Conversations on Mathematics

John Baez and James Dolan

• Minkowski and lattices - February 11, 2022.

• The kid who learned math on the street - February 18, 2022. On calculating the Néron–Severi group of an abelian surface.

• February 21, 2022 conversation. Topics include:
  • Néron–Severi groups of abelian varieties.
  • How to describe complex line bundles, or holomorphic line bundles, on a complex torus (e.g. an abelian variety) using Riemann forms.
  • The square tiling honeycomb {4,4,3} and its relation to the abelian surface that's the product of two copies of the elliptic curve given by the complex numbers mod the Gaussian integers.
  • The hexagonal tiling honeycomb {6,3,3} and its relation to the abelian surface that's the product of two copies of the elliptic curve given by the complex numbers mod the Eisenstein integers.

• Endomorphism rings of abelian varieties - February 24, 2022.
• Chern connection - February 24, 2022.

• February 28, 2022 conversation. Topics include:
  • Classifying holomorphic line bundles on an abelian variety using their Riemann forms.
  • Endomorphisms of abelian varieties, the Rosati involution, how to describe polarizations on an abelian variety as 'positive' endomorphisms, and the structure of the Néron–Severi group tensored with \(\mathbb{Q}\) as a Jordan algebra. For this it is useful to read:
    • Christina Birkenhake and Herbert Lange, Complex Abelian Varieties.
    • James Milne, Abelian Varieties.
  • The Kronecker–Weber theorem and Kronecker's Jugendtraum: the relation between elliptic curves with complex multiplication and abelian extensions of imaginary quadratic fields.

• Digression on Jordan cone of 3-by-3 hermitian complex matrixes - March 3, 2022.

• Stacks - March 5, 2022.

• March 7, 2022 conversation. Topics include:
  • The factor of automorphy of an arbitrary holomorphic line bundle over a complex abelian variety.
  • Jordan algebras and the Rosati involution
  • Kronecker's Jugendtraum

• Paracompact hyperbolic Coxeter groups - March 8, 2022.

• John Baez, Line bundles on complex tori (part 1), The n-Category Café, March 13, 2022.

• John Baez, Line bundles on complex tori (part 2), The n-Category Café, March 19, 2022.

• March 14, 2022 conversation. Topics include:
  • Hyperbolic Coxeter groups coming from rings of algebraic integers:
    • Norman Johnson and Asia Ivić Weiss, Quadratic integers and Coxeter groups.
    Three examples:
    1. The integers \(\mathbb{Z}\), the modular group and \(\mathrm{PGL}(2,\mathbb{Z})\), and the closely related Coxeter group \(\{\infty,3\}\) which acts as symmetries of the infinite-order triangular tiling of the hyperbolic plane.
    2. The Gaussian integers \(\mathbb{Z}[i]\), the groups \(\mathrm{PSL}(2,\mathbb{Z}[i])\) and \(\mathrm{PGL}(2,\mathbb{Z}[i])\), and the closely related Coxeter group \(\{4,4,3\}\) which acts as symmetries of the square tiling honeycomb in hyperbolic 3-space.
    3. The Eisenstein integers \(\mathbb{Z}[\omega]\), the groups \(\mathrm{PSL}(2,\mathbb{Z}[\omega])\) and \(\mathrm{PGL}(2,\mathbb{Z}[\omega])\), and the closely related Coxeter group \(\{6,3,3\}\) which acts as symmetries of the honeycomb tiling honeycomb in hyperbolic 3-space.
  • The classification of holomorphic line bundles on complex tori.

• Pic₀ (the Picard variety) - March 14, 2022.

• Hodge conjecture - March 24, 2022.

• John Baez, Holomorphic gerbes (part 1), The n-Category Café, March 27, 2022.

• John Baez, Holomorphic gerbes (part 2), The n-Category Café, April 10, 2022.

• March 28, 2022 conversation. Topics include:
  • The Appell–Humbert theorem classifying holomorphic line bundles on a complex torus, and Ben-Bassat's analogous result for holomorphic gerbes:
    • Eran Assaf, The Appell–Humbert theorem.
    • Oren Ben-Bassat, Gerbes and the holomorphic Brauer group of complex tori.
  • The "belief method" for describing algebras of monads on the bicategory of locally presentable \(k\)-enriched categories, where \(k\) is a symmetric monoidal locally presentable category.

• Belief method - March 28, 2022.

• April 4, 2022 conversation. Topics include:
  • Ben-Bassat's version of the Néron–Severi group for holomorphic gerbes:
    • Oren Ben-Bassat, Gerbes and the holomorphic Brauer group of complex tori.
  • Generalizations of the Jacobian, the Picard group and the Néron–Severi group from holomorphic line bundles to holomorphic n-gerbes:
  • Holomorphic gerbes (part 1), The n-Category Café, March 27, 2022.
  • Holomorphic gerbes (part 2), The n-Category Café, April 10, 2022.
  • An application of the 'belief method' to commutative quantales (which are cocomplete symmetric monoidal posets).

• April 11, 2022 conversation. Topics include:
  • Using sheaf cohomology to generalize the Jacobian, the Picard group and the Néron–Severi group from holomorphic line bundles to holomorphic \(n\)-gerbes:
    • Holomorphic gerbes (part 1), The n-Category Café, March 27, 2022.
    • Holomorphic gerbes (part 2), The n-Category Café, April 10, 2022.
  • The belief method applied to the doctrine of symmetric monoidal locally presentable \(k\)-linear categories, which gives 'algebro-geometric theories'.
  • The 'theory of an object' is the category of \(k\)-linear species with its Cauchy tensor product.
  • The theory of an object whose exterior cube is the initial object. The representations of \(\mathrm{GL}(2)\), \(\mathrm{SL}(2)\) and related groups.

• April 18, 2022 conversation. Topics include:
  • Young diagrams and representations of:
    • the general linear group \(\mathrm{GL}(N)\),
    • the special linear group \(\mathrm{SL}(N)\),
    • the projective general linear group \(\mathrm{PGL}(N)\),
    all illustrated in the case \(N = 2\).

• May 16, 2022 conversation. Topics include:
  • John Baez, Motivating motives, May 25, 2022.
  • J. S. Milne, Motives: Grothendieck's dream.
  • J. S. Milne, The Riemann Hypothesis over finite fields: from Weil to the present day.

May 23, 2022 conversation. Topics include:
• Counting points on elliptic curves, and motives.
• Supersingular elliptic curves over finite fields, which are those with a quaternion algebra of endomorphisms.
• Getting abelian varieties from cyclotomic fields, using the example of the 20th cyclotomic field
  

  This field has Galois group \(\mathrm{GL}(1,\mathbb{Z}/20) \cong \mathrm{GL}(1,\mathbb{Z}/4) \times \mathrm{GL}(1,\mathbb{Z}/5)\):

  

  so its poset of subfields is isomorphic to the opposite of the poset of subgroups of \(\mathrm{GL}(1,\mathbb{Z}/4) \times \mathrm{GL}(1,\mathbb{Z}/5)\):

  
  (Pictures by Simon Burton.)

• June 6, 2022 conversation. Topics include:
• The moduli stack of real elliptic curves.
• Generalizing Kronecker's Jugendtraum from elliptic curves to abelian varieties.
• Manin's 'Alterstraum' and elliptic curves with real multiplication.
• Heegner numbers.

• The figure-eight puzzle - June 8, 2022. On the lemniscate of Bernoulli, elliptic curves and lemniscate elliptic functions.

• June 13, 2022 conversation. Topics include:
• The category of pure numerical motives (with rational coefficients) over the field with \(q\) elements, \(\mathbb{F}_q\).
• The classification of simple objects in this category. The absolute Galois group of the rationals acts on the set of Weil \(q\)-numbers, and assuming the Tate Conjecture, simple objects correspond to orbits of this action. This is Proposition 2.6. in Milne's Motives over finite fields. I got the definition of Weil \(q\)-number wrong. Given a prime power \(q\), a Weil \(q\)-number is a complex number such that:
  1. for some integer \(n\), \(q^n z\) is an algebraic integer,
  2. for some integer \(m\), \(|gz| = q^{m/2}\) for every element \(g\) in the absolute Galois group of the rationals.
  The point, again, is that the absolute Galois group of the rationals acts on the set of Weil \(q\)-numbers, and the set of orbits is isomorphic to the set of simple numerical motives over \(\mathbb{F}_q\). This is Proposition 2.6 in Milne.

• June 20, 2022 conversation. Topics include:
• Correcting the definition of Weil \(q\)-number, which is explained in Definition 2.5 of Milne's Motives over finite fields.
• Toposes in number theory.
• The topos of species.
• Dirichlet species:
  • John Baez and James Dolan, Dirichlet species and the Hasse–Weil zeta function.
• Set-valued functors on the groupoid or category of finite fields of characteristic \(p\).
• The topos of actions of the Galois group.

• June 27, 2022 conversation. Topics include:
  • The exponential sheaf sequence of a complex abelian surface \(X\), and how the corresponding long exact sequence in sheaf cohomology lets us describe the Néron–Severi group as the image of \(H^2(X,\mathcal{O}^\ast)\) in \(H^2(X,\mathbb{Z})\), or equivalently the kernel of the map from \(H^2(X,\mathbb{Z}\)) to \(H^2(X,\mathcal{O})\). When the rank of the Néron–Severi group is maximal the image of the map from \(H^2(X,\mathbb{Z})\) to \(H^2(X,\mathcal{O})\) is a lattice in a complex vector space — but generically it appears to be a dense subgroup.
  • How the classification of simple objects in the category of pure numerical motives changes when we use the algebraic closure of the rationals as coefficients, rather than the rationals. This is Proposition 2.21 in Milne's Motives over finite fields. (I said tensoring irreps of \(\mathbb{R}\) amounts to multiplication of numbers, but it's really addition.)

• July 4, 2022 conversation. Topics include:
• The exponential sheaf sequence and a detailed study of the map from \(H^2(X,\mathbb{Z})\) to \(H^2(X,\mathcal{O})\) when \(X\) is an abelian surface. We can work out this map and its image explicitly in examples. Here we do the easiest case, when \(X\) the product of two identical elliptic curves, each being the complex numbers mod the lattice of Gaussian integers. We see that in this case the image of the map from \(H^2(X,\mathbb{Z})\) to \(H^2(X,\mathcal{O})\) is a lattice in a 1-dimensional complex vector space. But in other cases we expect the image is dense!
• A polarization on an abelian variety \(X = V/L\) puts a positive definite hermitian form on \(V\), and this lets us describe the Néron–Severi group in terms of self-adjoint operators on \(V\). This clarifies how a polarization puts a Jordan algebra structure on the Néron-Severi group tensored with \(\mathbb{R}\).

• July 11, 2022 conversation. Topics include:
• Classifying complex abelian surfaces in order of genericity, and the ranks of their Néron–Severi group:
  • The generic case gives a Néron–Severi group of rank 1,
  • the cartesian product of two distinct elliptic curves gives rank 2,
  • the cartesian square of a generic elliptic curve gives rank 3, and
  • the cartesian square of an elliptic curve with complex multiplication gives rank 4.
  To understand this, we should look at the endomorphism ring of the abelian surface and tensor it with the reals, giving the 'endomorphism algebra'. Upon picking an polarization this algebra gets a Rosati involution which makes the algebra into a star-algebra, allowing us to split endomorphisms into a self-adjoint and skew-adjoint part.
• The moduli stack of elliptic curves and the associated Coxeter group. How can we generalize this to higher-dimensional principally polarized abelian varieties? How do modular curves generalize to the higher-dimensional case, giving Siegel modular varieties?

• Jugendtraum - July 13, 2022.

• July 18, 2022 conversation. Topics include:
  • We work out the map from \(H^2(X,\mathbb{Z})\) to \(H^2(X,\mathcal{O})\) when \(X\) is the product of two elliptic curves, namely the complex numbers mod the Gaussian integers and the complex numbers mod the Eisenstein integers. We see that in this case the image of the map from \(H^2(X,\mathbb{Z})\) to \(H^2(X,\mathcal{O})\) is a abelian subgroup of rank 4 in a 1-dimensional complex vector space — and thus, dense in this vector space.
  • Six stages of understanding Kronecker's Jugendtraum:
    1. First, the patterns of how primes split in abelian extensions of the rational numbers:
       • Wikipedia, Quadratic reciprocity.
       • Wikipedia, Reciprocity law.
       • B. F. Wyman, What is a reciprocity law?
    2. Second, the Kronecker–Weber theorem saying that every abelian extension of the rationals is contained in a cyclotomic field.
    3. Third, the correspondence between subfields of the \(n\)th cyclotomic field and subgroups of its Galois group.
    4. Fourth, relativization: generalizing these ideas to abelian extensions of arbitrary number fields.
    5. Fifth: complications due to nonprincipal ideals. Artin reciprocity.
    6. Sixth, Kronecker's Jugendtraum.

• July 25, 2022 conversation. Topics include:
• The long exact exponential sheaf sequence for a generic abelian surface.
• Analogies between Artin reciprocity and the abelian case of children's drawings (dessins d'enfants). A notation for subfields of a cyclotomic field, and an analogous notation for abelian branched covers of the Riemann sphere.

• August 1, 2022 conversation. Topics include:
• The analogy between number fields and 3-manifolds, making primes analogous to knots:
  • Chao Li and Charmaine Sia, Knots and primes.
  • Barry Mazur, Thoughts about primes and knots (video).
  • Masanori Morishita, Analogies between knots and primes, 3-manifolds and number rings.
• Three Coxeter groups, each with a homomorphism onto the next: the child's drawing Coxeter group, the cartographic group and the modular Coxeter group.

• August 15, 2022 conversation. Topics include:
  • Lambda-rings, how the Grothendieck group of a 2-rig becomes a lambda-ring, and the big Witt comonad:
  • John Baez, Joe Moeller and Todd Trimble, Schur functors and categorified plethysm.
  • H. Lenstra, Construction of the ring of Witt vectors.
  • D. O. Tall and G. Wraith, Representable functors and operations on rings.
  • J. Borger and B. Weiland, Plethystic algebra.
  • Zeta functions and the big Witt ring:
  • Niranjan Ramachandran, Zeta functions, Grothendieck groups, and the Witt ring.

• August 22, 2022 conversation. Topics include:
  • An approach to children's drawings based on the free group generated by 3 involutions, a Coxeter group whose even subgroup is the free group on 2 generators (the fundamental group of the thrice punctured Riemann sphere).
  • Two examples: one connected to the Gaussian elliptic curve, and another connected to the symmetric group on 3 letters, which produces a branched covering of the Riemann sphere by itself.

• August 29, 2022 conversation. Topics include:
  • Zeta functions, lambda-rings and the big Witt ring, continued:
    • Niranjan Ramachandran, Zeta functions, Grothendieck groups, and the Witt ring.
  • Abelian children's drawings and the kagome lattice, continued: an approach based on the abelianization of the free group on 2 generators (the fundamental group of the thrice punctured Riemann sphere).

• September 9, 2022 conversation. Topics include:
  • An approach to children's drawings based on the free group on two generators: the fundamental group of the Riemann sphere punctured at \(0,1\) and \(\infty\), with the two generators corresponding to loops around \(0\) and \(1\). Any action of this group on finite sets gives a covering of the Riemann sphere branched at \(0,1\) and \(\infty\).
    • The case where one generator acts as the cycle (ABCDEFG) and the other acts as the permutation (AB)(CDE)(FG).
    • The case where one generator acts as the cycle (ABCDE) and the other acts as the permutation (A)(BCDE).
    Examples of this sort give a certain class of tree-shaped children's drawings, and branched coverings of the Riemann sphere by itself defined by polynomials in one variable.

• September 16, 2022 conversation. Topics include:
  • Dessin d'enfants, also known as children's drawings.
  • The Shabat polynomial of a tree-shaped child's drawing, and Chebyshev polynomials:
    • George Shabat and Alexander Zvonkin, Plane trees and algebraic numbers.
    • J. Bétréma and J. A. Zvonkin, Plane trees and Shabat polynomials.
    • Wikipedia, Chebyshev polynomials.
  • An analogy between Kronecker's Jugendtraum and Grothendieck's children's drawings.
  • The absolute Galois group of the rational numbers and the Grothendieck–Teichmüller group.

• September 23, 2022 conversation. Topics include:
  • The Hurwitz automorphism theorem, the (2,3,7) Coxeter diagram, and children's drawings:
    • John Baez, 42.
    • Wikipedia, Hurwitz surface.
  • Young diagrams and the classification of irreps of the symmetric groups \(S_n\): an approach using double cosets and Mackey's decomposition theorem:
    • Brian Conrad, Mackey theory and applications.
  • How a Young diagram gives a polynomial that gives the dimension of \(Y(V)\) as a function of the dimension of the vector space \(V\), where \(Y\) is the Schur functor corresponding to that Young diagram.
  • An analogy between Kronecker's Jugendtraum and Grothendieck's children's drawings.

• September 30, 2022 conversation. Topics include:
  • Children's drawings and 'designs'. Gaussian and Eisenstein children's drawings.
  • An analogy between Kronecker's Jugendtraum and Grothendieck's children's drawings.

• October 7, 2022 conversation. Topics include:
  • The connection between Kronecker's Jugendtraum and Grothendieck's children's drawings.

• October 14, 2022 conversation. Topics include:
  • The connection between Kronecker's Jugendtraum and Grothendieck's children's drawings, especially in the case of Gaussian children's drawings.

• October 21, 2022 conversation. Topics include:
  • The connection between Kronecker's Jugendtraum and Grothendieck's children's drawings, especially in the case of Gaussian children's drawings.

• October 28, 2022 conversation. Topics include:
  • The moduli stack of elliptic curves versus the moduli stack of acute triangles.
  • The connection between Kronecker's Jugendtraum and Grothendieck's children's drawings, especially in the case of Gaussian children's drawings.

• November 11, 2022 conversation. Topics include:
  • Torsion in ideal class groups, as explained by Mickaël Montessinos.
    • John Baez, Post on Mathstodon, November 8, 2022.
    • Mickaël Montessinos, Post on Mathstodon, November 8, 2022.
  • The moduli stack of elliptic curves versus the moduli stack of acute triangles (including right triangles):
    • Wikipedia, Moduli stack of elliptic curves.
    • nLab, Moduli stack of elliptic curves.
  • The Grothendieck topos of \(\mathrm{S}_3\)-equivariant sheaves on the equilateral triangle (which is itself the moduli space of acute triangles).

• November 18, 2022 conversation, with Simon Willerton. Topics include:
  • The moduli stack of elliptic curves versus the moduli stack of acute (including right) triangles:
    • Wikipedia, Moduli stack of elliptic curves.
    • nLab, Moduli stack of elliptic curves.
    • Kai Behrend, An introduction to algebraic stacks, in Moduli Spaces, eds. L. Brambila-Paz, P. Newstead, R. P. Thomas and O. García-Prada, Cambridge U. Press, Cambridge 2014, pp. 1–131. (This teaches the theory of stacks using the moduli stack of all triangles.)
    • Jaime Gaspar and O. Neto, All triangles at once, Amer. Math. Monthly 122 (2015), 982–982.
    • Ian Stewart, Why do all triangles form a triangle?, Amer. Math. Monthly 124 (2017), 70–73.
  • The connection between Kronecker's Jugendtraum and Grothendieck's children's drawings, especially in the case of Gaussian children's drawings.

• November 25, 2022 conversation. Topics include:
  • The moduli stack of elliptic curves versus the moduli stack of acute (including right) triangles: how to get from such a triangle to a tetrahedron with four copies of that triangle as faces, and then from that to an elliptic curve that's a branched double cover of the Riemann sphere, with the tetrahedron's four corners as branch points:
    • Wikipedia, Moduli stack of elliptic curves.
    • nLab, Moduli stack of elliptic curves.
  • The cross-ratio of these four points on the Riemann sphere is a modular function of level 2, sometimes called the 'lambda function':
    • Wikipedia, Modular lambda function.
  • The connection between Kronecker's Jugendtraum and Grothendieck's children's drawings, especially in the case of Gaussian children's drawings.

• December 2, 2022 conversation. Topics include:
  • Laplace transforms, Legendre transforms and the tropical rig (or idempotent analysis):
    • Hugo Touchette, Legendre–Fenchel transforms in a nutshell (especially Remarks 5 and 6).
    • John Baez, Quantization and cohomology (week 12).
    • John Baez, Quantization and cohomology (week 13).
  • The three-fold way and ten-fold way:
    • John Baez, Division algebras and quantum theory.
    • John Baez, The tenfold way.
  • The moduli stack of elliptic curves versus the moduli stack of acute (including right) triangles:
    • Wikipedia, Moduli stack of elliptic curves.
    • nLab, Moduli stack of elliptic curves.
  • 7 drawing styles for children's drawings.

• 7 drawing styles for designs, December 2, 2022.

• December 16, 2022 conversation. Topics include:
  • Children's drawings defined using the Gaussian integers and computing the rational function, a generalization of the Shabat polynomial, associated to a certain class of children's drawings:
    • Jean Bétréma and Alexander Zvonkin, Plane trees and Shabat polynomials.

• January 5, 2023 conversation. Topics include:
  • The tenfold way.
  • Two dessins d'enfant that are probably in the same orbit of the Grothendieck-Teichmüller group, built starting from the Gaussian integers.
  • Course notes on the Grothendieck-Teichmüller group:
    • Thomas Willwacher, The Grothendieck-Teichmüller group.

• January 12, 2023 conversation. Topics include:
  • Using software to study two dessins d'enfant that are probably in the same orbit of the Grothendieck-Teichmüller group, built starting from the Gaussian integers.
  • The tenfold way.

• January 19, 2023 conversation. Topics include:
  • Bott periodicity:
    • 8 categories of representations of Clifford algebras, related by adjoint functors:
      • The tenfold way (part 6), The n-Category Café, January 15, 2023.
    • A construction of symmetric spaces from Clifford algebras:
      • The tenfold way (part 7), The n-Category Café, January 16, 2023.
    • A second construction of symmetric spaces from Clifford algebras, giving infinite loop spaces, as explained in Milnor's book Morse Theory:
      • The tenfold way (part 8), The n-Category Café, January 18, 2023.
  • Using software to study two dessins d'enfant that are probably in the same orbit of the Grothendieck-Teichmüller group, built starting from the Gaussian integers.

• January 26, 2023 conversation. Topics include:
  • Rough thoughts on a 'diamond cutting' construction of Jordan algebras and symmetric spaces from 3-graded Lie algebras:
    • John Baez, This Week's Finds in Mathematical Physics (Week 193).
  • A category-theoretic examination of the construction of symmetric spaces from Clifford algebras in Milnor's book Morse Theory:
    • The tenfold way (part 8), The n-Category Café, January 18, 2023.
  • Symmetric spaces as involutory quandles.
  • Using software to study two dessins d'enfant that are now known to be in the same orbit of the Grothendieck–Teichmüller group, built starting from the Gaussian integers.

• February 9, 2023 conversation. Topics include:
  • John's talk on the tenfold way. You can see a video of this talk here, see the slides here, and see a longer treatment of the tenfold way here.

• February 16, 2023 conversation. Topics include:
  • Ideal class groups of algebraic number fields. The case of the 20th cyclotomic field illustrates the Kronecker–Weber theorem, which says that every extension of \(\mathbb{Q}\) with an abelian Galois group is contained in some cyclotomic field. The Galois group of the 20th cyclotomic field is \(\mathrm{GL}(1, \mathbb{Z}/20)\), generated by the Frobenius automorphism. This group \(\mathrm{GL}(1, \mathbb{Z}/20)\) is isomorphic to \(\mathbb{Z}/2 \times \mathbb{Z}/4\), with the two factors corresponding to the primes 2 and 5.
    

    The subfields of the 20th cyclotomic field correspond to the subgroups of its Galois group, and we can draw the lattice of these subfields, and indicate which steps up this lattice involve ramification at 2, 5, or the real prime.

    

    (Pictures by Simon Burton.)

  • The Hilbert class field of an algebraic number field \(F\) is its maximal unramified abelian extension, and its Galois group over \(F\) is the ideal class group of \(F\).
  • A false but interesting conjecture: trying to express an unramified abelian extension \(E\) of a number field as being directly 'built out of' the ideals in the corresponding subgroup \(S\) of the ideal class group, where \(E\) and \(S\) correspond to each other via the 'covariant Galois correspondence'.
  • Some applications of the affine group scheme of \(n\)th roots of unity, which corresponds to the ring \(\mathbb{Z}[x]/\langle x^n - 1 \rangle\).

• Class fields - February 17, 2023.

• February 23, 2023 conversation. Topics include:
  • A short description of Grothendieck–Teichmüller theory developed from Willwacher's notes:
    • Thomas Willwacher, The Grothendieck–Teichmüller group.
  • How the composite of normal covers gives a short exact sequence.
  • How this yields a map from the absolute Galois group of the rationals to the group of automorphisms of the profinite completion of the free group on 2 generators.

• March 2, 2023 conversation. Topics include:
  • Basics of Grothendieck–Teichmüller theory:
    • Thomas Willwacher, The Grothendieck–Teichmüller group.
    In a composite of two normal coverings, the group of deck transformations of the bottom stage acts on the group of deck transformations of the bottom stage. In Grothendieck–Teichmüller theory, these groups of deck transformations are Galois groups, and the bottom stage is 'arithmetic' while the top stage is 'geometric'. An example involving the two Gaussian designs James Dolan has been intensively studying.
  • Digression into the relation between degree and genus of an algebraic curve, and the gonality of an algebraic curve.

• March 9, 2023 conversation. Topics include:
  • Studying extensions of fields geometrically.
  • A principal \(G\)-bundle \(P\) over a space \(X\) gives a functor \(F\) from the category \(\mathsf{Rep}(G)\) of representations of \(G\) to the category \(\mathsf{Vect}(X)\) of vector bundles over \(X\): this functor sends any representation of \(G\) to its associated vector bundle. We can think of \(F\) as a 'model' of the 'theory' \(\mathsf{Rep}(G)\).
  • We can generalize this idea and make it more precise using ideas from algebraic geometry, where we think of groups as giving commutative Hopf algebras. To do this we define a 'torsor' of a commutative Hopf algebra as follows. Choose a field \(k\) and consider the doctrine of 2-rigs, by which we mean symmetric monoidal locally presentable \(k\)-linear categories. For any commutative Hopf algebra \(H\), the category \(\mathsf{Comod}(H)\) of comodules of \(H\) is a theory in this doctrine: that is, a 2-rig. We define a torsor of \(H\) to be a model of this theory in some other 2-rig \(\mathsf{R}\): that is, a 2-rig map \( F \colon \mathsf{Comod}(H) \to \mathsf{R}\). Thus, we have taken the associated bundle construction \(F \colon \mathsf{Rep}(G) \to \mathsf{Vect}(X)\) coming from a principal bundle \(P\) and used it to generalize the concept of principal bundle to the concept of 'torsor of a cococommutative Hopf algebra'.
  • We can generalize further by working with commutative Hopf algebroids over \(k\), which give a certain notion of 'stack' simultaneously generalizing the concept of 'space' associated to a commutative \(k\)-algebra (namely an affine scheme over \(k\)) and the notion of 'group' associated to a commutative Hopf algebra over \(k\) (namely an affine group scheme over \(k\)).
  • Returning to the Hopf algebra case, consider an example. For any finite group \(G\) we get a commutative Hopf algebra \(H = k^G\) so we can talk about a torsor of \(H\), defined as above. For short we call this a \(G\)-torsor over \(k\). Unraveling the definitions, a \(G\)-torsor over \(k\) is a 2-rig map \(F \colon \mathsf{Rep}(G) \to \mathsf{R}\) where \(\mathsf{R}\) is some 2-rig. We can describe this more explicitly when we take \(k = \mathbb{Q}\), take \(\mathsf{R} = \mathsf{Vect}\) and take \(G = \mathbb{Z}/2\). In this case a \(\mathbb{Z}/2\)-torsor over \(\mathbb{Q}\) is the same as a rational vector space \(L\) equipped with an isomorphism \( \alpha \colon L \otimes L \stackrel{\sim}{\longrightarrow} \mathbb{Q}\). Any quadratic extension \(\mathbb{Q}(\sqrt{n})\) of the rationals gives such an \(L\), namely the 1-dimensional subspace of rational multiples of \(\sqrt{n}\), where \(\alpha \colon L \otimes L \to \mathbb{Q}\) is multiplication. Conversely any \(\mathbb{Z}/2\)-torsor over \(\mathbb{Q}\) gives a quadratic extension of the rationals. We can check directly that \(\mathbb{Z}/2\)-torsors over \(\mathbb{Q}\) are indeed classified by square-free integers.

• March 16, 2023 conversation. Topics include:
  • Studying extensions of fields geometrically. Review of what was done on March 9, 2023.
  • Definition of group object \(G\) and group action in an arbitrary cartesian category. The definition of \(G\)-torsor can almost be done at this level of generality, but we need a way to rule out things such as the action of a group on the empty set.
  • Digression on the 'group with no elements':
    • John Baez, The group with no elements, The n-Category Café.
  • Studying abelian extensions of fields using torsors. The case of quadratic extensions of the rationals was relatively easy because \(\mathbb{Z}/2\) 'splits' over the rational numbers: that is, all irreducible representations of \(\mathbb{Z}/2\) on rational vector spaces are 1-dimensional. An attempt to extend the program to \(\mathbb{Z}/3\), which does not split over the rationals.

• Underlying real affine-algebraic group of a complex affine-algebraic group - March 17, 2023.

• March 23, 2023 conversation. Topics include:
  • Studying abelian extensions of fields using torsors. The case of quadratic extensions of the rationals was relatively easy because \(\mathbb{Z}/2\) splits over the rational numbers: that is, every irreducible representations of \(\mathbb{Z}/2\) on rational vector spaces are 1-dimensional. To extend the program to \(\mathbb{Z}/3\), which does not split over the rationals, we conduct some preliminary investigations of subfields of the 63rd cyclotomic field, which correspond to subgroups of \(\mathrm{GL}(1, \mathbb{Z}/63) \cong \mathrm{GL}(1, \mathbb{Z}/9) \times \mathrm{GL}(1, \mathbb{Z}/7) \cong \mathbb{Z}/6 \times \mathbb{Z}/6\):
    

    There are 30 such subgroups, and we construct a chart of 12 of them mimicking the method used on February 16th, 2023 for the 20th cyclotomic field.

• Pullback bundles - March 24, 2023.

• Maschke's theorem - March 24, 2023.

• April 6, 2023 conversation. Topics include:
  • \(G\)-torsors over a number field \(k\) for a finite group \(G\). We defined these in our March 9, 2023 conversation. Most abstractly these are 2-rig maps \(f \colon \mathsf{Rep}(G) \to \mathsf{Vect}\), where representations and \(\mathsf{Vect}\) are defined using finite-dimensional vector spaces over \(k\). We also gave some other ways of thinking about these. Today we study the case \(G = \mathbb{Z}/3\). In this case \(F\) maps the group algebra \(k[G]\) to some cubic extension \(K\) of \(k\) — and this is the whole point, since we are using them to study cubic extensions of \(k\). The idea is that a cubic extension of \(k\) can be seen as a 'twisted version' of the group algebra \(k[\mathbb{Z}/3]\).

    The theory works most simply when \(k\) contains (or in our conversation is) the Eisenstein field: the rationals with a primitive cube root of \(1\) adjoined. Then \(\mathbb{Z}/3\) splits over \(k\), so one can show that a \(\mathbb{Z}/3\)-torsor over \(K\) amounts to a line object \(L\) in the category of \(k\)-vector spaces together with a trivialization of its tensor cube: $$ F \colon 1_{\otimes} \to L^{\otimes 3} $$ where the tensor unit \(1_{\otimes} \in \mathrm{Vect}\) is just the field \(k\). Even more concretely, we get a \(\mathbb{Z}/3\)-torsor over \(k\) from a nonzero element \(A \in k\): to do this, we take \(L = k\) and take \(F\) to be multiplication by \(A\). Then two choices of \(A\), say \(A\) and \(A'\), give isomorphic \(\mathbb{Z}/3\)-torsors iff \(A' = B^3A\) for some nonzero \(B \in k\).

    When \(k\) is the Eisenstein field, any cubic extension \(K\) is a 6th-degree extension of \(\mathbb{Q}\). We erroneously claim that this is always a Galois extension of \(\mathbb{Q}\). When it actually is, the Galois group of \(K\) over \(\mathbb{Q}\) can thus be either \(\mathbb{Z}/6\), which is abelian, or \(S_3\), which is not. We thus consider this challenge: given \(A \ne 0\) in the Eisenstein field \(k\), determine whether the Galois group of \(K = \sqrt[3]{A}\) over \(\mathbb{Q}\) is abelian or not.

    Next, what if \(k\) does not contain the Eisenstein field? Then we can redo our previous discussion using a formal 'Eisenstein object' \(E\) replacing the tensor unit in \(\mathsf{Vect}\), and an 'Eisenstein line object' \(L_E\) replacing \(L\), with a trivialization of its tensor cube: $$ F \colon E \to L_E^{\otimes_E 3} $$

• April 13, 2023 conversation. Topics include:
  • \(\mathbb{Z}/3\)-torsors over \(\mathbb{Q}\). These are interesting because \(\mathbb{Z}/3\) doesn't split over \(\mathbb{Q}\). \(\mathbb{Z}/3\) does split over the Eisenstein field \(\mathbb{Q}(\omega)\), where \(\omega\) is a primitive cube root of unity. For this reason we've seen \(\mathbb{Z}/3\)-torsors over \(\mathbb{Q}(\omega)\) correspond to numbers \(x \in \mathbb{Q}(\omega)\) up to multiplication by nonzero cubes in \(\mathbb{Q}(\omega)\). We have a map sending \(\mathbb{Z}/3\)-torsors over \(\mathbb{Q}\) to \(\mathbb{Z}/3\)-torsors over \(\mathbb{Q}(\omega)\). Which \(\mathbb{Z}/3\)-torsors over \(\mathbb{Q}(\omega)\) do we get this way? Those coming from \(x\) such that \(x^2 = \overline{x}\) up to a cube factor in \(\mathbb{Q}(\omega)\).
  • Fixing a mistake from last time: most cubic extensions of \(\mathbb{Q}(\omega)\) are not Galois over \(\mathbb{Q}\). Analysis of what's really going on.
  • Analogy between what we're doing and exponentiating by a 'tiny object', like the 'infinitesimal object' \(D\) used to describe a tangent vector in synthetic differential geometry:
    • Wikipedia, Infinitesimal object.

• April 20, 2023 conversation. Topics include:
  • \(G\)-torsors over a number field \(k\) for a finite group \(G\). We defined these in our March 9, 2023 conversation. Most abstractly these are 2-rig maps \(F \colon \mathsf{Rep}(G) \to \mathsf{Vect}\), where representations and \(\mathsf{Vect}\) are defined using finite-dimensional vector spaces over \(k\). We also gave some other ways of thinking about \(G\)-torsors. Today we study the case \(G = \mathbb{Z}/3\). In this case \(F\) can map the group algebra \(k[G]\) to some cubic extension \(K\) of \(k\) — and this is the whole point, since we are using them to study cubic extensions of \(k\). The idea is that a cubic extension of \(k\) can be seen as a 'twisted version' of the group algebra \(k[\mathbb{Z}/3]\).

    The theory works most simply when \(k\) contains (or is) the Eisenstein field: the rationals with a primitive cube root of 1 adjoined. Then \(\mathbb{Z}/3\) splits over \(k\), so one can show that a \(\mathbb{Z}/3\)-torsor over \(K\) amounts to a line object \(L\) in the category of \(k\)-vector spaces together with a trivialization of its tensor cube. Even more concretely, we get a \(\mathbb{Z}/3\)-torsor over \(k\) from a cube root of nonzero element \(A \in k\). Then two choices of \(A\), say \(A\) and \(A'\), give isomorphic \(\mathbb{Z}/3\)-torsors iff \(A' = B^3A\) for some nonzero \(B \in k\).

    When \(k\) is the Eisenstein field, any cubic extension \(K\) is a 6th-degree extension of \(\mathbb{Q}\), and in fact (we claim) a Galois extension. The Galois group of \(K\) over \(\mathbb{Q}\) can thus be either \(\mathbb{Z}/6\), which is abelian, or \(S_3\), which is not. We thus consider this challenge: given \(A \ne 0\) in the Eisenstein field \(k\), determine whether the Galois group of \(K\) over \(\mathbb{Q}\) is abelian or not.

    Next, what if \(k\) does not contain the Eisenstein field? Then we can redo our previous discussion using a formal 'Eisenstein object' \(E\) replacing the tensor unit in \(\mathsf{Vect}\), and an 'Eisenstein line object' replacing \(L\).

• April 27, 2023 conversation. Topics include:
  • Two appearances of a 3 × 3 square in the study of cubic extensions of number fields, which may or may not be related.
  • An infinite-dimensional vector space in number theory. Let \(\omega\) be a primitive cube root of unity, so \(\mathbb{Z}[\omega]\) is the ring of Eisenstein integers and \(\mathbb{Q}(\omega)\) is the corresponding field, the 'Eisenstein field'. Then the abelian group \(\mathrm{GL}(1,\mathbb{Z}(ω))\) is the direct sum of countably many copies of \(\mathbb{Z}\), one for each Eisenstein prime mod units, and one copy of \(\mathbb{Z}/6\), consisting of the units in the Eisenstein integers, which are the powers of \(\omega\). If we tensor this abelian group \(\mathrm{GL}(1,\mathbb{Q}(\omega))\) with the field with 3 elements \(\mathbb{F}_3\) (thought of as another abelian group), we get an infinite-dimensional vector space over \(\mathbb{F}_3\). This has one basis element for each Eisenstein prime mod units, together with one for the units. How does Eisenstein conjugation act on this vector space? Since this operator squares to 1, we can split this vector space as a direct sum of the +1 eigenspace and the -1 eigenspace.
  • The work of Albert, Brauer, Hasse and Noether relating the Brauer group of a number field \(k\) to the second cohomology of its absolute Galois group:
    • Peter Roquette, The Brauer–Hasse–Noether theorem in historical perspective.
    This work relies on building central simple algebras over \(k\) as 'twisted' versions of the group algebra of the absolute Galois group of \(k\). The 'twisting' is done using a 2-cocycle on the Galois group, which Schreier called a 'factor system'. When this group is cyclic, these twisted algebras are called 'cyclic algebras'.

• May 4, 2023 conversation. Topics include:
  • Artin reciprocity for \(\mathbb{Z}/3\)-torsors. If \(k\) is a number field, we can characterize the \(\mathbb{Z}/3\)-torsors over \(k\) in two equivalent ways: the 'Tannakian' way and the 'Frobenius pattern' way.

    As already discussed in our April 20, 2023 talk, the 'theory of \(\mathbb{Z}/3\) torsors over \(k\)' is another way of talking about the category of representations of \(\mathbb{Z}/3\) on rational vector spaces. 'Theories' here are symmetric monoidal locally presentable categories, and we can also describe this theory by saying it's the free symmetric monoidal locally presentable category on an Eisenstein line object \(L\) with a trivialization of its tensor cube. We can further constrain this theory by saying that the Eisenstein conjugate of \(L\) is isomorphic to its tensor square (in a way compatible with the above trivialization). This extra constraint makes it so models of this theory in the category of vector spaces over \(k\) correspond to abelian cubic extensions of the number field \(k\).

    We can also state this all more concretely, 'gauge-fixing' by taking \(L\) to be the 'standard' Eistenstein line object \(E\) which can be taken to be the sum of two copies of the unit for the tensor product in our theory. We can actually put this into the theory, equipping our theory an isomorphism \(G \colon L \to E\). Then the models of the theory are given by solutions to some polynomial equations, which we outline.

• May 11, 2023 conversation. Topics include:
  • An attempt to understand some of James Dolan's recent thoughts on torsors. Fix a field \(k\) and a finite group \(G\). Let \(\mathrm{Vect}\) be the category of vector spaces over \(k\). Let \(\mathrm{Rep}(G)\) be the category of representations of G on vector spaces over k. \(\mathrm{Rep}(G)\) is a '2-rig', or more precisely a locally presentable 2-rig: that is, a symmetric monoidal locally presentable Vect-enriched category.

    \(\mathrm{Rep}(G)\) is generated, as a 2-rig, by the regular representation of G, which we could call \(r(G)\). But \(\mathrm{Rep}(G)\) also contains another special object \(k^G\): the algebra of functions on the finite set \(G\), treated as a trivial representation of \(G\). This is a coalgebra in \(\mathrm{Rep}(G)\) whose comultiplication encodes the multiplication in \(G\). This coalgebra \(k^G\) has a coaction on \(r(G)\), \(r(G) \to r(G) \otimes k^G\). This coaction makes \(r(G)\) into what James calls a '\(G\)-torsor in \(\mathrm{Rep}(G)\)', meaning that there is also a 'codivision' map \(k^G \to r(G) \otimes r(G)\) obeying a certain equation. All this is dual to how a nonempty set \(X\) is a torsor of \(G\) if it has an action \(G \times X \to X\) together with a division map \(X \times X \to G\) obeying a certain equation.

    Indeed, any 2-rig \(R\) contains a Hopf object that we could call \(k^G\): it's the direct sum of \(G\) copies of the unit object, with a comultiplication coming from multiplication in \(G\). We can use this to define a concept of a '\(G\)-torsor in R', copying the definition just given in \(\mathrm{Rep}(G)\).

    Such a \(G\)-torsor in a 2-rig \(R\) works out to be the secretly the same thing as a 2-rig map \(F \colon \mathrm{Rep}(G) \to R\) since we can determine such an \(F\) by choosing any \(G\)-torsor in \(R\), and vice versa. Thus, James usually shortcuts the whole process and defines a \(G\)-torsor in a 2-rig \(R\) simply to be a 2-rig map \(F \colon \mathrm{Rep}(G) \to R\).

    Finally, suppose \(L\) is any field extending \(k\) with Galois group \(G\). Then \(L\) gives an algebra object in \(\mathrm{Vect}\) which is also a \(G\)-torsor in \(\mathrm{Vect}\).

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