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Zhao-Di Liu helped design and implement an optical dephasing simulator, controlling a photon's initial polarization–frequency quantum state to engineer the subsequent loss and revival of quantum coherence.^[ref]

A physicist working on quantum information. Affiliation recorded at publication in 2018: University of Science and Technology of China, CAS Key Laboratory of Quantum Information.^[ref]

John Preskill's quantum computation notes develop the distinction between a quantum state ray, a normalized wave function, and a density operator. This makes a practical difference for a subsystem of an entangled state: its predictions are described by a reduced density operator obtained with the partial trace, even when the joint quantum state is pure. His discussion also shows why different statistical preparations can give the same density operator and therefore identical predictions for measurements confined to that subsystem.^[ref]

A theoretical physicist working on quantum information theory.^[ref]

Homepage: www.preskill.caltech.edu/

A mathematician working in computability theory, and the author of Discrete Mathematics: An Open Introduction.^[ref]

Ilse Ipsen works in numerical linear algebra, randomized algorithms, the theory of matrices and scientific computing. Her research also connects numerical methods with nonlinear problems, parameter estimation and statistics. These topics concern both efficient computation and how approximation errors or perturbations affect the reliability of a calculated result.^[ref]

C. T. Kelley studies numerical analysis, including iterative methods for nonlinear equations, mathematical optimization and linear systems. His research connects solver design with numerical stability and computational cost, including methods that perform different stages at different arithmetic precisions.^[ref]

Homepage: ctkelley.github.io/

K. I. Dickson is credited as a coauthor of Condition Estimates for Pseudo-Arclength Continuation by K. I. Dickson and others. The paper studies the linear systems arising when a solution branch is followed through a fold using pseudo-arclength continuation.

The mathematical contribution concerns conditioning: adding a continuation constraint can keep the augmented problem regular even when a fixed-parameter formulation is singular. The initials are retained from the verified paper; no full name or personal biography is inferred.^[ref]

Gerald Teschl is a mathematician working on spectral theory and inverse spectral theory for differential and difference operators, together with their connections to integrable systems. His books treat Schrödinger operators, ordinary differential equations and discrete models such as the Toda lattice.^[ref] A direct spectral problem asks for the spectrum of an operator specified by its coefficients. An inverse spectral problem asks how the coefficients can be reconstructed from spectral data. This turns observed frequencies or quantum energy data into information about the underlying mathematical model, subject to the uniqueness conditions of the particular problem.

Claude Berge was a mathematician whose work helped establish modern graph theory, with connections to combinatorics, game theory and optimization. His books gave common mathematical language to networks arising in different applications. He also formulated the strong perfect graph conjecture, a major stimulus to research on perfect graphs.^[ref] A perfect graph has equal chromatic number and clique number in every induced subgraph. The condition expresses when the obvious lower bound for graph coloring, supplied by mutually adjacent vertices, is always attainable; requiring it in every induced subgraph is essential.

George Pólya was a mathematician whose research spanned probability theory, analysis, combinatorics and geometry. He also became an influential writer about mathematical discovery, emphasizing the search for analogies, simpler cases and useful auxiliary constructions.^[ref] His work on random walks gives a striking example of dimension changing a qualitative answer: a simple symmetric random walk on the integer lattice returns to its starting point with probability one in dimensions one and two, while in dimension three that probability is less than one. The distinction connects local random steps with the distinction between recurrent graphs and transient graphs.

John Edensor Littlewood was a mathematician whose work with Godfrey Harold Hardy connected analysis, inequalities, series and the Riemann zeta function. Their collaboration helped develop the Hardy-Littlewood circle method for arithmetic counting problems. He also worked with Mary Cartwright on nonlinear ordinary differential equations motivated by radio engineering, including the Van der Pol equation.^[ref] Their investigations of strongly driven oscillators helped reveal how a deterministic dynamical system can have complicated long-term behavior. This makes his work a bridge between the estimates of classical analysis and the geometric study of dynamical systems.

Godfrey Harold Hardy was a mathematician whose work joined analysis with number theory, including Fourier series, the Riemann zeta function and the distribution of prime numbers. His long collaboration with John Edensor Littlewood and his work with Srinivasa Ramanujan produced influential methods and results in analytic number theory.^[ref] The Hardy-Littlewood circle method exemplifies this connection: it studies arithmetic counting problems through an integral of products of exponential sums, separating regions near rational points from the remaining contribution. His teaching also helped establish rigorous accounts of limits, functions and convergence in English undergraduate analysis.

David E. Joyce is a mathematician whose mathematical writing and online resources cover Euclidean geometry, algebra, complex numbers and dynamical systems. His annotated presentation of Euclid's Elements makes the structure of the ancient text accessible through explanations and geometric diagrams. His other resources explore symmetry, knot theory, hyperbolic geometry and pictures of Julia sets and Mandelbrot sets.^[ref] These topics share a visual route into abstract mathematics: a construction or a computer-generated picture suggests relationships, while definitions and proofs determine which of those relationships hold generally.

Homepage: mathcs.clarku.edu/~djoyce/

Herbert Robbins was a mathematician and statistician whose research ranged from topology to Empirical Bayes method methods and stochastic approximation. With Sutton Monro, he introduced an iterative approach to finding a zero of a function when observations of its values contain random error. Instead of requiring exact function evaluations, successive corrections use noisy measurements with diminishing step sizes. This idea became fundamental to stochastic optimization.^[ref] He also coauthored an influential mathematical exposition with Richard Courant, emphasizing explanations and examples that reveal why a proof works rather than a collection of calculation rules.

Nimrod Megiddo is a mathematician known for research on optimization and the computational complexity theory of algorithms. His fixed-dimensional linear programming algorithm solves a problem with $n$ constraints in $O(n)$ time when the dimension $d$ is fixed. The dependence on $d$ is absorbed into the constant, so this statement does not give a uniformly linear algorithm when the number of variables also grows.^[ref] This result illustrates a useful distinction in computational complexity theory: a geometric optimization problem can become much easier when its ambient dimension is a fixed parameter.

Homepage: theory.stanford.edu/~megiddo/

Refael Hassin is a mathematician working on combinatorial optimization and strategic behavior in queueing theory. His research asks how individual decisions change the operation of a service system: customers may decide whether to join a queue, while its operator chooses prices or what information to reveal.^[ref] One of his studies considers a service rate or quality known to the operator but uncertain to customers. Revealing that information can change customer decisions and the operator's expected profit, so greater information does not automatically have the same consequences for every participant.^[ref]

Homepage: english.tau.ac.il/profile/hassin

Marin Mersenne was a mathematician and scientific correspondent whose network helped Pierre de Fermat, René Descartes and other researchers exchange questions and results. He studied number theory and the acoustics of vibrations of mechanical strings.^[ref] A Mersenne number has the form $2^p-1$; for it to be a Mersenne prime, $p$ must be a prime number, although that condition is not sufficient: $2^{11}-1=23\cdot 89$. For an ideal stretched mechanical string, the fundamental frequency is $f=(2L{)}^{-1}\sqrt{T/\mu }$, relating the audible pitch to the length $L$, tension $T$ and mass per unit length $\mu$.

Euclid was a mathematician associated with Alexandria around 300 BCE. His Euclid's Elements organized geometry and number theory into definitions, assumptions and chains of proofs, drawing together earlier results as well as material attributed to him. Very little reliable biographical information survives, so the familiar portraits are later imagined likenesses.^[ref] The lasting attraction of the Euclid's Elements is its deductive structure: a construction or mathematical theorem becomes reusable in subsequent proofs, rather than remaining an isolated calculation. Its Euclidean algorithm and proof that there are infinitely many prime numbers remain basic tools of number theory.

Peter G. Doyle is a mathematician whose work connects geometry, combinatorics and probability theory. His book with J. Laurie Snell explains how a random walk on a graph can be studied through an electrical network: a probability of reaching one boundary before another satisfies the same averaging equations as an electrical potential.^[ref] His work on isospectral manifolds also investigates how much geometry can be recovered from the spectrum of a differential operator: equal vibration frequencies need not determine the shape.^[ref]

Homepage: faculty-directory.dartmouth.edu/peter-doyle

J. Laurie Snell worked on Markov chains and their applications and wrote Introduction to Probability by Charles M. Grinstead and J. Laurie Snell with Charles M. Grinstead. His autobiographical research statement also describes work on finite mathematics and teaching people to assess probability and statistical claims in the news.^[ref] A Markov chain replaces the full history by a specified present-state transition law; first-step analysis then turns hitting probabilities and expected hitting times into linear equations. That link between a stochastic model and solvable equations is central to the subject's practical usefulness.

Homepage: chance.dartmouth.edu/Laurie.x.html

Charles M. Grinstead coauthored Introduction to Probability by Charles M. Grinstead and J. Laurie Snell with J. Laurie Snell. The institutional publication record identifies the1997 second revised edition and its authors.^[ref] The text develops probability from concrete models through tools such as conditional probability, expectation and Markov chains. For example, counting length-$n$ sequences of independent fair coin tosses gives probability$2^{-n}\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ for exactly$k$ heads, connecting discrete counting with a probability model rather than treating either in isolation.

Homepage: works.swarthmore.edu/fac-math-stat/153/