Category: Mathematics

Primes in Arithmetic Progressions to Large Moduli. https://projecteuclid.org/journals/acta-mathematica/volume-156/issue-none/Primes-in-arithmetic-progressions-to-large-moduli/10.1007/BF02399204.full Primes in Arithmetic Progressions to Large Moduli. II. https://eudml.org/doc/164255Primes in Arithmetic Progressions to Large Moduli. III https://www.ams.org/journals/jams/1989-02-02/S0894-0347-1989-0976723-6/ Dipsersion, sums of Kloosterman sums

At the forefront of contemporary mathematics, the Langlands Program—initiated by Robert Langlands in the late 1960s—seeks to relate two seemingly distant worlds. On one side are Galois representations, which encode the symmetries of algebraic equations through continuous homomorphisms from absolute Galois groups into matrix groups. On the other side are automorphic forms, highly symmetric analytic […]

The transition from classical to quantum mechanics represents one of the most profound shifts in the history of science. It is not merely a refinement of existing laws but a wholesale reconstruction of the conceptual and mathematical framework used to describe physical reality. At the heart of this transition lies the procedure of quantization, a […]

In the vast landscape of theoretical physics, certain models, despite their apparent simplicity, serve as profound gateways to entire universes of mathematical structure. The Toda lattice is a preeminent example of such a system. Introduced by Morikazu Toda in 1967, it was conceived as a simple, one-dimensional model of a crystal, describing a chain of […]

The elegance of the Lagrangian formalism (from previous post) extends seamlessly from discrete particle systems to continuous physical systems, such as fields and fluids, which possess an infinite number of degrees of freedom. Lagrangian Density For classical field theory, the concept of a single Lagrangian is replaced by a Lagrangian density, denoted by . A […]

Newtonian formulation of classical mechanics in terms of forces, while a foundational pillar, often confronts significant analytical hurdles in systems with intricate constraints or non-Cartesian coordinates. Its vectorial foundation, though immensely successful in many domains, can lead to intractable equations when grappling with complex geometries or coupled motions. Analytical mechanics, specifically the Lagrangian formalism, represents […]

The Calogero–Moser–Sutherland system begins with a very concrete mechanical problem. We have particles on a line, with positions and momenta . Their phase space has canonical Poisson brackets , while brackets among two positions or two momenta vanish. The rational Calogero–Moser Hamiltonian is At first sight this is simply a many-body system with a repulsive […]

Let and let be a smooth map. We will prove that has a fixed point. The case of continuous functions can be obtained by reduction to smooth functions by mollification. Rather than trying to solve directly, we deform the simple map into the displacement map and construct an analytic signed count of their zeros. The […]

In classical mechanics, we often start with Newton’s laws (). But there’s a more profound and symmetrical formulation: Hamiltonian mechanics. Here, a system isn’t described by position and velocity, but by generalized coordinates and their conjugate momenta . The set of all possible pairs forms the phase space, a stage where the system’s entire history […]

The Transition Amplitude as a Path Integral In quantum mechanics, the probability amplitude for a particle to propagate from an initial position at time to a final position at time is given by the matrix element of the time-evolution operator: where is the system’s Hamiltonian. The path integral is derived by evaluating this matrix element […]

We prove the classical quantitative prime number theorem: there is an absolute constant for which Consequently, and The main issue is not to prove merely that , but to understand why the error has the particular shape . The answer is a balance between two costs in a contour shift. A zero-free region lets us […]

A recurring theme in mathematical logic is that certain problems which appear purely finite can nevertheless exceed the proving power of standard arithmetic. At first glance this seems paradoxical: if a statement concerns only ordinary integers and finite procedures, why should it require anything stronger than ordinary reasoning about numbers? Yet a number of striking […]

The WKB approximation estimates solutions of the one-dimensional Schrödinger equation when changes slowly over a wavelength. The idea is that the quantum wave is locally like a free-particle wave with slowly changing momentum In a classically allowed region, , WKB gives Thus the local wavelength is , and the amplitude grows where the particle moves […]

The two-dimensional Ising model is one of the central exactly solvable models in statistical mechanics. At first sight it looks quite different from the dimer model. The variables are spins, not matchings. The partition function is a sum over configurations, not over pairings of vertices. The interaction is local and energetic: neighboring spins prefer to […]

The dimer model is one of the most beautiful exactly solvable models in two-dimensional statistical mechanics. It begins with an elementary combinatorial question: in how many ways can one cover a finite graph by disjoint edges, so that every vertex is covered exactly once? But behind this innocent question lies a deep analytic mechanism. The […]

The Baker-Campbell-Hausdorff (BCH) formula addresses the fundamental question of how to combine exponentials of non-commuting operators. Given two operators (or matrices), and , the goal is to find an operator such that If and commute (i.e., ), the solution is simply . However, when they do not commute, is given by a more complex infinite […]

Consider the theta series We want to understand the classical half-integral weight transformation formula under an arbitrary matrix on . For , in with the result is Here is odd, and is the Jacobi symbol, understood as the signed Kronecker symbol when signs are allowed. The theta series is one of the first places where […]

The Landsberg–Schaar relation is a reciprocity formula for quadratic exponential sums. It transforms a finite sum with quadratic phase into another finite quadratic sum in which the roles of and are essentially reversed. The transformation includes the striking phase factor . This factor is not incidental: it is the same analytic phase that appears in […]

Quadratic reciprocity compares two apparently unrelated questions. For distinct odd primes and , one may ask whether is a square modulo , and whether is a square modulo . The quadratic reciprocity law says that these answers agree except in one case: when both primes are modulo . In Legendre-symbol notation, the theorem is The […]

Napoleon’s Theorem states that if equilateral triangles are constructed externally on the three sides of any triangle, then the centroids of those three equilateral triangles form an equilateral triangle. Classical proofs often use angle chasing or trigonometric identities. There is, however, a more structural proof: the construction respects cyclic symmetry, and the discrete Fourier transform […]

One of the most beautiful facts in quantum mechanics is that the existence of a magnetic monopole would force electric charge to be quantized. If a particle of electric charge (e) moves in the field of a magnetic monopole of magnetic charge (g), then consistency of the quantum theory requires Equivalently, if we define the […]

A modern student usually meets elliptic curves in a very polished form: Then one defines a group law by drawing lines, introduces divisors, Abel maps, Jacobians, period lattices, and so on. This is beautiful, but historically it is backwards. The older story began with much more concrete questions: How long is an arc of an […]

Primality is among the oldest questions in arithmetic: given a positive integer , is it prime, or does it factor into smaller integers? Using basic number theory, there is an immediate procedure. A composite integer has a prime divisor at most , so one may test every possible divisor up to that point. This always […]

The one-dimensional Ising model consists of a row of spins, each of which can point in one of two directions. We label these two directions by and . The essential feature is that neighboring spins interact: when the coupling is ferromagnetic, neighboring spins prefer to agree. Thus two adjacent positive spins and two adjacent negative […]

The class number formula is one of the beautiful results in number theory. It connects the arithmetic of a number field with the behavior of an analytic function at . On the arithmetic side stand the class number, the units, the regulator, the discriminant, and the roots of unity. On the analytic side stands the […]

For a real number , write for its fractional part. Assuming the prime number theorem, we shall prove that where is Euler’s constant. The quantities fluctuate in a seemingly irregular way as runs through the primes. The useful observation is that, after dividing by , the primes up to become uniformly distributed through with respect […]

We prove that every continuous map from the closed unit ball to itself has a fixed point. The theorem is usually presented in topological language: a fixed-point-free map would produce a retraction of the ball onto its boundary, but the ball cannot retract onto its boundary. The proof below makes that mechanism visible. The key […]

Abel’s proof that the general quintic cannot be solved by radicals is sometimes presented as though it ended the classical theory of algebraic equations. In fact, it posed a new and more precise problem. Once one knows that no universal radical formula can solve an arbitrary equation of degree five, the natural question is no […]

The QR algorithm is a method for finding eigenvalues. In the real symmetric case, the problem is especially clean. We are given and we want to find numbers and orthonormal vectors such that . Equivalently, we want an orthogonal matrix and a diagonal matrix such that Thus the eigenvalue problem is, at heart, a problem […]

For more than two centuries, the solution of polynomial equations had appeared to follow a compelling pattern. The quadratic equation had a formula. The cubic equation had yielded to the methods of the Italian algebraists, and the quartic had soon followed. Each success had the same general character: starting from the coefficients, one combined the […]

Let denote the natural logarithm. We consider the fractional parts as points on the interval . It is tempting to think that these points should spread uniformly around the interval, because : the sequence winds around the interval infinitely often, crossing each integer infinitely many times. But this intuition confuses two different facts. A sequence […]

Let us begin with the basic problem. Given an exponential sum what information about the coefficients is enough to ensure that its partial sums actually converge at almost every point? The first condition one naturally encounters is square summability: Because the exponentials are orthogonal, this condition immediately implies that the partial sums are Cauchy in […]

Consider the one-sided Fourier partial sums This is one of the first Fourier series in which several different notions of convergence visibly separate from one another. The coefficients satisfy but Thus the coefficient sequence belongs to , but not to . Square summability will give an limit. The failure of absolute summability means that uniform […]

One of the earliest and most influential applications of the probabilistic method in combinatorics was given by Paul Erdős in his 1947 paper Some Remarks on the Theory of Graphs. The argument is famous not because it is long or technically complicated, but because it introduced a powerful new way of proving existence. Instead of […]

For the harmonic oscillator, the motion is almost magically simple. Confined to a quadratic potential , a displaced particle oscillates as a sine or cosine forever. The system possesses a single, intrinsic clock: its period is strictly independent of amplitude. But if we add even the simplest nonlinear correction, the story changes in a deep […]

A geodesic is the correct replacement for a straight line on a curved space. If a curve lies in ordinary Euclidean space, being straight means that its acceleration vanishes: But if the curve is constrained to lie on a curved surface, its ambient acceleration need not vanish. For example, a great circle on a sphere […]

The equation already suggests grouping the four real variables into two Gaussian integers. Put and . Then Thus Lagrange’s theorem says that every positive integer is represented by the binary Hermitian form on . This reformulation changes the point of view. Rather than seeking four integers directly, we study positive definite Hermitian forms over the […]

Fermat’s theorem on sums of two squares says that an odd prime can be written as if and only if . The proofs of Heath-Brown and Zagier are striking because they prove this theorem by counting fixed points of involutions. The guiding principle is simple: when an involution acts on a finite set, all non-fixed […]

The familiar proof of Fermat’s two-square theorem through Gaussian integers has a very satisfying shape. For a prime , one first finds a solution of . This produces the Gaussian ideal The fact that has class number one, or more concretely that it is a principal ideal domain, turns this ideal into one generated by […]

The four-square theorem states that every nonnegative integer can be written as a sum of four integer squares: For instance, At first this is surprising. Squares are sparse, and there are genuine congruence obstructions to representing every integer by two or three squares. Yet four squares suffice for every nonnegative integer. The proof combines two […]

Let Lagrange’s four-square theorem says that every positive integer can be written as for some integers . Equivalently, every positive integer is the squared length of an integer vector in . At first this looks like a direct equation-solving problem: given , find four integers whose squares add to it. The classical descent proof instead […]

We prove Fermat’s two-squares theorem for primes. The theorem says that whenever a prime is congruent to one modulo four, it is a sum of two squares. The purpose of this exposition is not merely to collect five proofs. It is to explain how the first family of proofs is really organized around one object […]

The ordinary Dedekind reciprocity law already has two apparently different explanations. One is geometric: a rational line cuts a lattice rectangle into two complementary regions, and the staircases along their common boundary cancel except at the endpoints. The other is analytic: the same staircase information is encoded by cotangent poles, and the residue theorem says […]

We have already seen two ways to understand Dedekind reciprocity. The Carlitz generating-function proof starts with a rational line and follows its finite lattice staircase: horizontal and vertical edges cancel in pairs, leaving only endpoint terms. The direct Bernoulli proof unfolds the staircase into a finite rectangle and then determines the answer by its variation […]

Let be coprime. We want to prove the classical reciprocity law where the Dedekind sum is defined by Here for nonintegral , and for integral . Let be coprime integers. We use the periodic first Bernoulli function defined for every real . In particular, when . This is slightly different from the usual sawtooth function […]

Let be coprime integers. Begin with the line of rational slope , drawn from to . This small geometric object already contains several familiar reciprocity laws. Eisenstein’s lattice proof of quadratic reciprocity studies the lattice points on one side of this line and extracts only a parity from the count. Dedekind reciprocity comes from keeping […]

The quartic formula becomes much less mysterious once one sees what problem it is trying to solve. For a cubic, Cardano found a way to write the unknown as a sum of two quantities, arranged so that the mixed terms combine into the required linear term. For a quartic, the basic aim is different: one […]

We will prove Fundamental Theorem of Algebra which says that every nonconstant polynomial with complex coefficients has a complex root. Equivalently, every polynomial with complex coefficients can be broken completely into linear factors. For example, a polynomial of degree can be written in the form where The theorem says that once we have added the […]

Let where . We shall prove that has exactly complex roots, counted with multiplicity. The central observation is that the degree- term governs the polynomial at large scale. When is large, every lower-degree monomial is smaller than by at least one factor of . Thus, on a sufficiently large circle, the curve traced by is […]

The exponent-four case of Fermat’s Last Theorem says that there are no nonzero integers satisfying Fermat’s original method proves something stronger and, in a sense, more natural: has no solution in positive integers. Once this stronger statement is known, the exponent-four case follows at once, because a hypothetical equation would be an equation of the […]

We will prove that there do not exist nonzero integers such that Replacing by , this is the same as saying that has no nonzero integer solutions. The statement is the exponent-three case of Fermat’s Last Theorem. Euler’s proof is an early and striking example of a general mathematical strategy: begin with the ordinary factorization […]

Morley’s Trisector Theorem states that for any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle. Its beauty lies in the surprising emergence of a perfectly regular shape from an arbitrary starting triangle Let’s first look at direct proof by trigonometric computations. For a with angles , we have […]