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Johann Andreas von Segner trained in medicine and later worked in mathematics, physics and astronomy. His institutional biography emphasizes fluid mechanics and his development of a reaction water wheel, an early hydraulic turbine.^[ref] The apparatus turns a redirected water flow into torque: the exiting fluid carries angular momentum, and its change is balanced by a reaction on the rotor. This makes the device an example of how a conservation law becomes a mechanical design principle.

Homepage: www.catalogus-professorum-halensis.de/segner-johann-andreas.html

Eugène Charles Catalan studied counting problems including triangulations of a polygon, leading to the sequence now called the Catalan numbers.^[ref] The formula $C_n=(n+1{)}^{-1}(\genfrac{}{}{0ex}{}{2n}{n})$ counts several apparently different families: triangulations of a convex $(n+2)$-gon, balanced parenthesizations and plane binary-tree structures. The Catalan recurrence expresses how selecting a root operation divides such an object into two smaller objects. This illustrates why a historical counting problem can become a reusable language for combinatorics.

Homepage: 200.ulg.ac.be/catalan-bio.html

Igor Pak works in discrete mathematics, including enumerative combinatorics, algebraic combinatorics, polyhedral geometry, and probability and computation on groups. His research page presents these as interconnected subjects and lists lectures on convex polytopes and combinatorial inequalities.^[ref] A common theme is to explain a numerical counting identity through the structure of the objects being counted, such as a bijection, rather than relying only on algebraic manipulation. Polyhedral methods add geometric constraints to those discrete enumeration questions.

Homepage: www.math.ucla.edu/~pak/

Willem Mantel posed the1907 question now associated with the triangle-free extremal bound: a triangle-free graph with $n$ vertices has at most $\lfloor n^2/4\rfloor$ edges. A balanced complete bipartite graph attains the bound. The historical record distinguishes the proposer from the published solver, Willem Abraham Wythoff. Mantel also taught calculus to prospective engineers in Delft.^[ref] The result is a model of extremal graph theory: forbidding a small configuration imposes a global limit on edge density.

Homepage: www.nieuwarchief.nl/serie5/pdf/naw5-2016-17-4-275.pdf

Pál Turán worked in number theory and combinatorics. His1934 proof of the typical size of the number of prime divisors became an important precursor of probabilistic number theory: averaging a counting function can reveal the behaviour of almost all integers without computing it separately for each integer.^[ref] In extremal graph theory, the theorem carrying his name asks how many edges force a complete subgraph. The triangle case gives the sharp threshold$\lfloor n^2/4\rfloor$, attained by a balanced complete bipartite graph.

Homepage: mathshistory.st-andrews.ac.uk/Biographies/Turan/

Robbert Fokkink studies game theory, dynamical systems, ergodic theory and topology. His research statement highlights search games, symbolic sequences and the topology of inverse limits and tiling spaces.^[ref] A search-game model makes uncertainty strategic: the best search plan depends on where a target can be hidden and what information the searcher receives. His historical research on Willem Abraham Wythoff also connects the provenance of early extremal graph theory with the study of mathematical games.^[ref]

Homepage: diamhomes.ewi.tudelft.nl/~rfokkink/

Jan-Hendrik Evertse works in number theory, particularly Diophantine equations. His books study the unit equation over a number field, equations involving discriminants, and effective methods for Diophantine equations over finitely generated integral domains. The effective viewpoint asks for bounds and procedures that make solution sets accessible to explicit computation, beyond an abstract existence or finiteness statement.^[ref]

Homepage: pub.math.leidenuniv.nl/~evertsejh/

A German mathematician who made major contributions to elliptic functions, number theory and Hamiltonian mechanics. His 1829 work on elliptic functions developed a theory using theta functions, during the same period in which Niels Henrik Abel independently advanced the subject.

He also investigated first-order partial differential equations arising in dynamics and wrote extensively on determinants. The Jacobian determinant bears his name: it measures local volume scaling for a differentiable change of variables. This connects his work to both coordinate transformations in multivariable calculus and the study of dynamical equations. His Königsberg teaching helped establish research seminars as a way to train mathematicians. ^[ref]

Homepage: mathshistory.st-andrews.ac.uk/Biographies/Jacobi/

André Jürgen Massing works on numerical methods for partial differential equations, including finite element methods for fluid mechanics and problems with moving boundaries. His publications study both computational formulations and error estimates, connecting approximation on a mesh with the reliability of the simulated physical model.^[ref]

Homepage: www.ntnu.no/ansatte/andre.massing

John C. Butcher works on numerical methods for ordinary differential equations, particularly Runge-Kutta methods and general linear methods. His research connects the algebraic structure of a numerical method with its approximation and numerical stability properties. The Butcher tableau records the stages and weights of a Runge-Kutta method compactly.^[ref]

Homepage: www.math.auckland.ac.nz/~butcher/biography.html

A German mathematician who studied linear differential equations and later held a chair in technical mechanics at Karlsruhe. His work on second-order equations with four singular points led to the Heun equation, extending the family associated with the hypergeometric differential equation.

His name is also attached to Heun's method, a predictor-corrector Runge-Kutta method for numerical integration. The predictor estimates an endpoint slope and the correction averages it with the initial slope. The special-function equation and the numerical method are distinct contributions linked by the study of differential equations, rather than two names for the same construction. ^[ref]

Homepage: mathshistory.st-andrews.ac.uk/Biographies/Heun/

A German mathematician whose work combined numerical analysis with spectroscopy. He studied numerical solution of equations and the measured wavelengths of elemental spectral lines, rather than treating mathematical computation as separate from experimental questions.

His name is attached to the Runge-Kutta method for approximating ordinary differential equations, alongside Martin Kutta. Such methods replace a continuous evolution by a succession of computed time steps. His work on spectral regularities similarly sought structure in numerical measurements. These two interests make him an example of the exchange between practical calculation and mathematical models of physical observations. ^[ref]

Homepage: mathshistory.st-andrews.ac.uk/Biographies/Runge/

Rodolfo Ruben Rosales works in applied mathematics, including nonlinear waves, fluid mechanics, materials science and numerical partial differential equations. His research connects differential equations with physical wave propagation and the numerical methods for approximating those dynamics.^[ref]

Homepage: math.mit.edu/directory/profile.html?pid=228

Steven G. Krantz works in complex analysis, including several complex variables and their connections to harmonic analysis. His research includes boundary behavior and estimates for holomorphic functions, integral operators and the geometry of complex domains. His books develop these analytic and geometric viewpoints as well as introductory real analysis.^[ref]

Homepage: www.math.wustl.edu/~sk/

Daniel Dadush studies mathematical optimization, including linear programming, integer programming, geometry of numbers and discrepancy theory. His work connects the geometry of high-dimensional convex sets with algorithms for mathematical optimization.^[ref]

Homepage: homepages.cwi.nl/~dadush/

Albert W. Tucker worked in topology, game theory and nonlinear optimization. With Harold W. Kuhn he helped formulate the conditions now called the Karush-Kuhn-Tucker conditions, which extend the use of Lagrange multipliers to inequality constraints. He also supplied the familiar name and narrative for the prisoner's dilemma.^[ref] That example makes a mathematical distinction memorable: individually advantageous choices can lead to an outcome both players prefer less than a coordinated alternative. The payoff inequalities, rather than the story alone, define the strategic conflict.

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

Harold W. Kuhn connected game theory with continuous and discrete optimization. His work with Albert W. Tucker supplied the1951 nonlinear-programming formulation behind the Karush-Kuhn-Tucker conditions, while his1955 Hungarian algorithm addressed the assignment problem. He also developed extensive-form games and the relation between mixed strategies and behavioral strategies.^[ref] These contributions distinguish three mathematical questions: local constrained optimality, a discrete cost-minimizing assignment, and which randomized decisions can represent a player's information-dependent plan.

Biography: www.princeton.edu/news/2014/07/05/harold-kuhn-princeton-mathematician-who-advanced-game-theory-dies-88

William Karush obtained necessary conditions for inequality-constrained optimization in his1939 master's thesis. The result was independently rediscovered and published by Harold W. Kuhn and Albert W. Tucker, producing the name Karush-Kuhn-Tucker conditions.^[ref] Their distinguishing idea is that an inequality constraint can be active or inactive: a nonnegative Lagrange multiplier pairs with its constraint through complementary slackness. A constraint qualification is needed for the usual necessary-condition theorem; a stationary Lagrangian equation alone is insufficient.

Biography: www.math.u-szeged.hu/~hajnal/courses/MSc_Optimalizalas/2014/Cottle%20-%20William%20Karush%20and%20the%20KKT%20theorem.pdf

A French mathematician who studied linear differential equations, particularly equations with periodic coefficients. His 1883 memoir developed the structure now called Floquet theory: periodic changes can be separated from exponential evolution in a fundamental solution matrix. This reduces a periodically driven linear problem to information accumulated over one period.^[ref]

The Floquet multipliers then determine whether repeated periods amplify or suppress perturbations. The idea is useful in the stability analysis of periodic motions, where a coefficient matrix need not be constant for its long-term growth to be tractable. Floquet's teaching and research career was closely associated with Nancy. ^[ref]

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

Marco Gualtieri works in differential geometry and mathematical physics, with a focus on generalized geometry. His own research page identifies that focus rather than presenting him only as an instructor.^[ref] The geometric viewpoint concerns structures carried by a smooth manifold, where tangent directions and covectors can be studied together. It connects the study of complex geometry and symplectic geometry with mathematical descriptions of physical fields.

Homepage: www.math.toronto.edu/mgualt/

Instructor for the 2012 Topology I: smooth manifolds (MAT1300HF) course; the course information explicitly identifies his role as instructor.