Differential Equations – Migoro Media

Differential Equations: The Math Behind Motion, Change, and Chaos

Discover how equations that describe change shape the world—from physics and biology to finance and engineering

A differential equation is a mathematical equation that relates a function with its derivatives. In simple terms, it describes how a quantity changes in relation to another—most often how something evolves over time or space. These equations are the language of dynamic systems: they model motion, growth, decay, oscillations, and countless other processes that involve continuous change.

Differential equations are essential across nearly every scientific and engineering discipline.

They form the backbone of physics, where they describe how forces produce motion; in biology, where they model population dynamics and the spread of disease; in economics, where they track fluctuating markets; and in engineering, where they govern the behavior of circuits, fluids, structures, and more. Their ability to capture real-world change makes them one of the most powerful tools in the mathematical toolkit.

The concept of differential equations, however, is relatively young in the scope of human history. While ancient thinkers pondered change and motion philosophically and geometrically, the formal mathematical theory of differential equations only emerged in the 17th century with the invention of calculus. Since then, they have evolved into a vast and rich field—interwoven with mathematics, physics, technology, and the modern world.

🏛️ Pre-Calculus Foundations (Antiquity – 1600)

Key Focus: Conceptual groundwork before formal differential equations existed

Before differential equations emerged as a formal mathematical concept, many civilizations laid essential theoretical foundations. Ideas of motion, change, and continuous variation were explored long before calculus gave them precise mathematical expression. Though these early efforts lacked the tools of modern analysis, they formed the conceptual bridge between observation and theory.

Ancient Greek Mathematics

The Greeks were pioneers in using geometry to understand the natural world. While they did not develop algebra or calculus, their philosophical and mathematical inquiries into motion and change foreshadowed the ideas that would later be captured by differential equations.

Zeno’s Paradoxes (5th century BCE) raised profound questions about motion and infinity. In his paradoxes, Zeno argued that to move from one point to another, an infinite number of smaller distances must be crossed—posing a challenge to the very notion of continuous motion.
Archimedes (c. 287 BCE – c. 212 BCE) used an early form of infinitesimal reasoning in his method of exhaustion to calculate areas and volumes. Though not formal calculus, this approach resembles integration and laid critical groundwork for later developments.
Greek mathematics was rooted in static geometry, but their concerns with continuity, limits, and the infinite would deeply influence future mathematical thinking.

Medieval and Islamic Mathematics

Between the 9th and 15th centuries, Islamic mathematicians preserved and extended Greek mathematical knowledge, making notable advances in optics, mechanics, and algebra that subtly anticipated calculus and differential reasoning.

Ibn al-Haytham (Alhazen, 965–1040 CE), often regarded as the “father of optics,” investigated the laws of reflection and refraction using mathematical principles. His work on motion and light propagation required careful reasoning about changing quantities—an implicit forerunner to DE models.
Mathematicians like Omar Khayyam and Sharaf al-Din al-Tusi studied cubic equations and geometric curves with a level of abstraction that edged closer to functional relationships and change.
Though symbolic algebra had not yet emerged, scholars employed geometric arguments with infinitesimal quantities, helping form the intellectual bridge between the ancient world and the calculus revolution.

Renaissance Thinkers

As Europe emerged from the Middle Ages, new attention to science, measurement, and mechanics revived interest in motion and mathematical abstraction.

Thinkers such as Niccolò Tartaglia and Girolamo Cardano explored projectile motion and early physical dynamics, laying the groundwork for mechanics.
The invention of symbolic algebra by François Viète (1540–1603) was transformative. His use of letters to represent quantities allowed for general formulas—an essential step toward expressing mathematical relationships as equations.
Renaissance scholars, inspired by classical texts and firsthand experiments, began to view nature as a system governed by mathematical laws—a belief that would catalyze the formalization of calculus and differential equations in the century that followed.

Together, these developments created a fertile intellectual environment. While no differential equations had yet been written down, the mathematical and philosophical seeds of the discipline had been firmly planted.

🌟 Birth of Differential Equations (17th Century)

Key Focus: The invention of calculus and the first appearance of differential equations

The 17th century was a turning point in the history of mathematics. During this period, two towering figures—Isaac Newton and Gottfried Wilhelm Leibniz—independently developed the foundations of calculus, giving rise to the very first differential equations. For the first time, mathematicians could systematically describe how quantities change with respect to one another, opening the door to a new era in science.

Isaac Newton (1643–1727)

English mathematician and physicist Isaac Newton created his version of calculus under the name “fluxions,” focusing on quantities that flow or vary with time.

Newton formulated differential equations to express physical laws. One of the earliest and most famous is Newton’s Second Law of Motion, $F = ma$ , which, when written in terms of position as a function of time, becomes a second-order differential equation:
$\frac{d^2x}{dt^2}$
In his Philosophiæ Naturalis Principia Mathematica (1687), Newton used early DEs to describe planetary motion, gravitation, and the laws of mechanics.
His work in optics and motion required analyzing the rates at which variables changed—something only expressible through differential equations.

While Newton’s notation was cumbersome and not widely adopted, his physical intuition and mathematical methods made him a pioneer in applying DEs to real-world phenomena.

Gottfried Wilhelm Leibniz (1646–1716)

Working independently in Germany, Leibniz developed a more formal and symbolic version of calculus that ultimately shaped how we study DEs today.

He introduced the notation that became standard: $dydx\frac{dy}{dx}$ , $dx\int y \, dx$ , and other expressions that elegantly captured changing quantities.
Leibniz emphasized mathematical structure and symbolism, which made his version of calculus more accessible to other mathematicians and easier to apply broadly.
In the 1680s and 1690s, he and his contemporaries began writing and solving exact differential equations—often involving separable or linear first-order DEs.

Despite the heated dispute over who invented calculus, both Newton and Leibniz are now credited as co-founders. Leibniz’s influence was especially profound in Europe, where his notation became the foundation of differential calculus.

The First DE Texts

The invention of calculus sparked a wave of mathematical innovation across Europe. Among the first to build on Newton and Leibniz’s work were the Bernoulli brothers, Jacob and Johann.

Jacob Bernoulli studied separable and linear differential equations, exploring curves defined by such equations. He posed and solved the isoperimetric problem, an early example of applying calculus to geometry.
Johann Bernoulli, Jacob’s younger brother, made even more substantial contributions:
- In 1690, he published work on separable differential equations, introducing formal methods for solving them.
- His analysis of the exponential growth equation, now written as $dydx=ky\frac{dy}{dx} = ky$ , remains a cornerstone of DE study today.
The Bernoullis also explored integrating factors and homogeneous equations, laying the foundations for solving techniques still taught in modern classrooms.

By the end of the 17th century, differential equations had moved from abstract invention to practical mathematical tool. Thanks to Newton’s laws of motion, Leibniz’s notation, and the Bernoullis’ problem-solving methods, DEs began to spread throughout mathematics, physics, and engineering—poised to become a central pillar of modern science.

📚 Growth and Formalization (18th Century)

Key Focus: Refinement of theory and solving techniques

As the 18th century unfolded, differential equations evolved from isolated discoveries into a formalized mathematical subject. With the foundation of calculus firmly in place, mathematicians began classifying, analyzing, and applying differential equations to a wide range of scientific problems. This era witnessed the emergence of systematic methods for solving both ordinary and partial differential equations, many of which remain central to the field today.

Leonhard Euler (1707–1783)

Swiss mathematician Leonhard Euler was the first to bring structure and clarity to the study of differential equations. His prolific output laid the groundwork for modern DE theory.

Euler introduced a systematic classification of differential equations:
- Linear vs. nonlinear
- Homogeneous vs. non-homogeneous
- First-order vs. higher-order equations
He applied DEs to a vast array of physical problems, including:
- Pendulum motion
- Rigid body dynamics
- Orbital mechanics
- Fluid flow
Euler developed methodical techniques for solving DEs, such as the integrating factor for linear first-order equations and characteristic equations for linear second-order DEs with constant coefficients.
His influential textbook, Institutiones Calculi Differentialis (published in 1755), became one of the first major treatises to treat differential equations as a formal discipline, deeply influencing European mathematics.

Euler’s contributions were so expansive that he is often considered the true founder of the field of differential equations as we know it.

Jean le Rond d’Alembert & Joseph-Louis Lagrange

The 18th century also saw the development of partial differential equations (PDEs), which describe how quantities change in multiple variables—such as time and space.

Jean le Rond d’Alembert (1717–1783) derived the wave equation while studying the vibration of strings:

This became one of the first and most important PDEs in physics, describing phenomena such as sound waves and light.
Joseph-Louis Lagrange (1736–1813) furthered the mathematical study of mechanics. He:
- Developed Lagrangian mechanics, where the equations of motion are derived from an energy-based variational principle rather than Newton’s force laws.
- Laid the groundwork for the calculus of variations, a powerful tool for deriving DEs from optimization problems—a technique that still underpins modern physics.

These contributions formalized the idea that laws of nature can be expressed as differential equations, particularly PDEs.

Partial Differential Equations (PDEs) Emerge

While ordinary differential equations (ODEs) describe functions of a single variable, partial differential equations became essential for modeling systems with multiple continuous dimensions—space, time, or more.

PDEs became deeply connected to physical phenomena, including:
- Heat conduction
- Sound propagation
- Fluid dynamics
- Elasticity
Mathematicians and physicists began exploring boundary value problems, where solutions to PDEs must satisfy conditions at the edges of the domain—key to solving real-world problems.
The growing understanding of PDEs required new mathematical tools, such as Fourier series, which would come into full development in the following century.

By the end of the 18th century, differential equations had become a structured field with its own methods, terminology, and applications. With Euler’s formality, d’Alembert’s wave equation, and Lagrange’s mechanics, DEs were no longer just a mathematical curiosity—they were the engine driving a deeper understanding of the physical universe.

Mathematical Maturity (19th Century)

Key Focus: Rigorous foundations, deeper theory, and applications to physics

The 19th century marked the transition of differential equations from a developing mathematical tool to a fully matured and rigorously defined discipline. Mathematicians not only deepened the theoretical understanding of DEs but also connected them more intimately to real-world physical phenomena—from heat and sound to elasticity and electromagnetism. This century introduced the mathematical rigor still seen in modern analysis and made boundary value problems, existence theorems, and eigenfunction expansions essential parts of the DE toolkit.

Joseph Fourier (1768–1830)

French mathematician and physicist Joseph Fourier revolutionized the study of heat and waves through Fourier series—a method of expressing complex periodic functions as sums of sines and cosines.

In his seminal work Théorie analytique de la chaleur (1822), Fourier introduced the heat equation
This partial differential equation (PDE) modeled how heat diffuses through a material over time.
Fourier’s major innovation was representing arbitrary functions as trigonometric series, now called Fourier series—laying the groundwork for signal processing, quantum mechanics, and modern PDE solution methods.
His ideas encountered skepticism due to their intuitive rather than rigorous nature, but they were later placed on firm footing and became indispensable to mathematical physics.

Carl Gustav Jacobi & Augustin-Louis Cauchy

The 19th century also saw a strong shift toward mathematical rigor, particularly in calculus and the theory of differential equations.

Augustin-Louis Cauchy (1789–1857) formalized limits, continuity, and convergence, providing the first rigorous definitions for calculus. He:
- Introduced methods for analyzing existence and uniqueness of solutions to differential equations.
- Studied linear differential equations and systems of DEs.
- Emphasized conditions under which solutions behave well—foundational to modern analysis.
Carl Gustav Jacobi (1804–1851) contributed significantly to systems of differential equations, determinants (Jacobian matrices), and canonical forms in classical mechanics.

Their work helped transition differential equations from a practical science tool into a formal branch of pure mathematics.

Sophie Germain & Elasticity Theory

Sophie Germain (1776–1831), a self-taught mathematician, made pioneering contributions to elasticity and vibration theory, despite widespread gender-based barriers.

She studied vibrating plates and formulated equations for their motion—higher-order partial differential equations that extend the wave equation to two-dimensional surfaces.
Her work culminated in what’s now called the Germain plate equation, a fourth-order PDE:
$Δ2u=qD\Delta^2 u = \frac{q}{D}$
where $Δ2\Delta^2$ is the biharmonic operator, used in elasticity and structural engineering.
Though often overlooked in her time, Germain’s work laid the groundwork for structural mechanics and applied mathematics.

Riemann, Dirichlet, and Sturm-Liouville Theory

As DEs became central to mathematical physics, new tools emerged to analyze complex problems, particularly those involving boundary conditions.

Johann Peter Gustav Lejeune Dirichlet (1805–1859) formulated the now-famous Dirichlet conditions, which determine when a function can be represented by a Fourier series—crucial for PDE solutions.
Bernhard Riemann (1826–1866) explored the geometric interpretation of PDEs and advanced potential theory, contributing to what would become Riemannian geometry—a critical framework in general relativity.
The Sturm-Liouville theory, developed by Charles-François Sturm and Joseph Liouville, established a method for solving second-order linear differential equations with boundary conditions:
This theory led to eigenvalue problems and orthogonal functions, foundational concepts in quantum mechanics and vibrations.

By the close of the 19th century, differential equations had become deeply intertwined with physics and engineering while resting on solid mathematical ground. From heat and sound to elasticity and spectral theory, DEs had matured into a unified language for describing the continuous world—and would soon become central to 20th-century science and technology.

🧬 Modernization and Abstraction (20th Century)

Key Focus: Theoretical advances and integration into broader mathematics

The 20th century saw differential equations evolve beyond classical mechanics into a sophisticated, abstract, and computationally-driven field. Mathematicians developed new frameworks to understand the structure of DEs in infinite dimensions, while physicists applied them to the strange, probabilistic world of quantum mechanics. At the same time, the advent of computers revolutionized how DEs could be solved—turning theory into simulation and prediction. From chaos theory to functional analysis, this century permanently reshaped the landscape of mathematical modeling.

Hilbert & Functional Analysis

At the dawn of the century, mathematicians such as David Hilbert and Stefan Banach transformed the study of differential equations by embedding them into the broader field of functional analysis.

Hilbert introduced Hilbert spaces—infinite-dimensional vector spaces equipped with inner products—where solutions to PDEs could be studied as points in abstract geometric spaces.
Banach spaces, developed by Stefan Banach, extended this idea to more general function spaces, making it possible to analyze nonlinear and integral equations.
These spaces allowed DEs, especially PDEs, to be studied through operator theory—treating differentiation and integration as operators acting on functions.
Hilbert also pioneered variational methods, where the solution to a DE is framed as the function that minimizes (or maximizes) a certain integral—still a core idea in physics and optimization today.

This shift made DEs a part of a larger mathematical structure, leading to deep connections with topology, geometry, and modern analysis.

Numerical Methods

As DEs became more complex, exact solutions were often impossible to find. The 20th century witnessed an explosion in numerical techniques—ways to approximate solutions with increasing precision.

Classic methods such as:
- Euler’s Method: A stepwise approach to solving first-order ODEs.
- Runge-Kutta Methods: More accurate, higher-order generalizations for solving initial value problems.
- Finite Difference Methods: Approximating derivatives by discrete differences—ideal for solving PDEs numerically.
These techniques laid the foundation for computational mathematics. By mid-century, early computers began solving large systems of DEs in engineering, physics, and weather prediction.
This marked a profound shift: from purely symbolic work to data-driven simulations. DEs became not only theoretical tools but practical engines behind aircraft design, nuclear physics, climate modeling, and more.

Differential Equations in Quantum Mechanics

One of the most transformative uses of DEs in the 20th century came from quantum physics.

Erwin Schrödinger’s Equation (1926)
This linear PDE governs how a quantum system evolves over time and forms the cornerstone of quantum mechanics.
The Schrödinger equation is a second-order PDE involving a wavefunction $ψ\psi$ , which encodes the probabilities of a particle’s state.
Richard Feynman later introduced the path integral formulation, offering a new approach to solving quantum DEs by integrating over all possible paths a particle could take—deeply linked to functional analysis and variational principles.

Quantum mechanics showed that differential equations describe not only deterministic systems like pendulums, but also probabilistic, wave-like behaviors at the smallest scales.

Chaos Theory and Nonlinear DEs

While much early work in DEs focused on systems that were linear and predictable, the 20th century revealed the fragile, chaotic side of nonlinear dynamics.

Henri Poincaré, in the late 19th and early 20th centuries, laid the groundwork for studying qualitative behavior of differential equations. He showed that even simple nonlinear systems could behave unpredictably over time.
In the 1960s, Edward Lorenz discovered the Lorenz equations while modeling atmospheric convection:

These nonlinear ODEs demonstrated deterministic chaos—sensitive dependence on initial conditions, where small changes lead to drastically different outcomes.
This led to the birth of chaos theory, with far-reaching implications in meteorology, biology, economics, and beyond.

The discovery of chaos fundamentally changed our understanding of predictability in DE systems and highlighted the limits of analytical solutions.

By the end of the 20th century, differential equations had grown into a vast and interdisciplinary field. Whether driving the theory of quantum particles, powering computer simulations of planetary climates, or revealing the wild behavior of chaotic systems, DEs had become essential to both pure mathematics and applied science. Their modernization continues to shape every domain where change, uncertainty, and complexity converge.

🚀 21st Century and Beyond

Key Focus: Applications and interdisciplinary relevance

In the 21st century, differential equations are more important—and more interdisciplinary—than ever before. No longer confined to classical physics or engineering, DEs are now powering breakthroughs in artificial intelligence, epidemiology, climate science, finance, and biological modeling. At the same time, theoretical challenges such as the Navier–Stokes problem continue to push the boundaries of pure mathematics. Whether used for predicting pandemics or training deep neural networks, DEs are central to how we understand and shape the modern world.

Machine Learning & Neural DEs

One of the most exciting recent developments is the integration of differential equations with artificial intelligence.

In 2018, a landmark paper introduced Neural Ordinary Differential Equations (Neural ODEs), blending deep learning with continuous dynamical systems.
Neural ODEs model data as solutions to differential equations, offering a more flexible and mathematically grounded approach to time-series forecasting, physics-informed learning, and generative modeling.
These models are especially valuable in fields where data is sparse or noisy, but the underlying process follows continuous physical laws—like in medical imaging or molecular dynamics.

This convergence of DEs and machine learning is reshaping how models are trained, interpreted, and deployed in real-world systems.

Computational Modeling

With the explosion of computational power, differential equations have become the backbone of modern simulation and prediction across disciplines.

Climate modeling: Advanced PDE-based models simulate ocean currents, atmospheric dynamics, and long-term climate patterns to predict global warming and extreme weather.
Epidemiology: During the COVID-19 pandemic, SIR models (Susceptible-Infected-Recovered) and their variants—based on systems of nonlinear ODEs—helped governments forecast infection waves and assess public health interventions.
Biological systems: DEs are used to model everything from the spread of neurons’ electrical impulses to tumor growth and gene expression networks.
Finance and economics: The Black-Scholes equation, a PDE, underlies much of modern option pricing theory; newer models use DEs to capture market volatility and risk.

These models are often solved numerically using high-performance computing, bringing DEs from the chalkboard to real-time decision-making.

Ongoing Theoretical Work

While applications of DEs have exploded, deep theoretical challenges remain at the core of mathematical research.

The Navier–Stokes equations, which describe fluid flow, are among the most fundamental PDEs in physics. However, proving whether smooth, global solutions exist in three dimensions remains unsolved—one of the seven Millennium Prize Problems posed by the Clay Mathematics Institute.
Mathematicians continue to advance:
- PDE classification (elliptic, hyperbolic, parabolic)
- Nonlinear system analysis, including blow-up behavior, stability, and bifurcations
- Integrable systems, which admit exact solutions despite their complexity
New research often blends geometry, topology, and algebra with analysis, expanding how we think about differential equations as objects in abstract spaces.

Theoretical work ensures that the field of DEs continues to grow in depth, not just in breadth.

From modeling the atmosphere and financial markets to shaping AI architectures and solving grand theoretical puzzles, differential equations remain at the forefront of 21st-century science and mathematics. As our world grows more interconnected and data-driven, DEs will continue to be the language we use to understand change—across every scale, system, and discipline.

📅 Timeline: Key Moments in the History of Differential Equations

Date	Figure / Event	Contribution / Milestone
c. 250 BCE	🧠 Archimedes	Introduced the method of exhaustion—an early form of integration for calculating areas and volumes.
5th c. BCE	⚖️ Zeno of Elea	Proposed paradoxes of motion, highlighting early philosophical tension around continuity and infinite processes.
965–1040 CE	🔍 Ibn al-Haytham (Alhazen)	Applied geometric and early infinitesimal reasoning in optics and motion.
1591	🧮 François Viète	Pioneered symbolic algebra—essential groundwork for expressing differential equations.
1665–1675	⚙️ Isaac Newton	Developed calculus (“fluxions”) and used differential equations to model motion and gravity.
1675–1684	✒️ Gottfried Wilhelm Leibniz	Independently developed calculus with modern notation (dy/dx), publishing early differential equations.
1690	📘 Johann Bernoulli	Published one of the earliest formal problems involving a separable differential equation.
1755	📚 Leonhard Euler	Published Institutiones Calculi Differentialis, formalizing DE classification and solution methods.
1747	🎶 Jean le Rond d’Alembert	Derived the wave equation—a key PDE in physics.
1760s–1780s	⚖️ Joseph-Louis Lagrange	Developed Lagrangian mechanics and the calculus of variations.
1822	🔥 Joseph Fourier	Published Théorie analytique de la chaleur; introduced Fourier series and the heat equation.
1830s–1850s	📏 Cauchy & Jacobi	Introduced rigorous definitions for differential calculus and developed linear DE theory.
1816–1830s	🏗️ Sophie Germain	Developed higher-order PDEs for elasticity and vibrating plates.
1836–1850s	🧠 Sturm & Liouville	Developed eigenvalue theory and boundary value problem techniques.
1926	⚛️ Erwin Schrödinger	Introduced the Schrödinger equation, a cornerstone PDE in quantum mechanics.
1963	🌪️ Edward Lorenz	Created the Lorenz equations, launching modern chaos theory.
2000	🧩 Clay Institute	Named the Navier–Stokes existence and smoothness problem as one of the seven Millennium Prize Problems.
2018	🤖 Neural ODEs Introduced	First integration of differential equations into machine learning models—blending AI with continuous mathematics.

📚 References and Sources

The historical information presented on this page is based on well-established academic and primary sources. For students, educators, and curious readers who wish to explore further, the following references provide in-depth analysis and documentation of the development of differential equations:

Books

Boyer, Carl B.
A History of Mathematics. Wiley, 1991.
An authoritative and widely used source tracing the evolution of mathematics, including the development of calculus and differential equations.
Stillwell, John.
Mathematics and Its History. Springer, 3rd ed., 2010.
Offers a clear and chronological account of mathematical ideas, emphasizing the conceptual growth of differential equations and related areas.
Kline, Morris.
Mathematical Thought from Ancient to Modern Times. Oxford University Press, 1972.
A broad and detailed history that contextualizes the development of differential equations within the larger scope of mathematics and physics.

Primary Source Documents

Isaac Newton.
Philosophiæ Naturalis Principia Mathematica (1687).
The foundational text where Newton introduced his laws of motion, leading directly to the formulation of differential equations.
Leonhard Euler.
Institutiones Calculi Differentialis (1755).
A landmark work that laid the groundwork for the formal study of differential equations, classification methods, and solution techniques.

Academic Articles and Archives

JSTOR – www.jstor.org
Contains peer-reviewed journal articles on the history and applications of differential equations across mathematics, physics, and engineering.
SpringerLink – link.springer.com
Provides access to modern research and historical analyses of DE-related topics.
MacTutor History of Mathematics Archive – University of St Andrews
https://mathshistory.st-andrews.ac.uk
A valuable and accessible online resource for biographies and historical essays on major figures in the development of differential equations.
Clay Mathematics Institute – Millennium Prize Problems
www.claymath.org
Source of detailed information on ongoing theoretical challenges, such as the Navier–Stokes existence and smoothness problem

❓ Frequently Asked Questions (FAQs)

Q1: What exactly are differential equations?
A differential equation is a mathematical sentence that describes how something changes over time or space. It’s like a rulebook for motion, growth, decay, or waves.

Q2: Why do differential equations matter?
They show up in physics, biology, finance, AI—you name it. If something moves, grows, or spreads, a differential equation is probably behind it.

Q3: Do I need to understand calculus to appreciate this video?
Nope! The video is made for all curious minds. We tell the story in simple terms so even a 6th grader can follow along.

Q4: What’s the difference between Newton’s and Leibniz’s calculus?
Same core ideas, but different styles. Newton focused on motion (“fluxions”), while Leibniz created the notation we still use today (like dy/dx).

Q5: How are differential equations used in real life today?
Everything from predicting weather, modeling pandemics (like COVID-19), understanding brain signals, simulating black holes, and even training neural networks.

Q6: Will you make more videos like this?
Yes! Let us know in the comments what topics you want next—chaos theory, quantum mechanics, or maybe the math behind time travel?

Q7: Where can I learn more?
Check the video description for recommended books, references, and learning resources.