Mathematics on Chen Kai Blog

Product Thinking (5): Abstraction Thinking — From Math to Systems

Wed, 03 Jun 2026 09:00:00 +0000

The Instinct You Cannot Unlearn#

There is a moment in every abstract algebra course where the professor writes something like this on the board:

Let $\phi: G \to H$ be a group homomorphism. Then $\ker(\phi) \trianglelefteq G$ , and $G/\ker(\phi) \cong \text{im}(\phi)$ .

The first isomorphism theorem. When I first saw it, I thought it was a curiosity — a proof exercise to survive, then forget. I was wrong. That theorem planted something in my brain that never left: the instinct that every structure has a quotient, that what you throw away defines what you keep, and that two things that look nothing alike can be the same thing in disguise if you find the right map between them.

Kernel Methods (8): Deep Kernel Learning vs Deep Learning — A Practitioner's Guide

Thu, 30 Dec 2021 09:00:00 +0000

In 2026, why are you still reading about kernel methods? Aren’t transformers supposed to have eaten the entire ML stack? Yes and no. Transformers eat the headlines, but kernels still eat the corners — the regimes with 200 samples, the regimes where the model has to publish calibrated error bars, the regimes where a physicist needs to know which basis function caused the prediction. This final part is the field manual: when kernels actually win, how to debug them when they don’t, and how to bolt them on top of a neural network when you want the best of both worlds.

Kernel Methods (7): Large-Scale Kernels — Nystrom Approximation and Random Fourier Features

Fri, 24 Dec 2021 09:00:00 +0000

You want to train an RBF SVM on a million-image classification set. The Gram matrix is $10^6 \times 10^6$ doubles, which is 8 TB. That number alone — eight terabytes of RAM, just to store the kernel — is why most working data scientists who learned kernel methods in a stats class quietly never reach for them on real production workloads. The kernel trick gives you an infinite-dimensional feature space for the cost of one dot product per pair; the bill arrives when you have $$n^2$$ pairs.

Kernel Methods (6): Gaussian Processes — When Kernels Meet Bayesian Inference

Sun, 19 Dec 2021 09:00:00 +0000

Kernel ridge regression gives you a number. You feed it $$x_*$$ , it returns $\hat{y}_* = 23.7$ . End of story. But you wanted to act on that prediction — maybe schedule a delivery, dose a patient, place a bet — and the single number is not enough. Tomorrow’s temperature being “25°C” is useful; “very likely 25°C, 95% chance between 22 and 28” is actionable. Every decision under uncertainty needs the second one. Gaussian Processes are the cleanest way to upgrade a kernel method from “point predictor” to “distribution predictor”, and they do it without abandoning a single line of the kernel math from the previous five parts.

Kernel Methods (5): Kernel SVM, Kernel PCA, and Kernel Ridge Regression

Tue, 14 Dec 2021 09:00:00 +0000

Your features are two-dimensional, your data is clearly a circle inside a circle, and LinearSVC is at 50% accuracy with the wide-eyed look of an algorithm that genuinely believes a straight line is the answer. You stare at the scatter plot, you stare at the model, and somewhere in the back of your head the words kernel SVM surface. You type kernel='rbf', the accuracy jumps to 0.98, and the rest of the afternoon you wonder what exactly just happened — and why the same trick also gives you a Kernel PCA that unfolds a Swiss roll and a Kernel Ridge regressor that fits a sine wave with three lines of code.

Kernel Methods (4): Common Kernel Families — RBF, Matern, Polynomial, Periodic, and More

Thu, 09 Dec 2021 09:00:00 +0000

You type SVC(kernel='rbf') in scikit-learn for the first time. What did you set gamma to? 'scale'? 'auto'? You scrolled past those defaults without thinking. Three months later your model is overfitting, your Gram matrix looks like the identity, and you have no idea which knob is wrong. Most “kernel tuning” debt is really kernel choice debt — you picked the default kernel for the wrong reason, and now no amount of grid search will save you.

Kernel Methods (3): RKHS — The Theoretical Soul of Kernel Methods

Sat, 04 Dec 2021 09:00:00 +0000

If your eyes glaze over the moment a lecturer writes “RKHS” on the board, this part of the series is for you. RKHS is not a club of three intimidating letters — it is a function space, and once you see what lives inside it, kernel methods stop feeling like magic and start feeling like linear algebra you already know.

Kernel Methods (2): Mathematical Foundations — Positive-Definite Kernels and Mercer's Theorem

Mon, 29 Nov 2021 09:00:00 +0000

A week into kernel-SVM hacking I wrote what felt like a perfectly reasonable similarity function — tanh(1.5 * x.dot(y) - 2.0). It compiled, it ran, the math looked symmetric. Then sklearn coughed up ValueError: kernel matrix is not positive semidefinite and the optimiser produced a model that was worse than guessing.

That error message turned out to hide one of the deepest results in 20th-century analysis. “Positive-definite” is not a checkbox — it is the entire reason the kernel trick is allowed to exist. If your function is PSD, there exists a Hilbert space where it is a real inner product; if it is not, you are pretending to live in a space that nobody built. This post unpacks that statement, builds the operational tests, derives Mercer’s theorem, and works through enough numerical examples that the next time you see the failure message you will know exactly which line of math your kernel violated.

Kernel Methods (1): Why We Need Them — Hitting the Ceiling of Linear Algorithms

Wed, 24 Nov 2021 09:00:00 +0000

The first time I tried to fit a logistic regression to a dataset of two interlocking spirals, I burned an afternoon tweaking the regularizer, swapping solvers, and rescaling features — convinced that I was doing something wrong. The accuracy hovered around 50%. That is the noise floor of a coin flip; my model was, in a very literal sense, learning nothing.

The model was not buggy. The data was simply not the kind of object a straight line can describe. No amount of C, class_weight, or tol was going to change that. Once you have seen this failure mode once, you start noticing it everywhere — in customer-churn data with non-monotone relationships, in image classification before deep learning, in any regression where the trend bends. A linear algorithm has a hard ceiling, and you only break through that ceiling by changing the kind of object the algorithm operates on.

Differential Geometry (12): Fiber Bundles, Characteristic Classes, and Physics

Tue, 23 Nov 2021 09:00:00 +0000

Throughout this series, we have built differential geometry from the ground up: manifolds, tangent spaces, differential forms, integration, Riemannian metrics, connections, and curvature. A recurring theme has been the tangent bundle $$TM$$ — the collection of all tangent spaces glued together into a single geometric object. The Levi-Civita connection is a rule for differentiating sections of $$TM$$ (i.e., vector fields), and the Riemann curvature tensor measures the non-commutativity of this differentiation.

But the tangent bundle is just one example of a much more general construction: a fiber bundle. And the Levi-Civita connection is just one example of a connection on a vector bundle. This generalization is not merely aesthetic — it is the mathematical language of gauge theory, the framework underlying all of modern particle physics. Electromagnetism, the weak force, the strong force, and even gravity can all be described as connections on appropriate bundles, with their dynamics governed by curvature.

Differential Geometry (11): Curvature in Riemannian Geometry — Riemann, Ricci, and Scalar

Sun, 21 Nov 2021 09:00:00 +0000

Curvature is the central concept of Riemannian geometry. Intuitively, it measures how much a space deviates from being flat — how parallel lines converge or diverge, how triangles have angle excess or deficit, how volumes grow differently from Euclidean expectations. In the previous article we saw that the path-dependence of parallel transport signals the presence of curvature: a vector carried around a closed loop on $$S^2$$ returns rotated, while on $\mathbb{R}^n$ it does not. The next step is to make this precise, to extract a tensor that exactly captures this rotation, and to understand its various contractions.

Differential Geometry (10): Riemannian Geometry — Metrics, Connections, and Parallel Transport

Fri, 19 Nov 2021 09:00:00 +0000

Up to this point in the series we have studied smooth manifolds with their differentiable structure: charts, tangent vectors, differential forms, exterior calculus, integration. None of this required a notion of distance. We could differentiate functions, integrate top-degree forms, decide whether a distribution is integrable — all without ever measuring how long a curve is or what angle two tangent vectors make. The smooth structure is purely topological-with-derivatives. To do geometry in the classical sense — to measure, to compare, to speak of curvature, to recover the everyday meanings of “length” and “angle” — we need additional structure.

Differential Geometry (9): Integration on Manifolds and Stokes' Theorem

Wed, 17 Nov 2021 09:00:00 +0000

In single-variable calculus, the fundamental theorem says that integrating a derivative over an interval equals the boundary difference: $\int_a^b f'(x)\,dx = f(b) - f(a)$ . The “boundary” of $$[a, b]$$ is the two-point set $\{a, b\}$ , with $$b$$ counted positively and $$a$$ negatively. The right-hand side is the integral of $$f$$ over this signed boundary. The left-hand side is the integral of the derivative over the interval. This is, in essence, every “fundamental theorem” you have ever met — Green’s theorem in the plane, the divergence theorem in three dimensions, the classical Stokes’ theorem on surfaces. They are all instances of one statement on manifolds: the integral of $d\omega$ over $$M$$ equals the integral of $\omega$ over $\partial M$ .

Differential Geometry (8): Differential Forms — The Natural Language of Integration on Manifolds

Mon, 15 Nov 2021 09:00:00 +0000

In vector calculus on $\mathbb{R}^3$ , we have three derivative operations: gradient ( $\nabla f$ ), curl ( $\nabla \times F$ ), and divergence ( $\nabla \cdot F$ ). Each operates on a different type of object (scalar fields, vector fields). Two identities — $\nabla \times (\nabla f) = 0$ and $\nabla \cdot (\nabla \times F) = 0$ — sit there looking like happy coincidences. The three integral theorems (the fundamental theorem for line integrals, the classical Stokes’ theorem, the divergence theorem) appear unrelated and unmotivated except by their statements.

Differential Geometry (7): Vector Fields, Flows, and the Lie Bracket

Sat, 13 Nov 2021 09:00:00 +0000

A tangent vector lives at one point. It tells you “this direction, this speed, right here, right now.” It is fundamentally local — pluck it off the manifold and it remembers nothing about its neighbors. A vector field, by contrast, is what you get when you let one tangent vector at every point conspire smoothly. It is a velocity prescription on the entire manifold: stand anywhere, and the field tells you where to go. Follow the prescription, and you trace out an integral curve. Follow it from every starting point at once, and you get a flow — a one-parameter family of diffeomorphisms that drags the whole manifold along itself like a slow river.

Differential Geometry (6): Smooth Manifolds — Geometry Beyond Embedded Surfaces

Thu, 11 Nov 2021 09:00:00 +0000

The first five chapters of this series lived inside $\mathbb{R}^3$ . We had curves and surfaces, parametrized explicitly, with all the geometric data — first and second fundamental forms, principal curvatures, Christoffel symbols, the Theorema Egregium, Gauss-Bonnet — built up from coordinates we could write down. The Theorema Egregium revealed that the intrinsic story can be told without reference to the embedding. But “without reference to the embedding” still meant “the embedding exists; we just choose not to use it.”

Differential Geometry (5): The Gauss-Bonnet Theorem — Where Geometry Meets Topology

Tue, 09 Nov 2021 09:00:00 +0000

The Theorema Egregium of the previous chapter showed that Gaussian curvature is intrinsic — bend a surface without stretching and $$K$$ does not change. The Gauss-Bonnet theorem, which we develop here, says something equally remarkable in a different direction: integrate $$K$$ over a closed surface and you get a topological invariant. The total curvature of any sphere is $4\pi$ . The total curvature of any torus is $$0$$ . The total curvature of a double torus is $-4\pi$ . These facts are blind to the specific geometry — bend, twist, or smoosh the surface as you please, the total curvature does not change.

Differential Geometry (4): Intrinsic Geometry — Theorema Egregium and Geodesics

Sun, 07 Nov 2021 09:00:00 +0000

The previous two chapters set up a clear dichotomy. Chapter 2 introduced the first fundamental form $\mathrm{I}$ — the intrinsic metric, what an ant on the surface can measure. Chapter 3 introduced the second fundamental form $\mathrm{II}$ and the shape operator — the extrinsic data, how the surface bends in $\mathbb{R}^3$ . From $\mathrm{II}$ we computed Gaussian curvature $K = \det S$ and mean curvature $H = \mathrm{tr}\,S/2$ . By all appearances, both $$K$$ and $$H$$ should depend on the embedding. Bend the surface (without stretching) and you would expect both to change.

Differential Geometry (3): The Shape Operator — Curvature of Surfaces

Fri, 05 Nov 2021 09:00:00 +0000

The previous article gave us the intrinsic apparatus: the first fundamental form $\mathrm{I}$ , encoded as the symmetric matrix $\begin{pmatrix}E & F \\ F & G\end{pmatrix}$ . With it, an ant on the surface can measure lengths, angles, and areas without ever leaving. What an ant on a cylinder cannot do, equipped only with $\mathrm{I}$ , is detect that the cylinder is bent. The cylinder has the same first fundamental form as the plane, yet sits very differently in $\mathbb{R}^3$ .

Differential Geometry (2): Surfaces and the First Fundamental Form: Intrinsic Measurements

Wed, 03 Nov 2021 09:00:00 +0000

Curves were a one-dimensional warm-up. The geometry was governed by ODEs, and a single moving frame caught everything interesting. From now on we go up a dimension and the difficulty rises in three directions at once. Tangent vectors get replaced by tangent planes. The single arc-length parameter splits into two coordinates $$(u, v)$$ , and reparametrization becomes a $2\times 2$ Jacobian matrix instead of a scalar. And — the real change — we acquire two distinct kinds of geometry: intrinsic (what an ant living on the surface can measure) and extrinsic (how the surface bends in the surrounding $\mathbb{R}^3$ ). This article is the intrinsic story. We build the first fundamental form, the $2\times 2$ matrix-valued function that lets the ant measure lengths, angles, and areas without ever leaving the surface.

Differential Geometry (1): Curves in Space — Curvature, Torsion, and the Frenet Frame

Mon, 01 Nov 2021 09:00:00 +0000

I am going to start this series the way every honest course on differential geometry starts: with a single moving particle in $\mathbb{R}^3$ . Not a manifold, not a fiber bundle, not even a surface — just a dot tracing a path. Everything later — Gauss maps, second fundamental forms, connections, Riemannian curvature tensors — is in some sense an effort to do for higher-dimensional objects what we are about to do, very thoroughly, for a one-dimensional one. So bear with the warm-up. The pay-off is that by the end of this article we will own a complete local theory: two scalar functions ( $\kappa$ , $\tau$ ) and one orthonormal frame ( $\mathbf{T}, \mathbf{N}, \mathbf{B}$ ) that together pin down a curve up to rigid motion. The slogan is “curve = two numbers per point”, and that slogan deserves a proof.

Functional Analysis (12): Functional Analysis in Action — PDE and Quantum Mechanics

Sat, 23 Oct 2021 09:00:00 +0000

Eleven articles is a long time to spend on infrastructure. Normed spaces, Banach and Hilbert structure, dual spaces, weak topologies, bounded and unbounded operators, the spectral theorem, semigroups, distributions, Sobolev spaces — every one of those chapters paid for itself with a clean abstract result, but a reader could be forgiven for wondering when the abstraction was going to do anything. This final article is where I make good on the implicit promise of the series: every theorem we built was built because some concrete problem demanded it, and pulling those threads together gives us the modern toolkit for partial differential equations and quantum mechanics.

Functional Analysis (11): Distributions and Sobolev Spaces — Generalized Solutions

Thu, 21 Oct 2021 09:00:00 +0000

I want to start with a confession. For years I treated the Dirac delta the way an undergraduate physicist does: as a function that is zero everywhere except at the origin, where it is infinite, and whose integral is one. That description is, of course, mathematical nonsense. No measurable function can have those properties. Yet every quantum mechanics textbook uses $\delta$ on page one, every signal processing course writes $\delta(t)$ for an impulse, and every PDE book invokes Green’s functions $$E$$ satisfying $\Delta E = \delta$ . Either an entire scientific community has been making a fundamental error for a century, or there is a way to make this object rigorous. The latter, obviously — and the way is the theory of distributions.

Functional Analysis (10): Semigroups of Operators — Evolution Equations in Infinite Dimensions

Tue, 19 Oct 2021 09:00:00 +0000

The simplest interesting differential equation is $$u' = a u$$ , with $a \in \mathbb{R}$ . The solution $u(t) = e^{at} u_0$ is so familiar that it is easy to forget it is a piece of structure: the map $T(t) = e^{at}$ is a one-parameter family of operators on $\mathbb{R}$ satisfying $$T(0) = I$$ , $$T(t + s) = T(t) T(s)$$ , and continuity in $$t$$ . Replace $$a$$ with a self-adjoint matrix $$A$$ and you have $T(t) = e^{tA}$ , the matrix exponential, which solves the system $$u' = Au$$ . Replace $$A$$ with an unbounded operator on a Hilbert space — the Laplacian, the Schrödinger Hamiltonian, a Fokker-Planck operator — and you would like to do the same thing. But the matrix-exponential power series may not converge, the operator may not be defined on all of $$H$$ , and ordinary calculus stops working.

Functional Analysis (9): Unbounded Operators — When Boundedness Fails

Sun, 17 Oct 2021 09:00:00 +0000

Two articles ago I was talking about how spectral theory is the linear-algebraic infrastructure of quantum mechanics. The trouble is that nearly every operator a physicist actually cares about – the position operator, the momentum operator, the Laplacian, the Schrodinger Hamiltonian – is not bounded. They are not defined on the whole Hilbert space. They are densely defined, with domains that depend on the regularity or decay of the input function. None of the previous spectral apparatus applies directly. We need to extend it.

Functional Analysis (8): Spectral Theory — Decomposing Operators

Fri, 15 Oct 2021 09:00:00 +0000

When I first saw the word “spectrum” used for an operator I assumed it was a fancy synonym for “set of eigenvalues.” That is the right intuition for matrices and for compact operators, and it is exactly what one wants in introductory linear algebra. The trouble is that it is wrong as soon as the operator is not compact. The position operator $$(Mf)(x) = x f(x)$$ on $$L^2[0, 1]$$ has no eigenvalues: any eigenfunction would have to satisfy $x f(x) = \lambda f(x)$ a.e., which forces $$f = 0$$ everywhere away from a single point, hence $$f = 0$$ in $$L^2$$ . And yet the operator is clearly not invertible, since $\lambda I - M$ is multiplication by $x - \lambda$ , which fails to be boundedly invertible whenever $\lambda \in [0, 1]$ .

Functional Analysis (7): Compact Operators — The Bridge to Finite Dimensions

Wed, 13 Oct 2021 09:00:00 +0000

I owe my fondness for compact operators to a small embarrassment. As an undergraduate I assumed that infinite-dimensional linear algebra would feel exotic everywhere. It does not. There is a wide and well-mapped suburb of operator theory in which everything one learned about symmetric matrices – eigenvalues, orthogonal eigenvectors, the spectral decomposition – comes back almost unchanged, just with eigenvalues tailing off to zero instead of a finite list. That suburb is the world of compact operators, and the price of admission is a single condition: the operator must squeeze the unit ball into a relatively compact set. Once that condition is met, nearly everything follows: the spectrum is countable, nonzero eigenvalues are isolated with finite-dimensional eigenspaces, the Fredholm alternative holds, and integral equations of the second kind become as tractable as linear systems. The line between matrices and infinite-dimensional operators ceases to be a wall and becomes a permeable membrane.

Functional Analysis (6): Bounded Linear Operators and the Big Theorems

Mon, 11 Oct 2021 09:00:00 +0000

Why This Article Is Where the Theory Catches Fire#

For five articles I have been building scaffolding: metric and normed spaces, Hilbert spaces, dual spaces, weak topologies. None of those individually felt very impressive — I am, after all, just doing topology and linear algebra in slightly more general settings than usual. The point at which functional analysis genuinely delivers is right here, in the three great theorems of Banach space operator theory: the Uniform Boundedness Principle, the Open Mapping Theorem, and the Closed Graph Theorem. Each of these takes a piece of “pointwise” or “set-theoretic” data — pointwise boundedness, surjectivity, closedness of the graph — and concludes a global structural property — uniform boundedness, openness, continuity — that has no analog in finite dimensions because finite-dimensional linear algebra makes them all true automatically.

Functional Analysis (5): Weak and Weak-* Topologies — When Norm Convergence Is Too Strong

Sat, 09 Oct 2021 09:00:00 +0000

Why Weaker Topologies Exist and Why They Matter#

Article 1 ended with a depressing fact: in any infinite-dimensional normed space, the closed unit ball is not compact. No bounded sequence is guaranteed to have a norm-convergent subsequence. If you are trying to find a minimizer of an energy functional — say, the lowest-energy configuration of a vibrating membrane — you take a minimizing sequence, and you need a limit. In finite dimensions, Bolzano-Weierstrass delivers that limit. In infinite dimensions, it does not. The direct method of the calculus of variations appears dead on arrival.

Functional Analysis (4): Dual Spaces and the Hahn-Banach Theorem — Taming Linear Functionals

Thu, 07 Oct 2021 09:00:00 +0000

Why You Cannot Skip This Article#

Up to now, the theory has been about spaces and the elements that live in them. This article changes the perspective: it asks what you can say about a vector $$x$$ by measuring $$x$$ against a family of test functionals. The shift from “vectors” to “vectors plus functionals” is what turns Banach spaces into a serviceable analogue of finite-dimensional linear algebra. In finite dimensions, every linear functional is continuous and the dual space is the same dimension as the original — so there is nothing to prove. In infinite dimensions, continuity is a real constraint, and the existence of enough continuous functionals to separate points or extend partial data is not obvious. The Hahn-Banach theorem is what guarantees this, and it is the result that makes functional analysis possible.

Functional Analysis (3): Hilbert Spaces — Geometry in Infinite Dimensions

Tue, 05 Oct 2021 09:00:00 +0000

Inner Products and the Geometry They Create#

If a Banach space is a normed space that has agreed to be complete, a Hilbert space is a Banach space that has further agreed to admit angles. That extra agreement — an inner product — is what restores almost all of finite-dimensional geometry to the infinite-dimensional setting. Orthogonality, projection, the Pythagorean theorem, the notion of “closest point in a subspace” — all come back unchanged. The price of admission is a single axiom; the reward, geometric and computational, is enormous.

Functional Analysis (2): Normed Spaces and Banach Spaces

Sun, 03 Oct 2021 09:00:00 +0000

Why a Norm Is More Than a Metric Wearing a Hat#

In Article 1, the metric was a free-standing function on a set with no algebraic structure. That generality bought us topology and completeness, but it gave nothing back to the algebra. The moment I am willing to assume the underlying set is a vector space, a more rigid object becomes available: a norm, a single nonnegative function on the space whose induced metric $d(x,y) = \|x - y\|$ is translation-invariant and positively homogeneous.

Functional Analysis (1): Metric Spaces — Distance, Convergence, and Completeness

Fri, 01 Oct 2021 09:00:00 +0000

Why I Had to Stop Trusting My Finite-Dimensional Intuition#

The first thing graduate analysis did to me was take away my picture. Up to that point, “distance” had always been the length of an arrow drawn from the origin to a point — Pythagoras, three coordinates, done. Then somebody asked me how far two functions are from each other and the arrow disappeared.

Abstract Algebra (12): Algebra in the Wild — Cryptography, Coding Theory, and Beyond

Thu, 23 Sep 2021 09:00:00 +0000

For eleven articles, we have built algebra from the ground up: groups, rings, fields, Galois theory, modules, representations, categories. At times, the material may have felt like pure abstraction — beautiful, perhaps, but detached from the “real world.” This final article corrects that impression. The structures we have studied are not just mathematically elegant; they are the backbone of technologies and theories that shape modern life. By the end of this article, the question “is abstract algebra useful?” should feel about as well-posed as “is calculus useful?” — the answer is so overwhelmingly yes that the question itself sounds quaint.

Abstract Algebra (11): Category Theory — The Language of Mathematical Structure

Tue, 21 Sep 2021 09:00:00 +0000

Throughout this series, we have seen a recurring pattern: define a type of algebraic structure, define the “right” maps between structures (homomorphisms), and study the interplay between objects and maps. Groups have group homomorphisms. Rings have ring homomorphisms. Modules have module homomorphisms. Vector spaces have linear maps. In every case, we proved isomorphism theorems, constructed products and quotients, and identified “free” objects. The proofs were structurally identical, differing only in the specific axioms being preserved. The first time you notice this is mildly interesting; the tenth time, it starts to feel like there ought to be a uniform framework.

Abstract Algebra (10): Representation Theory — Groups Acting on Vector Spaces

Sun, 19 Sep 2021 09:00:00 +0000

An abstract group is a set with a binary operation satisfying certain axioms. This is elegant but sometimes hard to work with — how do you compute with elements of a group defined by generators and relations, or extract numerical invariants from a multiplication table? The solution, going back to Frobenius and Burnside over a century ago, is to represent group elements as matrices. Matrices are concrete: you can multiply them, take traces, compute determinants, decompose them into eigenspaces. Representation theory is the systematic study of this idea, and it has become one of the most powerful tools in modern algebra, number theory, and mathematical physics.

Abstract Algebra (9): Modules — Generalizing Vector Spaces

Fri, 17 Sep 2021 09:00:00 +0000

In every linear algebra course, you learn to work over a field: real numbers, complex numbers, or perhaps a finite field. The resulting theory is remarkably clean — every subspace has a complement, every finitely generated vector space has a basis, and all bases have the same cardinality. But what happens when we replace the field with a ring?

The answer is modules: the natural generalization of vector spaces, where scalars come from a ring rather than a field. The theory is richer, the pathologies more interesting, and — perhaps most importantly — modules turn out to encompass an enormous range of mathematical objects: abelian groups (modules over $\mathbb{Z}$ ), vector spaces with a linear endomorphism (modules over $$K[x]$$ ), ideals (modules over a ring), and group representations (modules over a group ring). What initially feels like a technical generalization is actually a unifying framework that organizes much of algebra.

Abstract Algebra (8): Galois Theory — The Bridge Between Fields and Groups

Wed, 15 Sep 2021 09:00:00 +0000

In 1832, a twenty-year-old mathematician named Evariste Galois, on the eve of a duel he would not survive, wrote down the final versions of his mathematical ideas in a letter to a friend. Those ideas — connecting the symmetries of polynomial roots to the structure of groups — would take more than a decade to be understood and published, but they would reshape algebra forever. Galois theory, as we now call it, establishes a precise dictionary between intermediate fields of a field extension and subgroups of a group of symmetries. It explains, in one elegant framework, why the quadratic formula exists, why there is no analogous formula for degree five, and what “solvability” really means.

Abstract Algebra (7): Field Extensions — Building Bigger Number Systems

Mon, 13 Sep 2021 09:00:00 +0000

Every mathematician, at some point, encounters a polynomial that refuses to be solved within the number system at hand. The ancient Greeks discovered that $\sqrt{2}$ is irrational — that is, $$x^2 - 2$$ has no solution in $\mathbb{Q}$ . The resolution was not to abandon the polynomial, but to enlarge the field. Field extensions formalize this enlargement and give us the structural scaffolding on which Galois theory is built.

I find it useful to think of a field extension as a kind of controlled inflation of a number system. We pump in just enough new elements to solve the equations we care about, and the tower law tells us exactly how much air we used. The resulting picture is much cleaner than I expected when I first met it: every step has a finite degree, the degrees multiply along chains, and the whole thing turns into linear algebra over the base field. This article develops the theory from the ground up: degrees and bases, simple extensions and minimal polynomials, the tower law, splitting fields, and separability. By the end, we will have the full toolkit needed to state and prove the Fundamental Theorem of Galois Theory in the next article.

Abstract Algebra (6): Polynomial Rings — Factorization and Unique Decomposition

Sat, 11 Sep 2021 09:00:00 +0000

Polynomials are the laboratory of algebra. Nearly every concept in ring theory — ideals, quotients, factorization, irreducibility — was first understood through polynomial examples before being abstracted. This is no coincidence: polynomial rings are rich enough to exhibit all the interesting phenomena yet structured enough to permit explicit computation.

In this article we study the ring $$R[x]$$ of polynomials over a ring $$R$$ . We develop the division algorithm, establish irreducibility criteria (Eisenstein, reduction mod $$p$$ , the rational root test), define Unique Factorization Domains, and prove Gauss’s Lemma — the bridge between factorization over $\mathbb{Z}$ and factorization over $\mathbb{Q}$ . The payoff is a clear picture of when and why unique factorization holds, and what goes wrong when it fails.

Abstract Algebra (5): Rings and Ideals — When Multiplication Enters the Picture

Thu, 09 Sep 2021 09:00:00 +0000

Groups capture symmetry through a single operation. But most of the number systems we actually compute with — integers, polynomials, matrices — carry two operations that interact: addition and multiplication. The moment you want to talk about divisibility, factorization, or solving equations, one operation is not enough. You need a ring.

This article develops ring theory from scratch: the axioms, the key examples, the pathologies that make ring theory richer (and harder) than group theory, and the central concept of an ideal — the ring-theoretic analogue of a normal subgroup. By the end you will have the language to state the First Isomorphism Theorem for rings and to understand why “modding out by an ideal” is the right way to build new rings from old ones.

Abstract Algebra (4): Sylow Theorems — Dissecting Finite Groups

Tue, 07 Sep 2021 09:00:00 +0000

Lagrange’s theorem tells you the order of any subgroup must divide $$|G|$$ . That is a necessary condition, and it is famously not sufficient — the alternating group $$A_4$$ has order $$12$$ , but no subgroup of order $$6$$ . So the moment you start asking “given $$|G| = n$$ , what does $$G$$ actually look like?”, Lagrange leaves you holding an empty bag.

The Sylow theorems are what go inside that bag. They say: for every maximal prime power $$p^a$$ dividing $$|G|$$ , a subgroup of order $$p^a$$ exists, all such subgroups are conjugate, and their count $$n_p$$ is sharply constrained ( $n_p \equiv 1 \pmod p$ , $n_p \mid [G:P]$ ). Ludwig Sylow proved this in 1872, and 150 years later it is still the first thing you reach for when somebody hands you a finite group of unknown order and asks what it is.

Abstract Algebra (3): Quotient Groups and Homomorphisms: The Art of Collapsing Structure

Sun, 05 Sep 2021 09:00:00 +0000

A group can be enormous — millions of elements, intricate multiplication tables, symmetries that take pages to describe. Yet hidden inside every group are natural compression points where you can collapse entire chunks of the group into single elements, producing a smaller group that still remembers something essential about the original. This article develops the machinery for doing that: normal subgroups, quotient groups, homomorphisms, and the isomorphism theorems that tie everything together.

Abstract Algebra (2): Group Actions — How Groups Move Things Around

Fri, 03 Sep 2021 09:00:00 +0000

From Abstract Groups to Concrete Actions#

Mental picture before any definitions: a group acting on a set is a collection of moves, and the set is the playground those moves permute. Pick up an object, do a move from the group, set it down. The orbit of an object is everywhere you can take it. The stabilizer is every move that puts it back exactly where it started.

Abstract Algebra (1): Groups — Your First Encounter with Algebraic Structure

Wed, 01 Sep 2021 09:00:00 +0000

Why Algebraic Structure Matters#

Before any definitions, here is the picture I want you to keep in mind. A group is a set in which you can combine any two elements to get a third, undo any element you have produced, and rearrange parentheses without consequence. That is the entire idea, dressed up. The rest of this article is a slow unpacking of that one sentence.