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Concepts39

Groups

📐Linear Algebra15 📈Calculus & Differentiation10 🎯Optimization14 🎲Probability Theory12 📊Statistics for ML9 📡Information Theory10 🔺Convex Optimization7 🔢Numerical Methods6 🕸Graph Theory for Deep Learning6 🔵Topology for ML5 🌐Differential Geometry6 ∞Measure Theory & Functional Analysis6 🎰Random Matrix Theory5 🌊Fourier Analysis & Signal Processing9 🎰Sampling & Monte Carlo Methods10 🧠Deep Learning Theory12 🛡️Regularization Theory11 👁️Attention & Transformer Theory10 🎨Generative Model Theory11 🔮Representation Learning10 🎮Reinforcement Learning Mathematics9 🔄Variational Methods8 📉Loss Functions & Objectives10 ⏱️Sequence & Temporal Models8 💎Geometric Deep Learning8

Category

🔷All ∑Math ⚙️Algo 🗂️DS 📚Theory

Level

All Beginner Intermediate

∑MathAdvanced

Lie Groups & Lie Algebras

Lie groups model continuous symmetries like rotations and rigid-body motions, combining algebra (group law) and calculus (smooth manifolds).

#lie group#lie algebra#so(3)+12

∑MathAdvanced

Curvature

Curvature measures how a geometric object bends, and it comes in several flavors: Gaussian, sectional, and Ricci curvature.

#gaussian curvature#sectional curvature

#ricci curvature

+11

∑MathAdvanced

Geodesics & Exponential Map

Geodesics are the “straightest possible” paths on curved spaces (manifolds) and locally minimize distance.

#geodesic#exponential map#riemannian metric+12

∑MathAdvanced

Smooth Manifolds & Tangent Spaces

A smooth manifold is a space that looks like ordinary Euclidean space when you zoom in, glued together using charts that transition smoothly.

#smooth manifolds#tangent space#chart+11

∑MathAdvanced

Betti Numbers

Betti numbers count independent k-dimensional holes: β₀ counts connected components, β₁ counts independent loops/tunnels, and β₂ counts voids.

#betti numbers#homology#simplicial complex+12

∑MathAdvanced

Persistent Homology

Persistent homology tracks how topological features (components, loops, voids) appear and disappear as you grow a scale parameter over a filtered simplicial complex.

#persistent homology#filtration#vietoris-rips+12

∑MathAdvanced

Manifolds & Manifold Hypothesis

A manifold is a space that locally looks like Euclidean space, stitched together by coordinate charts and smooth transition maps.

#manifold#topological manifold#smooth manifold+12

∑MathAdvanced

Topological Spaces & Continuity

A topological space abstracts the idea of “closeness” using open sets instead of distances, allowing geometry without measuring lengths.

#topological space#open set#continuity+12

∑MathAdvanced

KKT Conditions

KKT conditions generalize Lagrange multipliers to handle inequality constraints in constrained optimization problems.

#kkt conditions#lagrangian#complementary slackness+12

∑MathAdvanced

Floor Sum Formula

The floor sum computes S(n,m,a,b) = sum_{i=0}^{n-1} floor((a i + b)/m) efficiently in O(log(min(a,m))) time.

#floor sum#atcoder library#euclidean algorithm+12

∑MathAdvanced

Generating Functions - EGF

Exponential generating functions (EGFs) encode a sequence (a_n) as A(x) = \sum_{n \ge 0} a_n \frac{x^n}{n!}, which naturally models labeled combinatorial objects.

#exponential generating function#egf#binomial convolution+11

∑MathAdvanced

Pólya Enumeration

Pólya Enumeration Theorem generalizes Burnside’s Lemma by turning counting under symmetry into a polynomial substitution problem.

#pólya enumeration#cycle index#burnside lemma+12