## Conversations on Mathematics

#### John Baez and James Dolan

Here are some conversations on mathematics: some in email, some on YouTube. I have not corrected all the mistakes, some of which we eventually catch. If you want to see new ones as they appear, you can subscribe to my YouTube channel.

• March 28, 2022 conversation. Topics include:

• 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:
• 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.

• 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:
• 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?

• 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.

• 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.

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

• 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:
• The cross-ratio of these four points on the Riemann sphere is a modular function of level 2, sometimes called the 'lambda function':
• The connection between Kronecker's Jugendtraum and Grothendieck's children's drawings, especially in the case of Gaussian children's drawings.

• January 19, 2023 conversation. Topics include:

© 2023 John Baez and James Dolan
baez@math.removethis.ucr.andthis.edu