"Moonshine is profoundly connected with physics (namely conformal field theory and string theory). String theory proposes that the elementary particles (electrons, photons, quarks, etc.) are vibrational modes on a string of length about 10^−33 cm. These strings can interact only by splitting apart or joining together – as they evolve through time, these (classical) strings will trace out a surface called the world-sheet. Quantum field theory tells us that the quantum quantities of interest (amplitudes) can be perturbatively computed as weighted averages taken over spaces of these world-sheets. Conformally equivalent world-sheets should be identified, so we are led to interpret amplitudes as certain integrals over moduli spaces of surfaces. This approach to string theory leads to a conformally invariant quantum field theory on two-dimensional space-time, called conformal field theory (CFT). The various modular forms and functions arising in Moonshine appear as integrands in some of these genus-1 (‘1-loop’) amplitudes: hence their modularity is manifest." (Terry Gannon, "Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics", 2006)
"Although one speaks nowadays of the determinant of a matrix, the two concepts had different origins. In particular, determinants appeared before matrices, and the early stages in their history were closely tied to linear equations. Subsequent problems that gave rise to new uses of determinants included elimination theory (finding conditions under which two polynomials have a common root), transformation of coordinates to simplify algebraic expressions (e.g., quadratic forms), change of variables in multiple integrals, solution of systems of differential equations, and celestial mechanics." (Israel Kleiner, "A History of Abstract Algebra", 2007)
"Linear algebra is a very useful subject, and its basic concepts arose and were used in different areas of mathematics and its applications. It is therefore not surprising that the subject had its roots in such diverse fields as number theory (both elementary and algebraic), geometry, abstract algebra (groups, rings, fields, Galois theory), anal ysis (differential equations, integral equations, and functional analysis), and physics. Among the elementary concepts of linear algebra are linear equations, matrices, determinants, linear transformations, linear independence, dimension, bilinear forms, quadratic forms, and vector spaces. Since these concepts are closely interconnected, several usually appear in a given context (e.g., linear equations and matrices) and it is often impossible to disengage them." (Israel Kleiner, "A History of Abstract Algebra", 2007)
"Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values" (the bifurcation parameters) of a system causes a sudden"qualitative" or topological change in its behavior. Bifurcations can occur in both continuous systems" (described by ODEs, DDEs, or PDEs) and discrete systems" (described by maps)." (Tianshou Zhou, "Bifurcation", 2013)
"The integrands of the integrals on a differential manifold (on its curves, surfaces, etc.; all of them oriented) constitute a structure called an algebra. To be more specific, it is a graded algebra. [...] The subspaces constituted by all the elements of a graded algebra thathave the same grade constitute modules (generalization of the concept of vector space)." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)
"The important point about e is that an exponential function with this base grows at a rate precisely equal to the function itself. Let me say that again. The rate of growth of ex is ex itself. This marvelous property simplifies all calculations about exponential functions when they are expressed in base e. No other base enjoys this simplicity. Whether we are working with derivatives, integrals, differential equations, or any of the other tools of calculus, exponential functions expressed in base e are always the cleanest, most elegant, and most beautiful." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)
"Thus, calculus proceeds in two phases: cutting and rebuilding. In mathematical terms, the cutting process always involves infinitely fine subtraction, which is used to quantify the differences between the parts. Accordingly, this half of the subject is called differential calculus. The reassembly process always involves infinite addition, which integrates the parts back into the original whole. This half of the subject is called integral calculus." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)
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