# Channel URL

Click here: VisualMath

# 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, newest first:

- What are...my favorite theorems?
- What is...algebraic geometry?
- Magma in a nutshell.
- What is...algebraic graph theory?
- What is...geometric topology?
- 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

# TeX files

The TeX files for the presentations are freely available at: https://github.com/dtubbenhauer/My-TeX-files.git

# A list of my videos

- Topic: What is...a coin toss run? - Why is this difficult? Slides
- Topic: What is...the drunken bird constant? - Coming home, or not...? Slides
- Topic: What is...the else function? - Counting else. Slides
- Topic: What are...multiplicative compositions? - Additive < multiplicative. Slides
- Topic: What is...Golomb–Dickman's constant? - Cycles and primes. Slides
- Topic: What is...counting of abelian groups? - Commutative = easy. Slides
- Topic: What is...the tree constant? - Trees are difficult. Slides
- Topic: What is...tree counting? - n+1 and n-1. Slides
- Topic: What is...the Cayley–Bacharach theorem? - Nine and nine! Slides
- Topic: What is...Nagata's theorem? - Ill-behaved invariants. Slides
- Topic: What are...Dehn invariants? - Cutting polyhedra. Slides
- Topic: What is...the Gelfond–Schneider? - Logarithms and transcendence. Slides
- Topic: What are...Mahler functions? - Self-similarity and transcendence. Slides
- Topic: What is...Hilbert's Satz 90? - Number 90 is it! Slides
- Topic: What is...the Yang–Mills mass gap? - One comes after zero. Slides
- Topic: What is...the Hodge conjecture? - Topology and algebraic geometry. Slides
- Topic: What are...the Navier–Stokes equations? - Fluids are difficult. Slides
- Topic: What is...the Birch–Swinnerton-Dyer conjecture? - Counting, of course. Slides
- Topic: What is...the 3d Poincaré conjecture? - Its a sphere? Its a sphere! Slides
- Topic: What is...the Riemann hypothesis? - Critical strip free (almost). Slides
- Topic: What are...crystal graphs? - Low temperature behavior. Slides
- Topic: What is...Freudenthal's magic square? - Exceptional! Slides
- Topic: What are...octonions? - Or: Division = good, associativity = bad. Slides
- Topic: What is...planarity testing? - Or: Finding planar embeddings. Slides
- Topic: What are...Borcherds–Kac–Moody algebras? - Or: Still emerging from matrices. Slides
- Topic: What are...sporadic groups? - Or: Weird, but interesting. Slides
- Topic: What are...Kac–Moody algebras? - Or: Emerging from matrices. Slides
- Topic: What is...the complexity of embeddings? - Or: How difficult is drawing? Slides
- Topic: What are...Hecke algebras? - Or: Counting, or rather not counting?! Slides
- Topic: What is...the Bruhat decomposition? - Or: Lower and upper. Slides
- Topic: What is...recurrent versus transient? - Or: Leaving or staying. Slides
- Topic: What are...tree rotations? - Or: A surprisingly small bound. Slides
- Topic: What are...IBIS groups? - Or: Perfectly broken symmetries. Slides
- Topic: What are...Coxeter complexes? - Or: Spheres and points. Slides
- Topic: What is...the partition number? - Or: To count or not to count... Slides
- Topic: What are...Kronecker coefficients? - Or: Terribly difficult. Slides
- Topic: What is...a Hopf algebra? - Or: Why we comultiply. Slides
- Topic: What is...Padé approximation? - Or: Better than Taylor. Slides
- Topic: What are...the groups of Galois theory? - Or: An ocean of symmetric groups. Slides
- Topic: What is...Bass-Serre theory? - Or: Trees and groups. Slides
- Topic: What are...computer conjectures? - Or: A computer can write poems, so... Slides
- Topic: What are...computer proofs? - Or: Who needs mathematicians? Slides
- Topic: What is...homotopy type theory? - Or: HoTT = Ho + TT. Slides
- Topic: What is...the prime number theorem? - Or: Let us not count! Slides
- Topic: What is...the balanced graph theorem? - Or: Sudden appearance. Slides
- Topic: What is...the matrix equivalence theorem? - Or: One is ok. Two is, well... Three is impossible. Slides
- Topic: What is...Drozd's theorem? - Or: The Jordan normal form doesn't get better... Slides
- Topic: What is...Artin-Wedderburn's theorem? - Or: Matrices, of course. Slides
- Topic: What is...convolution? - Or: Area, even without area. Slides
- Topic: What is...symbolic integration? - Or: Symbolic integration rules! Slides
- Topic: What is...Karatsuba's algorithm? - Or: Faster than expected. Slides
- Topic: What are...alpha shapes and complexes? - Or: Topology and its applications. Slides
- Topic: What are...Frobenius algebras and 2d TQFTs? - Or: Physics and math walk hand in hand. Slides
- Topic: What is...Fagin's theorem? - Or: Almost always true, almost always false. Slides
- Topic: What is...the Novikov-Boone-Britton theorem? - Or: Sometimes life is not decidable. Slides
- Topic: What are...expander graphs? - Or: Sparse and connected. Slides
- Topic: What is...Stone's duality? - Or: Graphs and logic. Slides
- Topic: What is...Kasteleyn's theorem? - Or: Difficult, yet easy. Slides
- Topic: What is...the commuting matrix theorem? - Or: Around 25%. Slides
- Topic: What is...the Garden of Eden theorem? - Or: Ignored by life? Slides
- Topic: What is...the Ax-Grothendieck theorem? - Or: Polynomials rule! Slides
- Topic: What is...the choice zoo? - Or: No general choice, please! Slides
- Topic: What is...a Hadamard matrix? - Or: Orthogonality exists! Or not? Slides
- Topic: What are...independence results? - Or: Its true and its not true. Slides
- Topic: What are...skein relations? - Or: Life knows somehow... Slides
- Topic: What are...Riemann surfaces? - Or: Avoid choices! Slides
- Topic: What is...counting of Tait colorings? - Or: Evaluating graphs. Slides
- Topic: What is...a Tait coloring? - Or: Towards the four color theorem. Slides
- Topic: What is...the Rado graph? - Or: The law of large numbers for graphs. Slides
- Topic: What is...the LKB representation? - Or: Braids are linear. Slides
- Topic: What is...Dirac's belt trick? - Or: 720 degree is it! Slides
- Topic: What is...Fortune's algorithm? - Or: Parabolas and distance. Slides
- Topic: What is...the genus of a graph? - Or: Building bridges. Slides
- Topic: What is...the infinitude of prime knots? - Or: Euclid’s theorem for knots. Slides
- Topic: What are...torus knots? - Or: Wind p times, wind q times. Slides
- Topic: What is...Fáry's theorem? - Or: Straight lines everywhere. Slides
- Topic: What is...the circle packing theorem? - Or: From circles to graphs and back. Slides
- Topic: What is...exotic four space? - Or: Dimension four is weird!? Slides
- Topic: What is...homotopy of spheres? - Or: Surprisingly hard!? Slides
- Topic: What is...Hamming distance? - Or: Cubes! Slides
- Topic: What is...the jeu de taquin? - Or: Teasing permutations. Slides
- Topic: What is...coloring of numbers? - Or: From colors to set theory. Slides
- Topic: What is...the dimension of a graph? - Or: 1d objects can be high dimensional? Slides
- Topic: What is...crossing number additivity? - Or: Adding crossings adds crossings? Slides
- Topic: What is...the Erdős–Gabbai theorem? - Or: Realizing graphs. Slides
- Topic: What is...\((236c)^{11}\)? - Or: That is a polynomial in c! Slides
- Topic: What is...Whitney's embedding theorem? - Or: Projective spaces are nasty. Slides
- Topic: What is...the square ice constant? - Or: Ice and 1.539601... Slides
- Topic: What is...Bondy's theorem? - Or: Forgetting without loss. Slides
- Topic: What is...P versus NP? - Or: Building a chair versus recognizing it. Slides
- Topic: What is...a pointwise coloring of the plane? - Or: The chromatic number of the plane. Slides
- Topic: What is...Hadwiger's conjecture? - Or: Coloring is difficult... Slides
- Topic: What is...group growth? - Or: Small, medium-sized and big groups? Slides
- Topic: What is...a limit line graph? - Or: Line graphs converge; in some sense... Slides
- Topic: What is...card shuffling mathematically? - Or: Random walks. Slides
- Topic: What are...zeta functions of languages? - Or: Counting voodoo. Slides
- Topic: What is...a demon knot? - Or: Why wireless is great. Slides
- Topic: What are...diagram algebras? - Or: Pictures in algebra and algebra in pictures. Slides
- Topic: What are...spectra of Cayley graphs? - Or: Eigenvalues and characters. Slides
- 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 don’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 are...regular functions? - Or: Analogs of continuous functions. Slides
- Topic: What is...the dimension of a variety? - Or: A space in a space in a space... Slides
- Topic: What is...the Zariski topology in algebra? - Or: From topology to algebra. Slides
- Topic: What is...the Zariski topology? - Or: Varieties are closed, by definition. Slides
- Topic: What is...the coordinate ring? - Or: Polynomial functions and varieties. Slides
- Topic: What is...Hilbert's Nullstellensatz? - Or: The zero-locus-theorem. Slides
- Topic: What are...ideals of sets? - Or: Enter, algebra! Slides
- Topic: What are...conic sections? - Or: Light cones and walls. Slides
- Topic: What are...algebraic varieties? - Or: Zeros! Slides
- Topic: What is...algebraic geometry? - Or: Polynomials, and more. Slides

- Topic: Magma in a nutshell - part 5.
- Topic: Magma in a nutshell - part 4.
- Topic: Magma in a nutshell - part 3.
- Topic: Magma in a nutshell - part 2.
- Topic: Magma in a nutshell - part 1.
- Topic: Magma in a nutshell - part 0. Slides

- Topic: What are...representable matroids 2? - Or: How to rule out matrices. Slides
- Topic: What are...representable matroids? - Or: Matrices and matroids. Slides
- Topic: What are...matroid embeddings? - Or: This actually only about greedoids... Slides
- Topic: What is...a greedy algorithm 2? - Or: Greedy for matroids. Slides
- Topic: What is...a greedy algorithm 1? - Or: Greedy in general. Slides
- Topic: What is...the lattice of flats? - Or: The properties of inclusion. Slides
- Topic: What are...cryptomorphisms? - Or: Many definitions = good! Slides
- Topic: What are...transversal matroids? - Or: More examples from graph theory. Slides
- Topic: What are...linearly independent sets? - Or: All bases are of the same size. Slides
- Topic: What is a...matroid? - Or: Bases, forests, partitions and friends. Slides
- Topic: What are...colorings of random graphs? - Or: Concentrated colors. Slides
- Topic: What are...cliques in random graphs? - Or: Peaks! Slides
- Topic: What is...the diameter of random graphs? - Or: The same diameter!? Slides
- Topic: What are...symmetries of random graphs? - Or: There are no symmetries! Slides
- Topic: What is...true for almost all graphs? - Or: Many, many edges... Slides
- Topic: What are...subgraphs of random graphs? - Or: A lot of edges. Slides
- Topic: What are...random graph models? - Or: Different, yet the same. Slides
- Topic: What are...random graphs? - Or: Random is maybe not so random... Slides
- Topic: What is...the Tutte polynomial counting? - Or: Its counting a lot! Slides
- Topic: What is...the complexity of the Tutte polynomial? - Or: Easy or difficult? Slides
- Topic: What is...Tutte universality? - Or: Everyone discovers the same polynomial. Slides
- Topic: What is...the Tutte polynomial? - Or: More counting! Slides
- Topic: What is...chromatic detection? - Or: Determined by colorings. Slides
- Topic: What is...the chromatic polynomial? - Or: Polynomials and colors. Slides
- Topic: What are...graph polynomials? - Or: Easy problems, not hard problems. Slides
- Topic: What is...graph drawing? - Or: The Laplacian draws graphs. Slides
- Topic: What are...clustering methods? - Or: Partitioning made easy. Slides
- Topic: What is...the Laplace matrix? - Or: Taking the degree into account. Slides
- Topic: What is...spectral Hamiltonicity? - Or: The second largest - part 3. Slides
- Topic: What is...the edge expansion constant? - Or: The second largest - part 2. Slides
- Topic: What are...the second eigenvalue's contributions? - Or: Not quite canonical. Slides
- Topic: What are...applications of the spectrum? - Or: The spectrum knows a lot! Slides
- Topic: What are...graphs with small spectrum? - Or: ADE is it! Slides
- Topic: What are...example of spectra? - Or: Let us compute! Slides
- Topic: What is...path counting via eigenvalues? - Or: Paths “=” matrix powers “=” largest eigenvalue. Slides
- Topic: What is...path counting? - Or: Paths “=” matrix powers. Slides
- Topic: What are...adjacency matrices? - Or: Graphs “=” matrices. Slides
- Topic: What is...algebraic graph theory? - Or: It's a matrix! Slides

- Topic: What is...braid group cryptography? - Or: Applications 2 (topology in cybersecurity). Slides
- Topic: What are...knots and DNA? - Or: Applications 1 (topology in biochemistry). Slides
- Topic: What is...the unknotting problem? - Or: How to detect the unknot?! Slides
- Topic: What is...the h-cobordism theorem in action? - Or: Towards the Poincaré conjecture. Slides
- Topic: What is...the h-cobordism theorem? - Or: Nice manifolds between manifolds. Slides
- Topic: What is...the Poincaré conjecture? - Or: Spheres will be spheres. Slides
- Topic: What is...4d Kirby calculus again? - Or: Knots and four manifolds, part 2. Slides
- Topic: What is...4d Kirby calculus? - Or: Knots and four manifolds, part 1. Slides
- Topic: What is...a handle decomposition? - Or: Handles and even more handles! Slides
- Topic: What are...four manifolds? - Or: Movies and colors! Slides
- Topic: What is...a Heegaard diagram? - Or: Handle diagrams. Slides
- Topic: What is...a Heegaard splitting? - Or: Attaching handles. Slides
- Topic: What are...lens spaces? - Or: The birth of geometric topology!? Slides
- Topic: What are...homology spheres? - Or: Spheres, but not really... Slides
- Topic: What is...the linking matrix? - Or: Kirby calculus on matrices. Slides
- Topic: What is...Kirby calculus? - Or: Knots and three manifolds, part 2. Slides
- Topic: What is...Dehn surgery? - Or: Knots and three manifolds. Slides
- Topic: What are...three manifolds? - Or: A glimpse of 4d. Slides
- Topic: What is...the chromatic number of a surface? - Or: Graphs on surfaces. Slides
- Topic: What are...knotted surfaces? - Or: Codimension 2. Slides
- Topic: What is...the classification of surfaces? - Or: Punctures, handles, projective planes. Slides
- Topic: What are...words for surfaces? - Or: How to “read” surfaces. Slides
- Topic: What are...combinatorial surfaces? - Or: Words and friends. Slides
- Topic: What are...surfaces? - Or: Spheres and friends. Slides
- Topic: What are...homeomorphisms? - Or: Rubber geometry. Slides
- Topic: What is...a knot homology? - Or: Vector spaces, not numbers. Slides
- Topic: What is...the bracket polynomial? - Or: Jones and co. Slides
- Topic: What are...knot groups? - Or: Never forget the complement. Slides
- Topic: What is...the knot genus? - Or: Minimal surfaces and knots. Slides
- Topic: What is...the Alexander polynomial? - Or: Alexander-Conway skein calculus. Slides
- Topic: What is...the knot determinant? - Or: Enter, linear algebra. Slides
- Topic: What is...a knot coloring? - Or: A colorful approach. Slides
- Topic: What are...knots? - Or: Why knot?! Slides
- Topic: What is...geometric topology? - Or: It's a disc! Slides

- Topic: What is...chemical representation theory? - Or: Applications (rep theory in chemistry). Slides
- Topic: What is...a local-global principle? - Or: Small determines big. Slides
- Topic: What is...a defect? - Or: Measuring complexity. Slides
- Topic: What is...a block? - Or: Decomposing a problem. Slides
- Topic: What are...Brauer characters? - Or: Ignore the ground field. Slides
- Topic: What is...modular representation theory? - Or: Division made hard. Slides
- Topic: What is...the Tsetlin library? - Or: Sorting books made easy. Slides
- Topic: What is...the Ehrenfest model? - Or: Rep theory and diffusion. Slides
- Topic: What are...the simples of the transformation monoid? - Or: Through strands, again. Slides
- Topic: What are...the simples of the Brauer monoid? - Or: Through strands. Slides
- Topic: What is...the Brauer monoid? - Or: 1d cobordisms! Slides
- Topic: What is...the CMP theorem? - Or: Clifford-Munn-Ponizovskiĭ theorem a.k.a. H-reduction. Slides
- Topic: What are...Green cells? - Or: Measuring information loss. Slides
- Topic: What is...representation theory of monoids? - Or: What to expect. 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 Kronecker–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