A Universe of Sorts
§ Siddharth Bhat
 Why quaternions work better
 DFA to CFG via colimits?
 Why pointless topology is powerful
 Denotational semantics in a few sentences
 Monge Matrix
 Fixpoint as decorator
 Combinatorial generation algorithms
 Perform DP on measures, not indexes.
 Alternative version of MyhillNerode
 Polya Enumeration
 Weighted Burnside Lemma
 Cycle index polynomial
 Mnemonics For Symmetric Polynomials
 Uses of minimal string rotation
 Suffix Automata
 Simpson's Paradox
 Myhill Nerode Theorem
 Linearity of expectation for sampling
 Dedekind MacNiellie
 Good and bad combinatorics: intro to counting
 Expected number of turns to generate all numbers
1..N
(WIP)  Diameter in single DFS (WIP)
 Min cost flow (WIP)
 Clojure: minimal makefile for REPL driven dev with Neovim
 Delimited continuations
 Never forget monic again
 Weird canonical example of monic and epic: left/right shift
 Playing guitar: being okay with incorrect chords
 Sparse table
 Duval's algorithm
 Amortized complexity from the verifier perspective
 Relationship betwee permutations and runs
 Brouwer's fixed point theorem
 XOR on binary trie
 Inconvergent: beautiful generative art
 Prefix/Border function
 Shortest walk versus shortest path
 Minimal tech stack
 FFT
 Zscoder codeforces rating
 Continuum TTRPG
 Words to know in target language
 DP on subarrays
 Vis editor cheat sheet
 Mean, Median and Jensen's
 The similarity between labellings and representations
 L1 norm is greater than or equal to L2 norm
 Z algorithm
 For a given recurrence, what base cases do I need to implement?
 Number of distinct numbers in a partition
 Splitting $f(x) = y$ into indicators
 Why searching for divisors upto
sqrt(n)
works  Heuristics for the prime number theorem
 Sum of absolute differences of an array
 GCD is at most difference of numbers
 implementing GCD and LCM
 Centroid of a tree
 Center of a tree
 Image unshredding as hamiltonian path
 Distance between lines in nD

lower_bound
binary search with closed intervals  Sliding window implementation style
 Kawaii implementation of
x = min(x, y)
 CSES: Counting Towers
 Smallest positive natural which can't be represented as sum of any subset of a set of naturals
 Example of RVs that are pairwise but not 3way independent.
 Notes on Liam O Connor's thesis: Cogent
 C++
lower_bound
, upper_bound
API  Books that impart mental models
 Subarrays ~= prefixes
 Operations with modular fractions
 Modular inverse calculation
 The number of pairs
(a,b)
such that ab≤x
is O(xlogx)
 DP as path independence
 Binary search to find rightmost index which does not possess some property
 Correctness of
lower_bound
search with halfopen intervals  Greedy Coin change: proof by probing
 Clean way to write burnside lemma
 The groupoid interpretation of type theory
 Mnemonics for free = left adjoint
 Where to scratch a cat
 Mnemonic for Specht module actions
 Quotes from 'Braiding Sweetgrass'
 Transfinite recursion: Proof
 Transfinite induction: Proof
 Thoughts on playing EmBm
 An explanation for why permutations and linear orders are not naturally isomorphic
 We can't define choice for finite sets in Haskell!
 Geomean is scale independent
 Thoughts on playing Em Bm.
 Induction on natural numbers cannot be derived from other axioms
 Ordinals and cardinals
 Musing about Specht modules
 Every continuous function on $[a, b]$ attains a maximum
 Invisible cities
 Associativity of addition in cubicaltt
 Etymology of fiber bundle $F \rightarrow E \rightarrow B$
 Galois correspondence, functorially
 CubicalTT: sharpening thinking about indexed functions
 Functors to motivate adjuntions
 Madoka Magica: plot thoughts
 Chain rule functorially
 Lagrange multipliers by algebra
 Specht module construction
 Even and odd functions through representation theory
 Greg egan: Orthogonal
 Simplicial approximation: maps can be approximated by simplicial maps (TODO)
 Excision (TODO)
 Marshall: Andrej (TODO)
 Limit is right adjoint to diagonal
 Working out why right adjoints preserve limits.
 Limit/Colimit/Cone/Cocone: the arrows are consistent!
 Representable Functors
 Why terminal object is a limit
 Excluded middle is not false in intuitionistic logic
 Yoneda Lemma and embedding
 GHCID
 Character theory
 Cofibration
 Emily Riehl Contrability as uniqueness
 Cofactor as derivative of determinant
 Homology, the big picture
 Normaliztion by evaluation (TODO)
 Legal Systems very different from ours
 Lefschetz fixed point theorem (TODO)
 Shrinking wedge of circles / Hawaiian earring (TODO)
 Simplicial approxmation of maps (TODO)
 Lebesgue number lemma (TODO)
 Lean internals Cheat Sheet
 MicroUI
 Proof of tree having (V1) edges
 Creating PDFs to read code
 Bias and gain
 Barycentric subdivision: edge length decreases
 Homotopic maps produce same singular homology: Intuition
 Singular homology: induced homomorphism
 Demoscene tools
 Binaural Beat
 Low pass filter by delaying
 Octaves are double frequency apart (TODO)
 Bias and gain
 Show, don't tell
 Try and think of natural transformations as intertwinings
 Subobject classifier measures how much we need to pay to access fact
 Spectral norm of Hermitian matrix equals largest eigenvalue (TODO)
 Penrose cohomology [TODO ]
 Bicycle wheel proof of Gauss Bonnet (TODO)
 What is Levi Cevita trying to describe (TODO)
 Torsion as giving monodromy of path lifts (TODO)
 Cartan's spiral staircase (TODO)
 Dupin indicatrix (TODO)
 Rodrigues curvature formula (TODO)
 Weingarten map (TODO)
 When maps cannot be lifted to the universal cover
 Nets from Munkres (TODO)
 Limit point compactness from Munkres
 Proof of Heine Borel from Munkres (compact iff closed, bounded)
 Alexandrov topology
 Zeroth singular homology group: Intuition
 Examples of fiber products / pullbacks
 Covariant derivative
 Clackety sounds:
bucklespring
 Submersions and immersions
 Ehrsmann connection
 Quotes from the culture
 Lie bracket commutator from exponentiation (TODO)
 Thoughts on proof of fundamental group of unit circle
 Pasting lemma
 Tensoring with base ring has no effect
 Seeing the semidirect product of the dihedral group.
 Animating rotations with quaternion curves
 Mnemonic for homtensor and leftright adjoints
 Construction of tensor product: Atiyah macdonald
 Recovering topology from sheaf of functions: Proof from Atiyah Macdonald
 Urhyson's lemma
 Compact Hausdorff spaces are normal
 Stone representation theorem: Proof from Atiyah Macdonald
 Covariant Hom is left exact
 Internal versus External semidirect products
 Splitting of semidirect products in terms of projections
 Tensor is right exact
 Semidirect product as commuting conditions
 Exact sequences for semidirect products; fiber bundles
 Semidirect product is equivalent to splitting of exact sequence
 Intro to topological quantum field theory
 Non examples of algebraic varieties
 Nilradical is intersection of all prime ideals
 Exactness of modules is local
 Quotient by maximal ideal gives a field
 Ring of power series with infinite positive and negative terms
 Mean value theorem and Taylor's theorem. (TODO)
 Cayley Hamilton
 Nakayama's lemma
 Vector fields over the 2 sphere
 Learning to talk with your hands
 Lovecraftisms
 Hairy ball theorem from Sperner's Lemma (TODO)
 CS and type theory: Talks by vovodesky
 Hilbert basis theorem for polynomial rings over fields (TODO)
 Covering spaces
 Wedge Sum and Smash Product
 Quotient topology
 CW Complexes and HEP
 Stable homotopy theory
 Simply connected spaces
 Finitely generated as vector space v/s algebra:
 Weak and Strong Nullstllensatz
 Screen recording for kakoune pull request
 Intuition for why finitely presented abelian groups are isomorphic to product of cyclics
 Euler characteristic of sphere
 John Conway: The symmetries of things
 Semidirect product mnemonic
 Non orthogonal projections
 Why did maxwell choose his EM wave to be light?
 Fast string concatenation in python3
 Split infinitive
 Yoneda from string concatenation
 Right Kan extensions as extending the domain of a functor
 Non standard inner products and unitarity of representations
 take at most 4 letters from 15 letters.
 Flat functions
 Hopf Algebras and combinatorics
 Butcher group
 Neovim frontends
 A semidirect product worked on in great detail
 Direct and Inverse limits
 LEAN 4 overfrom from LEAN together 2021
 BLM master thesis
 Hololive subculture
 RSK correspondence for permutations
 Djikstra's using a segtree
 Markov and chebyshev from a measure theoretic lens
 Among any 51 integers, that are 2 with squares having equal value modulo 100
 $1^n + 2^n + \dots + (n1)^n$ is divisible by $n$ for odd $n$
 $10^{3n+1}$ cannot be written as sum of two cubes
 Coqclub: the meaning of a specification
 SQLite opening
 Old school fonts
 Stalking
syzigies
on hackernews  Conditional probability is neither causal nor temporal
 Hook length formula
 The tyranny of light
 Muirhead's inequality
 Rearrangement inequality
 Triangle inequality
 The Heather subculture
 Frobenius Kernel
 Galois theory by "Abel's theorem in problems and solutions"
 Galois theory perspective of the quadratic equation
 Burnside lemma by representation theory.
 Contributing to SAGEmath
 Shadow puppet analogy for entanglement
 Books for contest math
 Analysing simple games
 Linear algebraic proof of the handshaking lemma
 Historical contemporaries
 Rota's twelvefold way
 Counting necklackes with unique elements
 Decomposition of projective space
 Childhood: Playing pokemon gold in japanese
 Tensor is a thing that transforms like a tensor
 Tensor Hom adjunction
 Schur's lemma
 Daughters of destiny
 Stuff I learnt in 2020
 Line bundles, a high level view as I understand them today
 Conversations with a wood carver
 Discrete Riemann Roch
 Conversation with Olaf Klinke
 Topological groups and languages
 The mnemonica stack (TODO)
 Conversation with Alok about how I read
 KMP (Knuth, Morris, Pratt) (TODO)
 Reading C declarations
 Make mnemonics
 Vandermonde and FFT
 Thoughts on blitz chess: 950 ELO
 Periodic tables and make illegal states unrepresentable
 questions on the structure of graphs
 Combinations notation in bijective combinatorics
 Arguments for little endian
 Expectiles
 Depth first search through linear algebra (TODO)
 2SAT
 Longest increasing subsequence, step by step (TODO)
 On reading how to rule (TODO)
 Strongly Connected Components via Kosaraju's algorithm
 Articulation points
 Disjoint set union
 Making GDB usable
 Bouncing light clock is an hourglass
 Euler tours
 Representation theory of the symmetric group (TODO)
 Maximum matchings in bipartite graphs
 padics, 2's complement, intuition for bit fiddling
 Diameter of a tree
 Catalan numbers as popular candidate votes (TODO)
 The chromatic polynomial (TODO)
 Structure theory of finite endofunctions
 Number of paths in a DAG
 Set partitions
 Integer partitions: Recurrence
 Stars and bars by direct bijection
 DFS and topological sorting
 Tournaments
 Matching problems (TODO)
 Four fundamental subspaces
 WHO list of essential medicines (TODO)
 why is
int i = i
allowed in C++?  Kakoune cheatsheet
 Assembly IDE
 Cohomology is like holism
 Flows (TODO)
 Amortized analysis
 Shelly Kegan: death  Suicide and rationality (TODO)
 Sam harris and jordan peterson: Vancouver 1 (TODO)
 Correctness of binary search

readlink f
to access file path  rank/select as compress/decompress
 Remembering Eulerian and Hamiltonian cycles
 Nice way to loop over an array in reverse
 Dynamic Programming: Erik Demaine's lectures
 Accuracy vs precision
 Why is the gradient covariant?
 Politicization of science
 Multi ꙮ cular O: ꙮ / Eye of cthulu
 You can't measure the one way speed of light
 Show me the hand strategy
 Words that can be distinguished from letters if we know the sign of the permutation
 Easy times don't create weak people, they just allow weak people to survive.
 Multiplicative weights algorithm (TODO)
 How to fairly compare groups
 Bijection from
(0, 1)
to [0, 1]
 Rene Girard
 Noam Chomsky on anarchism (TODO)
 Slavoj Zizek: Violence
 Poverty: Who's to blame?
 Learn Zig in Y minutes
 The algebraic structure of the 'nearest smaller number' question
 Why loss of information is terrifying: Checking that a contextfree language is regular is undecidable
 Sciences of the artificial
 Numbering nodes in a tree
 Number of vertices in a rooted tree
 Median minimizes L1 norm
 LISP quine
 A slew of order theoretic and graph theoretic results
 Thebes
 Beethoven
 Neko to follow your cursor around
 Non commuting observables: Light polarization
 Statement expressions and other GCC C extensions
 A quick look at impredicativity
 Data oriented programming in C++
 Retro glitch
 SSA as linear typed language
 Nix weirdness on small machines
 Autodiff over derivative of integrals
 Proof of projective duality
 Preventing the collapse of civilization
 Violent deaths in ancient societies (TODO)
 An elementary example of a thing that is not a vector
 Elementary probability theory (TODO)
 The handshaking lemma
 Git for pure mathematicians
 Mutorch
 Computing the smith normal form
 Laziness for C programmers
 Exact sequence of pointed sets
 What is a syzygy?
 Under the spell of Leibniz's dream
 Normal operators: Decomposition into Hermitian operators
 Readable pointers
 The grassmanian, handwavily
 Lie bracket as linearization of conjugation
 Computational Origami
 Katex in duktape
 Kebab case
 Localization: Introducing epsilons (TODO)
 NaN punning: Storing integers in doubles in JavaScript
 Offline Documentation
 Using Gurobi
 osqp: convex optimizer in 6000 LoC
 stars and bars by generating functions
 This is not a place of honor
 Topological proof of infinitude of primes
 Burnside Theorem
 The Ise Grand shrine
 Edward Kmett's list of useful math
 Cokernel is not sheafy
 Von neumann: foundations of QM
 Discrete schild's ladder
 Derivative of step is dirac delta
 Extended euclidian algorithm
 In a PID, all prime ideals are maximal, geometrically
 Prime numbers as maximal among principal ideals
 Axiom of Choice and Zorn's Lemma
 Local ring in terms of invertibility
 Nullstellensatz for schemes
 Perspectives on Yoneda
 Germs, Stalks, Sheaves of differentiable functions
 Connectedness in terms of continuity
 Intuition for limits in category theory
 Finite topologies and DFS numbering
 Categorical definition of products in painful detail
 Why is the spectrum of a ring called so?
 Ergo proxy
 Satisfied and frustrated equations
 Combinatorial intuition for Fermat's little theorem
 An incorrect derivation of special relativity in 1D
 The geometry and dynamics of magnetic monopoles
 Sanskrit and Sumerian
 Writing Cuneiform
 The code of hammurabi
 The implicit and inverse function theorem
 Whalesong hyperbolic space in detail
 Motivating Djikstra's
 Intuitions for hyperbolic space
 Product of compact spaces in compact
 Hyperbolic groups have solvable word problem
 Elementary uses of Sheaves in complex analysis
 Snake lemma
 Kernel, cokernel, image
 The commutator subgroup
 Simplicity of A5 using PSL(2, 5)
 A5 is not solvable
 Complex orthogonality in terms of projective geometry
 Arithmetic sequences, number of integers in a closed interval
 The arg function, continuity, orientation
 Odd partitions, unique partitions
 Continued fractions, mobius transformations
 Permutationsandlyndonfactorization
 Graphs are preorders
 Crash course on DCPO: formalizing lambda calculus
 Parallelisable version of maximum sum subarray
 Thoughts on implicit heaps
 Discriminant and Resultant
 Polynomial root finding using QR decomposition
 A hacker's guide to numerical analysis
 Mobius inversion on Incidence Algebras
 Finite differences and Umbral calculus
 Permutahedron
 Lyndon + Christoffel = Convex Hull
 Geometric proof of
e^x >= 1+x
, e^(x) >= 1x
 Ranking and Sorting
 Proof of minkowski convex body theorem
 Burrows Wheeler
 Intuitionstic logic as a Heytig algebra
 Edit distance
 Evolution of bee colonies (TODO)
 Best practices for array indexing
 Algebraic structure for vector clocks
 Networks are now faster than disks
 Einsteinde Haas effect
 Rankselect as adjunction
 Bounding chains: uniformly sample colorings
 Coupling from the past
 Word problems in Russia and America
 Encoding mathematical hieararchies
 Learning code by hearing it
 Your arm can be a spinor
 Self modifying code for function calls: Look ma, I don't need a stack!
 Adjunctions as advice
 Reversible computation as groups on programs
 Blazing fast math rendering on the web
 VC dimension
 Symplectic version of classical mechanics
 Theorems for free
 How to reason with halfopen intervals
 How does one build a fusion bomb?
 Christoffel symbols, geometrically
 A natural vector space without an explicit basis
 Cache oblivious B trees
 KrohnRhodes decomposition
 Proving block matmul using program analysis
 Why I like algebra over analysis

using
for cleaner function type typedefs  A walkway of lanterns (TODO)
 Natural transformations
 The hilarious commentary by dinosaure in OCaml git
 How to link against MLIR with CMake
 Energy as triangulaizing state space
 The cutest way to write semidirect products
 My Favourite APLisms
 Proof of chinese remainder theorem on rings
 monic and epic arrows
 The geometry of Lagrange multipliers
 Efficient tree transformations on GPUs (TODO)
 Things I wish I knew when I was learning APL
 Every ideal that is maximal wrt. being disjoint from a multiplicative subset is prime
 Getting started with APL
 SpaceChem was the best compiler I ever used
 Mnemonic for Kruskal and Prim
 Legendre transform
 Cartesian Trees
 DFS numbers as a monotone map
 Self attention? not really
 Coarse structures
 Matroids for greedy algorithms (TODO)
 Grokking Zariski
 My preferred version of quicksort
 Geometric proof of Cauchy Schwarz inequality
 Dataflow analysis using Grobner basis
 Fenwick trees and orbits
 Dirichlet inversion
 Incunabulum for the 21st century: Making the J interpreter compile in 2020
 An example of a sequence whose successive terms get closer together but isn't Cauchy (does not converge)
 Krylov subspace method
 Good reference to the Rete pattern matching algorithm
 Leapfrog Integration
 Comparison of forward and reverse mode AD
 An invitation to homology and cohomology, Part 1  Homology
 An invitation to homology and cohomology, Part 2  Cohomology
 Stuff I learnt in 2019
 A motivation for padic analysis
 Line of investigation to build physical intuition for semidirect products
 Topology is really about computation  part 2
 Topology is really about computation  part 1
 PSLQ algorithm: finding integer relations between reals
 Geometric characterization of normal subgroups
 Handy characterization of adding an element into an ideal, proof that maximal ideal is prime
 Radical ideals, nilpotents, and reduced rings
 My disenchantment with abstract interpretation
 Computing equivalent gate sets using grobner bases
 The janus programming language  Time reversible computation

A = B
 A book about proofs of combinatorial closed forms  Generating
k
bitsets of a given length n
:  Bondi kcalculus
 Topology as an object telling us what zerolocus is closed:
 Vivado toolchain craziness
 What the hell is a Grobner basis? Ideals as rewrite systems
 Lie bracket versus torsion
 Blog post: Weekend paper replication of STOKE, the stochastic superoptimizer
 Collapsing
BlockId
, Label
, Unique
:  Spatial partitioning data structures in molecular dynamics
 Vector: Arthur Whitney and text editors
 Representing CPS in LLVM using the
@coro.*
intrinsics  Bug in the LLVM code generator: Lowering of
MO_Add2
and MO_AddWordC
 Discrete random distributions with conditioning in 20 lines of haskell
 Everything you know about word2vec is wrong
 Hamiltonian monte carlo, leapfrog integrators, and sympletic geometry
 Small Haskell MCMC implementation
 The smallest implementation of reverse mode AD (autograd) ever:
 Timings of passes in GHC, and low hanging fruit in the backend:
 Varargs in GHC:
T7160.hs
 Debugging debug info in GHC
 GHC LLVM code generator: Switch to unreachable
 Concurrency in Haskell
 Handy list of differential geometry definitions
 Lazy programs have space leaks, Strict programs have time leaks
 Presburger arithmetic can represent the Collatz Conjecture
 Using compactness to argue about covers
 Japanese Financial Counting system
 Stephen wolfram's live stream

Cleave
as a word has some of the most irregular inflections  McCune's single axiom for group theory

Word2Vec
C code implements gradient descent really weirdly  Arthur Whitney: dense code
 How does one work with arrays in a linear language?
 Linear optimisation is the same as linear feasibility checking
 Quantum computation without complex numbers
 Linguistic fun fact: Comparative Illusion
 Longform posts:
 Emacs Cheat Sheet
 Coq Cheat Sheet
 Writing Cheat Sheet
 Latex Cheat Sheet
 Architecture Cheat Sheet
 Recipes Cheat Sheet
 History Cheat Sheet
 Words Cheat Sheet
 Clojure Sheat Sheet
 Big list of quotes
 Empathy
 Vim Cheat Sheet
 Big list of Chess
 Big list of shitposting
 Cardistry
 Poems to memorize
 X86 Cheat Sheet
 Common Lisp Cheat Sheet
 Agda Cheat Sheet
 Big list of Hacker news
 Favourite Demoscenes