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Alexander Friedmann derived evolving homogeneous cosmological solutions of the Einstein field equations. The Friedmann equations relate cosmic expansion to energy density, pressure, spatial curvature and the cosmological constant.

The conceptual change is that general relativity permits the scale of a homogeneous universe to evolve, rather than requiring a static geometry. Observational evidence and the choice of matter content are additional inputs needed to identify which mathematical solution describes the Universe.^[ref]

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

Ana Cannas da Silva authored an exposition of symplectic geometry. A nondegenerate closed two-form organizes phase-space geometry and Hamiltonian motion, giving a geometric framework shared by mechanics and global mathematical questions.^[ref]

The Bay of Fundy is a coastal basin in eastern Canada notable for large tidal ranges. Its geometry and the dynamical response of the connected bay system strongly amplify the incoming semidiurnal tide.

The important physical idea is forced motion of a basin: water depth, shape and friction determine its natural response, while the external astronomical tide supplies the drive. Large range is not a direct local measurement of an unusually strong gravitational field. Range also varies with location and the spring-neap cycle, so one quoted maximum does not characterize every observation.

Brian Harvey developed the Berkeley Logo implementation with student collaborators and used programming as a medium for exploring ideas. His project page identifies the interpreter and its collaborators; his teaching homepage describes work in introductory computing and pre-college education.^[ref][ref]

The command repeat 4 [forward 100 right 90] draws a square by repeating one movement-and-turn rule. Its compact structure connects computer programming, geometry and a visible result, while changing the number of repetitions and angle provides an immediate experiment with the procedure. The language also supports symbolic data and recursion, so its use goes beyond drawing pictures.^[ref]

Carl Friedrich Gauss (1777–1855) connected number theory, geometry, astronomy and measurement. His work on quadratic reciprocity and the straightedge-and-compass construction of the regular 17-gon made concrete arithmetic structure control a geometric construction. His prediction of the position of Ceres from a short observed orbital arc helped recover the object in 1801, showing why orbit determination and least squares belonged together.^[ref]

The least squares viewpoint illustrates the bridge: observations rarely satisfy every model equation exactly, so one selects parameters minimizing a sum of squared discrepancies. In differential geometry, Gaussian curvature instead detects an intrinsic property of a surface. These contributions link discrete arithmetic questions, continuous geometry and inference from imperfect observations.

The Casimir effect is a force associated with the change in electromagnetic field fluctuations produced by material boundaries. For ideal perfectly conducting parallel plates separated by $a$ at zero temperature, the attractive pressure isReal measurements require finite-conductivity, temperature and geometric corrections. A classic experiment used a torsion-pendulum electromechanical system to measure the force between nearby conducting surfaces. The experimental comparison concerns the force difference produced by a specified geometry; it does not make an arbitrary absolute vacuum-energy offset observable.^[ref]

For $z=r{e}^{i\theta }\ne 0$, the complex logarithm consists of all values $\log r+i(\theta +2\pi k)$, $k\in \mathbb{Z}$. They give the same exponential because $e^{2\pi ik}=1$. In particular, the logarithms of one are $2\pi ik$, not just zero.

Choosing a single-valued branch of the complex logarithm requires a domain and an argument convention. A holomorphic branch has derivative $1/z$, but no such branch exists on all of $\mathbb{C}\setminus \{0\}$: integrating that derivative once around the origin gives $2\pi i$, whereas a globally defined primitive would give zero. This is why branch geometry matters in contour calculations.

Constantin Teleman authored course notes on Riemann surfaces. The notes connect local holomorphic coordinates with the global topology and geometry of complex curves.^[ref]

A cup anemometer infers wind speed from the rotation rate of cups mounted around a vertical axis. The cup geometry produces different aerodynamic forces on the two sides of the rotor, giving a net driving torque.

A calibration relates rotation frequency to flow speed over the useful range. Bearing friction affects low speeds, and rotor inertia limits response to rapid fluctuations. The cup speed is not simply equal to wind speed. Comparing the output with a reference flow establishes the transfer relation needed for quantitative observations.

Coauthor with Joseph Rabinoff of an interactive linear algebra textbook. The text connects computations with the geometry of linear systems, eigenvectors and orthogonal projection.^[ref]

Dan Romik works in number theory, combinatorics and probability theory.^[ref] His Complex Analysis Lecture Notes by Dan Romik also develops holomorphic functions, contour integrals and the geometry of the Riemann sphere, providing an expository connection between complex analysis and geometry.

Identity source: www.math.ucdavis.edu/people/general-profile?fac_id=romik

Edward Witten works on theoretical physics and its connections with geometry and topology. His Institute for Advanced Study profile identifies the researcher.^[ref]

His Monopoles and Four-Manifolds by Edward Witten used ideas from supersymmetric gauge theory to introduce a new description of smooth four-manifold invariants. The Seiberg–Witten equations replace a nonabelian instanton-counting problem with equations involving an Abelian gauge field and a spinor. The contribution demonstrates how a physical dual description can reveal a more tractable mathematical structure.^[ref]

Emission is the production of electromagnetic radiation by a physical system. Mechanisms include accelerated charges, transitions between quantum states and thermal fluctuations.

For an isolated atomic transition, a photon carries energy $h\nu =E_i-E_f$. In thermal emission, a macroscopic distribution of microscopic motions and states produces a spectrum controlled by temperature and material response. Emission must be distinguished from scattering of incident radiation, though a measured signal can contain both contributions. The process and observation geometry determine the relevant spectrum and angular distribution.

Emmanuel Breuillard works on group theory and its connections with geometry and additive structure. With Ben Green and Terence Tao, he proved the structure theorem for approximate groups.^[ref]

An approximate group is a finite symmetric set whose products can be covered by a bounded number of translates of itself. The theorem finds controlled nilpotent structure behind that bounded growth. It lets a local combinatorial restriction on products constrain the larger algebraic organization of the set.^[ref]

An endoscope is an instrument that brings illumination and imaging into a region that is difficult to view directly. Optical designs can use a relay-lens train, an ordered fibre bundle or a camera sensor near the tip.

An ordered image bundle must preserve relative fibre positions to transmit an image; a randomly arranged illumination bundle need not. Resolution, viewing angle, illumination and contrast depend on the complete optical system. The geometry that makes internal access possible also constrains aperture and image quality.

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.

Fabian Haiden studies connections among geometry, dynamical systems and category theory. His course notes Dynamics, Stability, and Noncommutative Algebra by Fabian Haiden develop topological entropy using covers and separated or spanning sets of trajectories, making orbit complexity into a quantitative invariant.

His research page connects these themes with stability conditions and categorical constructions. This illustrates how different mathematical descriptions can carry information about the same dynamics.^[ref]

The first overtone is the lowest oscillatory normal mode above the fundamental mode. It is an ordering by frequency, not a claim that its frequency is an exact integer multiple of the fundamental. If the fundamental is labeled $n=0$, the first overtone is $n=1$; another common convention labels them one and two.

For a string fixed at both ends, the frequencies are $f_n=nc/(2L)$ with $n\ge 1$, so the first overtone is the second harmonic at twice the fundamental. For an ideal pipe closed at one end, the next allowed mode is at three times the fundamental. Geometry and boundary conditions determine the relation.

The design and construction of defensive works. Surveying, geometry and practical mechanics connected this work with early modern mathematical expertise.