Projective geometry begins with a simple repair to Euclidean intuition. Parallel lines are an artifact of refusing to look far enough away. If you enlarge the plane by adding “directions” as legitimate points, then parallel lines meet, and many case distinctions disappear. This enlargement is not a trick; it is the natural setting for incidence, […]

Convexity is the geometric expression of “no dents.” It is a simple condition with a surprisingly rigid algebra around it. Once a set is convex, linear functionals see it cleanly, intersections behave predictably, and many existence questions reduce to finite combinatorics. Convex geometry is therefore less a catalog of shapes than a toolkit: a small […]

Geometry does not begin with coordinates. It begins with a notion of distance and with the idea that shortest paths should exist and behave predictably. Metric geometry isolates those ideas from any particular ambient space. It asks what can be proved from the metric axioms alone, what additional structure is needed to talk about “straightness,” […]

Differential geometry often feels difficult for two opposite reasons. At first, the subject can seem overloaded with coordinates and formulas. Later, after one learns enough invariant language, the subject can seem so abstract that geometric texture disappears. The method of moving frames is one of the rare tools that helps with both problems at once. […]

A large portion of differential geometry becomes much clearer when you understand one idea well: a connection is the device that tells you how to compare vectors living in different tangent spaces. Without that comparison rule, words like derivative, constant vector field along a curve, curvature, and even acceleration on a manifold become ambiguous. This […]

Differential geometry becomes much easier to hold in your mind when you stop treating it as a museum of definitions and start treating it as a disciplined way of reading shape. The central question is simple to say and difficult to answer well: how does a geometric object bend, and what can be known from […]

Algebraic geometry can feel like a mountain of definitions: varieties, schemes, morphisms, sheaves, divisors, line bundles, cohomology, moduli. Yet most effective problem solving in the subject runs on a small core. The core is not a list of theorems to memorize. It is a compact system of translations and a handful of structural lemmas that […]

Algebraic geometry has a reputation for proofs that feel like magic: a claim about geometry turns into a ring computation, a local argument becomes global by gluing, and a subtle fiber statement becomes a clean inequality about dimensions. Most successful arguments in the subject are built from a small number of reusable proof patterns. You […]

Algebraic geometry is precise enough to prove deep theorems, and subtle enough that a small mismatch of hypotheses can invalidate an argument without any obvious sign. Most recurring errors are not about missing a clever idea. They are about silently switching languages: treating a geometric statement as if it were a ring-theoretic statement, or treating […]

Differential geometry rewards precision and punishes casual habits. Many mistakes are not “small errors” but category errors: confusing intrinsic and extrinsic data, confusing tensors with their coordinate expressions, or silently changing conventions mid-proof. The good news is that most recurring mistakes are predictable. If you learn the failure modes, you start writing and reading arguments […]

Moduli is where algebraic geometry stops being “a dictionary between equations and shapes” and becomes a discipline about families. Instead of studying a single curve, a single surface, or a single vector bundle, you study all of them at once in a controlled way, and you ask for a parameter space that records how they […]

Algebraic geometry is often introduced as “the study of solutions to polynomial equations.” That is true, but it undersells what the subject becomes once you learn its grammar. The real power is that algebraic geometry gives you a language for turning vague geometric intuitions into statements that are rigid enough to prove, stable enough to […]