research connects partially ordered sets and triangulations of convex polytopes with the structure of finite-dimensional algebras, including quivers and homological algebra. This interaction allows combinatorial order and geometric decomposition to encode relationships between algebraic objects.^[ref]

Identity source: nchlswllms.github.io/

Constantin Carathéodory was a mathematician whose work connected measure theory, complex analysis and the calculus of variations. He began his career as an engineer and studied mathematics while working on a dam in Egypt, before entering mathematical research. His later contributions included links between the calculus of variations and partial differential equations, as well as conformal maps and their boundary behavior.^[ref] His Caratheodory measurability criterion is a basic construction principle for measures: it selects those sets that split the outer measure of every test set additively, producing a domain on which countable additivity holds.

Paul Garrett works on automorphic forms, L-functions, representation theory, harmonic analysis and number theory.^[ref] These subjects bring together functions constrained by symmetry, linear actions of groups and analytic descriptions of arithmetic. His research and expository materials explore the interaction of those viewpoints.

Identity source: cse.umn.edu/math/paul-garrett

James Joseph Sylvester was a mathematician whose work with Arthur Cayley helped develop linear algebra and invariant theory. He studied algebraic forms, their relations and quantities that detect special behavior of polynomials; his terminology included the discriminant.^[ref] His Sylvester's law of inertia explains why the numbers of positive and negative squares in a real quadratic form are unchanged by an invertible linear change of variables. Thus matrices that look different can describe the same underlying geometry, and the signs provide invariants that survive a change of coordinates.

Identity source: mathshistory.st-andrews.ac.uk/Biographies/Sylvester/

Claude Brezinski works in numerical analysis, particularly extrapolation, Padé approximation, continued fractions, orthogonal polynomials and numerical linear algebra. His research also treats systems of nonlinear equations. These subjects connect different representations of an approximation with ways to accelerate or stabilize a numerical computation.^[ref]

Identity source: claude-brezinski.fr/

André-Louis Cholesky was a mathematician and military surveyor whose computational work grew out of geodesy. During the revision of French triangulation surveys, he developed the method now known as the Cholesky decomposition.^[ref] For a real symmetric positive-definite matrix $A$, the Cholesky decomposition writes $A=L{L}^{\mathsf{T}}$ with $L$ lower triangular matrix and positive diagonal entries. A linear system can then be solved by two triangular substitutions. The link with surveying is natural: reconciling redundant, imperfect measurements by least squares leads to such systems when the design matrix has full column rank.

Identity source: www.polytechnique.edu/bibliotheque/actualites/cholesky-andre-x1895