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.
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.
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.
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 "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.
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:
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:
for some integer \(n\), \(q^n z\) is an algebraic integer,
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.
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 \(\mathbb{R}\).
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.
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:
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:
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:
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.
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.
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:
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: