# Disclaimer

Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references or other sources. Moreover, after upload I am not able to change the YouTube videos anymore, so take everything with a pinch of salt. I can change the slides, so let me know if you have spotted any typos. In any case, here is a list all my YouTube videos, including slides. The order below is by playlist/topic, newest first:

- My favorite theorems
- What is...representation theory?
- What is...category theory?
- What is...algebraic topology?
- What is...algebra?
- What is...linear algebra?
- For my talks see here, for quantum topology see here, for Soergel bimodules see here

# A list of my videos

- Topic: What is...Kronecker's theorem? - Or: Darts and polynomials. Slides
- Topic: What is...the Robinson-Schensted correspondence? - Or: Boxes and permutations. Slides
- Topic: What is...a greedy algorithm? - Or: Greed is good! Well... Slides
- Topic: What is...the BEST theorem? - Or: de [B]ruijn, van Aardenne-[E]hrenfest, [S]mith, [T]utte. Slides
- Topic: What are...set partitions? - Or: Prototypical lattices!? Slides
- Topic: What is...Frucht's theorem? - Or: Graphs and symmetries. Slides
- Topic: What are...randomized algorithms? - Or: Randomness rocks! Slides
- Topic: What is...the Gauss-Bonnet theorem? - Or: Curvature and Euler. Slides
- Topic: What are...prime knots? - Or: The prime numbers of knot theory!? Slides
- Topic: What are...operads? - Or: Topological algebra. Slides
- Topic: What is...Maschke's theorem? - Or: Blocks out of the blue. Slides
- Topic: What is...Kazhdan-Lusztig combinatorics? - Or: It is nonnegative!? Slides
- Topic: What is...a Voronoi diagram? - Or: Distance diagrammatically. Slides
- Topic: What is...the Tutte polynomial? - Or: Counting using polynomials. Slides
- Topic: What is...a Seifert surface? - Or: Soap and knots. Slides
- Topic: What is...sphere packing? - Or: Honeycombs in higher dimensions. Slides
- Topic: What is...Kolmogorov complexity? - Or: Shakespeare's work is not random. Slides
- Topic: What is...the gnu function? - Or: Symmetries are wild. Slides
- Topic: What is...a tropical curve? - Or: Straight curves. Slides
- Topic: What is...Bézout's theorem? - Or: Counting intersections. Slides
- Topic: What is...fuzzy logic? - Or: This video is 90% crap. Slides
- Topic: What is...the Gauss-Wantzel theorem? - Or: Why 65537? Slides
- Topic: What is...the small-world effect? - Or: Everyone knows everyone!? Slides
- Topic: What are...hyperbolic groups? - Or: From groups to Cantor sets. Slides
- Topic: What are...the three geometries? - Or: 0, 1 and \(\infty\). Slides
- Topic: What are...Chaitin's constants? - Or: Computing a glimpse of randomness. Slides
- Topic: What is...a coherence theorem? - Or: Don't underestimate associativity. Slides
- Topic: What is...Diffie-Hellman key exchange? - Or: How not to transfer the encryption key. Slides
- Topic: What are...Ramsey numbers? - Or: Counting made difficult. Slides
- Topic: What is...the Gessel-Viennot lemma? - Or: Determinants are graphs?! Slides
- Topic: What is...elliptic addition? - Or: Torus games. Slides
- Topic: What is...the fifteen theorem? - Or: What is special about 1,2,3,5,6,7,10,14,15? Slides
- Topic: What is...roundness? - Or: My circle has corners! Slides
- Topic: What is...the Jordan curve theorem? - Or: Come on, that's trivial... Slides
- Topic: What is...a Coxeter group? - Or: What is...1,$\infty$,3,5,3,4,4,4,3,3,3,3,...? Slides
- Topic: What is...the difference between Eulerian and Hamiltonian graphs? - Or: They are not dual!? Slides
- Topic: What is...the curse of dimensionality? - Or: Hyperballs do not exist! Well, kind of... Slides
- Topic: What is...Polya's theorem? - Or: Birds get lost. Slides
- Topic: What are...Chebyshev polynomials? - Or: Fibonacci with signs? Slides
- Topic: What is...reverse mathematics? - Or: Thinking backwards. Slides
- Topic: What is...the axiom of choice? - Or: It works even if you donâ€™t believe in it. Slides
- Topic: What is...the Mason-Stothers theorem? - Or: The ABC of polynomials. Slides
- Topic: What is...the pea and the sun paradox? - Or: One orange equals two oranges. Slides
- Topic: What is...the Robertson-Seymour theorem? - Or: Minors are majors. Slides
- Topic: What is...the Kuratowski-Wagner theorem? - Or: Forbidden pieces. Slides
- Topic: What is...the Jones polynomial? - Or: Resolving crossings. Slides
- Topic: What are...Catalan numbers? - Or: In praise of counting. Slides
- Topic: What is...the Abel-Ruffini theorem? - Or: Loops and roots. Slides
- Topic: What is...the art gallery theorem? - Or: How many cameras suffice? Slides
- Topic: What is...the finite Kakeya problem? - Or: A finite filling. Slides
- Topic: What is...Pick's theorem? - Or: Surprisingly simple. Slides
- Topic: What is...Kneser's conjecture? - Or: Coloring and topology. Slides
- Topic: What are...incarnations of the Borsuk-Ulam theorem? - Or: Topology or combinatorics or...? Slides
- Topic: What is...the Borromean impossibility theorem? - Or: Borromean rings do not exist! Slides
- Topic: What are...p-adic integers? - Or: Climbing infinite trees. Slides
- Topic: What are...Steiner systems? - Or: Finite geometry and puzzles. Slides
- Topic: What is...Heawood's conjecture? - Or: I need more than four colors. Slides
- Topic: What is...the Strassen algorithm? - Or: Divide and conquer. Slides
- Topic: What are...Gröbner bases? - Or: Minimal intersections. Slides
- Topic: What is...the chaos game? - Or: Deterministic versus random. Slides
- Topic: What is...the art of locating roots? - Or: Finding roots without finding them. Slides
- Topic: What is...the Schwartz-Zippel lemma? - Or: The art of not solving equations. Slides
- Topic: What are...random numbers? - Or: Compressible? Slides
- Topic: What is...Matiyasevich's theorem? - Or: I can't decide... Slides
- Topic: What is...the Burnside problem? - Or: Obviously not! Ah, well...? Slides
- Topic: What is...a cellular automaton? - Or: From simple rules to life. Slides
- Topic: What is...the diamond lemma? - Or: Can you win? Slides
- Topic: What is...Goodstein's theorem? - Or: Killing hydras. Slides
- Topic: What are...Monte Carlo experiments? - Or: The shotgun method. Slides
- Topic: What are...Artin braid groups? - Or: Topology via algebra. Slides
- Topic: What is...Frobenius' density theorem? - Or: Does it factor? Slides
- Topic: What is...an ADE classification? - Or: A strange reoccurring pattern. Slides
- Topic: What is...the discrete periodic table? - Or: Finite simple groups. Slides
- Topic: What is...the smooth periodic table? - Or: Simple Lie groups. Slides
- Topic: What is...the strong law of small numbers? - Or: There are not enough small numbers. Slides
- Topic: What is...Arrow's theorem? - Or: Why voting is always flawed (and is still a good idea). Slides
- Topic: What are...the seventeen wallpaper groups? - Or: Mathematics, architecture and decorative art. Slides
- Topic: What is...the philosophy of generating functions? - Or: How to encode counting problems. Slides
- Topic: What is...the Reidemeister theorem? - Or: Knotting stings in 2d. Slides
- Topic: What is...Euler's polyhedron formula? - Or: 2000 years of not doing the count. Slides
- Topic: What is...the Brouwer fixed point theorem? - Or: Why lines need to cross. Slides
- Topic: What is...the Lucas theorem? - Or: Playing with digits. Slides
- Topic: What is...the Gershgorin circle theorem? - Or: Circling the eigenvalues. Slides
- Topic: What is...the Perron-Frobenius theorem? - Or: The leading terms. Slides

- Topic: What is...the Young lattice? - Or: Representations of symmetric groups, part 4. Slides
- Topic: What is...Frobenius' character formula? - Or: Representations of symmetric groups, part 3. Slides
- Topic: What are...Specht modules? - Or: Representations of symmetric groups, part 2. Slides
- Topic: What are...Young diagrams? - Or: Representations of symmetric groups, part 1. Slides
- Topic: What is...Frobenius reciprocity? - Or: Back-and-forth. Slides
- Topic: What are...induction and restriction? - Or: How to find lost information! Kind of... Slides
- Topic: What is...Burnside's theorem? - Or: Groups and representations. Slides
- Topic: What is...Lagrange's theorem for representations? - Or: Divisibility. Slides
- Topic: What is...finite Fourier analysis? - Or: Fourier and finite groups. Slides
- Topic: What is...the dual group? - Or: Groups of characters. Slides
- Topic: What is...the regular representation? - Or: Action on itself. Slides
- Topic: What are...Schur's orthogonality relations? - Or: An orthonormal basis. Slides
- Topic: What are...examples of character tables? - Or: The ATLAS of finite groups and co. Slides
- Topic: What is...a character table? - Or: The gist of the matter!? Slides
- Topic: What are...characters? - Or: Polynomials! Slides
- Topic: What is...Schur's lemma? - Or: Matrices rarely commute. Slides
- Topic: What is...semisimplicity? - Or: The finite group miracle. Slides
- Topic: What is...the Jordan-Hölder theorem? - Or: Like prime numbers. Slides
- Topic: What are...indecomposable representations? - Or: The elements! Well, kind of... Slides
- Topic: What is...a simple representation? - Or: The elements! Slides
- Topic: What are...intertwiners? - Or: Maps between representations. Slides
- Topic: What are...modules? - Or: Linear symmetries. Slides
- Topic: What are...representations? - Or: Matrices are cool! Slides
- Topic: What is...representation theory? - Or: Lets make life linear! Slides

- Topic: What are...props? - Or: Applications 2 (category theory in signal processing). Slides
- Topic: What are...TQFTs? - Or: Applications 1 (category theory in mathematics). Slides
- Topic: What are...2-categories? - Or: Categories in categories. Slides
- Topic: What are...braided categories? - Or: Braids and categories. Slides
- Topic: What is...duality in categories? - Or: Rigid and pivotal categories. Slides
- Topic: What are...string diagrams, take 2? - Or: Two-dimensional algebra. Slides
- Topic: What is...strictification? - Or: MacLane's coherence theorem. Slides
- Topic: What are...monoidal categories? - Or: Monoids categorified. Slides
- Topic: What are...Kan extensions? - Or: Recover information; kind of... Slides
- Topic: What is...Beck's monadicity theorem? - Or: Categories via actions. Slides
- Topic: What are...algebras? - Or: Actions in action. Slides
- Topic: What are...monads? - Or: The natural numbers as functors. Slides
- Topic: What are...adjoint functors diagrammatically? - Or: Zigzag! Slides
- Topic: What are...string diagrams? - Or: Diagrammatic algebra. Slides
- Topic: What is...the adjoint functor theorem? - Or: Identities are free! Slides
- Topic: What are...examples of adjoint functors? - Or: Adjoints occur almost everywhere. Slides
- Topic: What are...adjoint functors? - Or: Life is not invertible. Slides
- Topic: What are...continuous functors? - Or: Functors not functions. Slides
- Topic: What is...a complete category? - Or: Real numbers!? Slides
- Topic: What are...limits? - Or: Universal diagrams. Slides
- Topic: What are...universal properties? - Or: Abstract nonsense in perfection. Slides
- Topic: What are...(co)equalizers? - Or: Why kernels? Slides
- Topic: What are...pushouts and pullbacks? - Or: Faces “equal” relations. Slides
- Topic: What are...products and coproducts? - Or: Vector spaces rock! Slides
- Topic: What are...initial and terminal objects? - Or: Another reason why zero is great! Slides
- Topic: What is...a concrete category? - Or: Set-based is cool, well sometimes ;-) Slides
- Topic: What is...the Yoneda lemma? - Or: Shoot on sight. Slides
- Topic: What is...a subcategory? - Or: The symmetric groups and cobordisms. Slides
- Topic: What is...the skeleton? - The bare minimum. Slides
- Topic: What is...equivalence? - Or: Equal means...? Slides
- Topic: What are...natural transformations? - Maps between functors. Slides
- Topic: What are...properties of functors? - Or: Preserving diagrams. Slides
- Topic: What are...functors? - Maps between categories. Slides
- Topic: What is...monic-epic-iso? - Or: Not quite injective-surjective-bijective. Slides
- Topic: What is...the duality principle? - Flipping arrows. Slides
- Topic: What are...(commuting) diagrams? - Or: Graphs and paths. Slides
- Topic: What is...a category? - Arrows in action. Slides
- Topic: What is...category theory? - Or: Organizing ideas. Slides

- Topic: What is...persistent homology? - Or: Applications 2 (topology in data analysis). Slides
- Topic: What are...some applications of topology? - Or: Applications 1 (topology in mathematics). Slides
- Topic: What is...the Hurewicz theorem? - Or: Homotopy and homology. Slides
- Topic: What are...Eilenberg-MacLane spaces? - Or: They are not spheres. Slides
- Topic: What is...cellular approximation of spaces? - Or: My polynomials have cells. Slides
- Topic: What is...cellular approximation? - Or: I miss my space-filling curves... Slides
- Topic: What is...Whitehead's theorem? - Or: A perfect invariant!? Slides
- Topic: What are...homotopy groups? - Or: Spheres in spaces. Slides
- Topic: What are...examples of (co)homology groups? - Or: Another of my favorite lists. Slides
- Topic: What is...Alexander duality? - Or: Horned spheres!? Slides
- Topic: What is...Poincaré duality? - Or: My face is 0-dimensional. Slides
- Topic: What is...orientability? - Or: A homological definition. Slides
- Topic: What is...the Künneth formula? - Or: Multiplication vs. tensor product. Slides
- Topic: What is...the cohomology ring? - Or: Polynomials, of course. Slides
- Topic: What is...the cohomology ring intuitively? - Or: Counting intersections. Slides
- Topic: What is...the universal coefficient theorem? - Or: Working integrally rocks. Slides
- Topic: What is...the Mayer-Vietoris sequence? - Or: More than the sum of its parts!? Slides
- Topic: What is...a (co)homology theory? - Or: Shut up and calculate. Slides
- Topic: What is...relative homology? - Or: Calculations modulo subspaces. Slides
- Topic: What is...cohomology? - Or: Reversing arrows. Slides
- Topic: What is...homology categorifying? - Or: Modules, polynomials and numbers. Slides
- Topic: What is...cellular homology? - Or: Winding around. Slides
- Topic: What are...simplicial and singular homology? - Or: Cycles modulo boundaries. Slides
- Topic: What are...simplicial complexes? - Or: Triangles everywhere. Slides
- Topic: What is...homology intuitively? - Or: What is a hole? Slides
- Topic: What are...examples of fundamental groups? - Or: One of my favorite lists. Slides
- Topic: What is...a Cayley complex? - Or: Groups geometrically. Slides
- Topic: What are...covering space action? - Or: Deck transformations and friends. Slides
- Topic: What is...a covering space? - Or: A topological Galois correspondence. Slides
- Topic: What is...the Seifert-Van Kampen theorem? - Or: Cut and compute. Slides
- Topic: What is...the fundamental group? - Or: How not to hang pictures on walls. Slides
- Topic: What are...operations on cell complexes? - Or: Cell by cell. Slides
- Topic: What are...cell complexes? - Or: Constructed from discs. Slides
- Topic: What is...homotopy? - Or: The same shape!? Slides
- Topic: What is...algebraic topology? - Or: Translating from topology to algebra. Slides

- Topic: What are...solvable polynomials? - Or: Field and Galois theory, application 3. Slides
- Topic: What is...the Kroneckerâ€“Weber theorem? - Or: Field and Galois theory, application 2. Slides
- Topic: What are...the limits of straightedge and compass? - Or: Field and Galois theory, application 1. Slides
- Topic: What are...the Galois groups of finite fields? - Or: Finite fields are easy. Slides
- Topic: What is...the fundamental theorem of Galois theory? - Or: From roots to groups and back. Slides
- Topic: What are...Galois extensions? - Or: Shuffling roots. Slides
- Topic: What are...group actions on fields? - Or: The beginning of Galois theory. Slides
- Topic: What are...normal and separable extensions? - Or: Linear factors matter. Slides
- Topic: What are...minimal polynomials? - Or: Minimal relations. Slides
- Topic: What are...field extensions? - Or: Adjoining roots. Slides
- Topic: What are...(finite) fields? - Or: Fields are relatively rare. Slides
- Topic: What are...irreducible polynomials? - Or: Analogs of primes. Slides
- Topic: What is...a unique factorization domain? - Or: Primes! Slides
- Topic: What is...an Euclidean domain? - Or: Generalizing division with remainder. Slides
- Topic: What is...the Chinese remainder theorem? - Or: Arranging rectangles. Slides
- Topic: What is...Hilbert's basis theorem? - Or: Algebra meets geometry (once again). Slides
- Topic: What is...localization? - Or: Why fractions matter. Slides
- Topic: What is...the calculus of ideals? - Or: Ideals are like numbers. Slides
- Topic: What are...subrings and ideals? - Or: Two valid substructures. Slides
- Topic: What is...a ring? - Or: Generalizing the integers. Slides
- Topic: What is...the Jordan-Hölder theorem? - Or: The elements of group theory. Slides
- Topic: What are...matrix groups? - Or: The most important groups?! Slides
- Topic: What is...the classification of abelian groups? - Or: Factoring numbers. Slides
- Topic: What are...the Sylow theorems? - Or: Canonical substructures? Slides
- Topic: What is a...Cayley graph? - Or: Graphs and groups. Slides
- Topic: What are...actions, orbits and stabilizers? - Or: How many ways are there to...? Slides
- Topic: What is...the symmetric group? - Or: Why strings can shuffle numbers. Slides
- Topic: What are...the isomorphism theorems? - Or: Basic rules of algebra generalize. Slides
- Topic: What is...a (normal) subgroup? - Or: Why care about the difference? Slides
- Topic: What is...a group? - Or: Abstract symmetries. Slides
- Topic: What is...algebra? - Or: How to not solve polynomial equations. Slides

- Topic: What is...a projective space? - Or: My lines cross. Slides
- Topic: What is...an affine map? - Or: Translations in action. Slides
- Topic: What is...an affine space? - Or: I lost my origin. Slides
- Topic: What is...the symmetric algebra? - Or: Polynomials in vector spaces. Slides
- Topic: What is...the exterior algebra? - Or: Anticommuting polynomials. Slides
- Topic: What are...tensor products? - Or: How to multiply vector spaces and matrices. Slides
- Topic: What is...a quotient vector space? - Or: Identifying information. Slides
- Topic: What is...the dual vector spaces? - Or: Two flipped sides. Slides
- Topic: What is...the LU decomposition? - Or: Lower-upper. Slides
- Topic: What is...Gaussian elimination? - Or: How to find intersections. Slides
- Topic: What is...a system of linear equations? - Or: Intersections matter. Slides
- Topic: What is...Sylvester's law of inertia? - Or: Signatures is linear algebra. Slides
- Topic: What is...the spectral theorem? - Or: My eigenvalues are real! Slides
- Topic: What is...an isometry? - Or: Tessellations of my walls. Slides
- Topic: What is...orthogonality? - Or: Flavors of 90 degrees. Slides
- Topic: What is...an inner product? - Or: Generalizing angles and lengths. Slides
- Topic: What is...a linear subspace? - Or: Hyperplanes and friends. Slides
- Topic: What is...the exponential of a matrix? - Or: Nilpotent vanishing. Slides
- Topic: What is...the Jordan normal form again? - Or: How to find and compute a gold standard. Slides
- Topic: What is...the Jordan normal form? - Or: Why (almost) all matrices are diagonalizable. Slides
- Topic: What is...diagonalization? - Or: Finding the right coordinate system. Slides
- Topic: What are...traces? - Or: The best? matrix invariant. Slides
- Topic: What are...eigenvalues and eigenvectors? - Or: Linear fixed points. Slides
- Topic: What is...the characteristic polynomial? - Or: Matrix roots. Slides
- Topic: What is...the permanent? - Or: The trivial representation. Slides
- Topic: What is...the determinant? - Or: Signed permutations. Slides
- Topic: What are...direct sums? - Or: How to add vector spaces and matrices. Slides
- Topic: What is...a change of basis? - Or: Moving, rotating and scaling axes. Slides
- Topic: What is...a linear map? - Or: Linear algebra done right. Slides
- Topic: What is...matrix multiplication? - Or: Why the definition is not random. Slides
- Topic: What is...a matrix? - Or: Incarnations of the same beast. Slides
- Topic: What is...a basis? - Or: A notion of dimension. Slides
- Topic: What is...a vector space? - Or: Geometry à la Descartes. Slides
- Topic: What is...linear algebra? - Or: Everything is linear. Slides