Svein Linge coauthored a text on finite difference methods with Hans Petter Langtangen. The book links mathematical discretization with implementations for partial differential equations, including diffusion equations. Deriving an update formula alongside its stability conditions explains why a physically smoothing equation can nevertheless produce an unstable numerical calculation if the time step is too large.^[ref]

Hans Petter Langtangen coauthored a text on finite difference methods for partial differential equations with Svein Linge. Its treatment of diffusion equations derives discrete updates and investigates their stability. The connection between an explicit Euler method and a spatial second-difference operator makes the restriction on the time step visible in a reproducible computational model.^[ref]

Carl de Boor authored notes on approximation theory developing best approximation, B-splines and spline interpolation. The notes connect concrete recurrence formulas with structural criteria such as the Schoenberg–Whitney theorem. This makes it possible to analyze when a spline representation exists and is unique, rather than relying only on numerical fitting.^[ref]

Vahid Shirbisheh authored lectures on C-star algebras connecting Banach algebras, spectral theory and functional calculus. The notes develop the Gelfand transform and the correspondence between commutative C-star algebras and appropriate topological spaces. This explains how an abstract algebra can encode continuous functions and recover geometric information from its multiplicative structure.^[ref]

Gerald Beresford Whitham developed an averaged variational principle for slowly varying dispersive waves. The method averages a Lagrangian over a rapid wave cycle and derives equations for the slower evolution of the wave parameters. It supplies a common framework for linear and nonlinear wave modulation instead of treating every wave train as a separate perturbation calculation.^[ref]

Germund Dahlquist established fundamental stability restrictions for linear multistep methods for ordinary differential equations. His 1963 paper analyzes A-stability and the associated order barrier, while identifying the special role of the trapezoidal rule. The result explains why increasing formal accuracy alone cannot guarantee stable integration of rapidly decaying differential equations.^[ref]

Guillaume Bal studied reconstruction from the attenuated Radon transform with full and partial measurements. His analysis distinguishes complete-data inversion from partial-angle reconstruction requiring additional conditions. The work uses a Riemann-Hilbert problem and operator decompositions, showing how incomplete ray measurements can still determine a source when attenuation and admissible functions are suitably controlled.^[ref]

C. Sidney Burrus authored notes on the Fast Fourier transform. The treatment includes the prime-factor algorithm, which exploits coprime transform lengths to reorganize a discrete Fourier transform. A suitable index map separates the calculation into smaller transforms without the interstage twiddle multiplications required by the usual Cooley-Tukey arrangement.^[ref]

Sinai Robins coauthored a text on integer-point enumeration with Matthias Beck. Their exposition develops Ehrhart polynomials, generating functions and Ehrhart reciprocity for polyhedra. The subject gives a precise bridge between discrete counting and continuous geometry: the leading growth of lattice-point counts records volume.^[ref]

Matthias Beck coauthored a text on integer-point enumeration in polyhedra with Sinai Robins. It connects generating functions, Ehrhart polynomials and Ehrhart reciprocity. Counting lattice points in dilated shapes thereby reveals geometric information such as volume, while reciprocity relates counts in the interior to counts in the closed shape.^[ref]

D. Hugh Redelmeier developed an enumeration algorithm for polyominoes. It grows connected collections of lattice squares while carrying a controlled set of candidate additions and inherited exclusions. This avoids reaching the same fixed polyomino through multiple growth histories, allowing exact counting without storing every smaller shape as a separate comparison object.^[ref]

Scott A. Vanstone coauthored the Handbook of Applied Cryptography by Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone. The book develops the mathematics and algorithms underlying cryptography, including modular square roots and number-theoretic computational problems. It makes explicit the distinction between an algorithm's mathematical correctness and the hardness assumptions used by a cryptographic system. ^[ref]

Alfred J. Menezes coauthored the Handbook of Applied Cryptography by Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone. Its number-theoretic algorithms include computing modular square roots and implementing public-key cryptography. The presentation distinguishes computational problems, algorithmic procedures and correctness conditions, which is essential when a plausible-looking arithmetic operation is not by itself a secure cryptographic construction. ^[ref]

Alin Bostan developed a fast algorithm with Ryuhei Mori for computing a distant term of a linear recurrence. The method extracts coefficients of a rational generating function by repeatedly reducing the target index through parity. Its arithmetic cost grows logarithmically with that index, avoiding sequential computation of all preceding terms.^[ref]

Hal Schenck coauthored a text on toric varieties with David A. Cox and John B. Little. It develops the bridge between the geometry of polyhedra and algebraic geometry, including the construction of varieties from a fan in toric geometry and the study of their divisors and line bundles. Combinatorial data thereby become tools for geometric calculations.^[ref]

John B. Little coauthored a text on toric varieties with David A. Cox and Hal Schenck. Its organizing idea is that cones, a fan in toric geometry and lattice data describe substantial parts of the geometry. This provides explicit examples and calculations for algebraic varieties that can otherwise be difficult to visualize or compute with.^[ref]

David A. Cox coauthored a systematic text on toric varieties with John B. Little and Hal Schenck. The subject connects algebraic geometry with the geometry of polyhedra: convex cones and a fan in toric geometry encode affine pieces and their gluings. The book develops this combinatorial description alongside divisors, homogeneous coordinates and line bundles.^[ref]

Rémi Carles studied critical nonlinear Schrödinger equations with and without a harmonic trapping potential. His work gives a lens transform relating the two evolutions in the mass-critical case. This permits information about existence and concentration in one equation to be transferred to the other, while keeping track of the transformed time interval.^[ref]

Elliott H. Lieb proved sharp forms of the Hardy–Littlewood–Sobolev inequality and related inequalities. The work determines optimal constants in bounds for singular integral expressions. Sharp constants distinguish an estimate that merely controls an integral from one that identifies its exact limiting strength, with consequences for analysis and mathematical physics.^[ref]

Sonja Žunar presented joint work on topological aspects of smooth automorphic forms. Her 2022 slides explain the use of locally convex spaces and the strict inductive limit topology in organizing spaces of such functions. This supplies a topology compatible with an increasing sequence of function spaces, rather than treating automorphic forms only as an unstructured collection of functions.^[ref]