Euclid Book V

David Fowler and C. Zeeman have made many important contributions to understanding this book of Euclid's which is assumed to be reorganization of the work of Eudoxus.

The theory is quite abstract and can be framed in terms of an action of the natural numbers $\mathbb{N}$ on any linearly ordered structure $\mathcal{A}$. We write $a : b = c : d$ for $a,b,c,d \in \mathcal{A}$ iff the following three conditions are satisfied:\[\forall n,m\in \mathbb{N}( (n.a = m.b) \Leftrightarrow ( n.c = m.d))\]\[\forall n,m\in \mathbb{N}( (n.a > m.b) \Leftrightarrow ( n.c > m.d))\]\[\forall n,m\in\mathbb{N}( (n.a < m.b) \Leftrightarrow ( n.c < m.d))\] 

In the case of $\mathbb{Q}$ and the action is the usual multiplication the first condition is sufficient. It just says that $\forall q \in \mathbb{Q}(\frac{a}{b} = q \Leftrightarrow \frac{c}{d} = q)$.  What is amazing is that Eudoxus' definition works for the real numbers too ! Consider the particular case in which $b = d = 1$. Then $a: 1 = c :1$ is just equality in $\mathbb{R}$. Eudoxus' definition is saying in this case that $a$ and $b$ determine the same Dedekind cut.

Borcherds' lectures on Elementary Number Theory

 https://www.youtube.com/playlist?list=PL8yHsr3EFj52Qf7lc3HHvHRdIysxEcj1H

 https://www.youtube.com/playlist?list=PL8yHsr3EFj53L8sMbzIhhXSAOpuZ1Fov8

 

Foundations of Elementary Pure Mathematics

The foundations of pure mathematics are, for arithmetic,  the elementary theory of numbers,  field (or Galois) theory and some basic finite group theory and for geometry,  complex analysis and basic algebraic topology  - to which we add advanced linear algebra and some combinatorics.

Pure mathematics took a wrong turn with the development of algebraic geometry in the 20th century.  We mean abstract algebraic geometry based on commutative algebra (after Weil's attempt using field theory) via scheme theory (to allegedly bridge the gap with algebraic number theory). The correct mathematica communis was later revealed to be not commutative algebra but category theory and specially topos theory (more generally from the perspective of homotopy theory and higher category theory). It is absurd  and sterile to study algebraic varieties  merely from the point of view of commutative algebra; rather they must be studied in the context of the interdisciplinary richness of complex analytic geometry (Hodge theory, D-modules, Kähler and Symplectic structures...); also semi- and subanalytic and real algebraic geometry must not be neglected. Thus to approach pure mathematics you should do either number theory (which is heavily grounded on field theory) or pure geometry i.e. complex analytic geometry - though the two are connected in interesting ways (modular forms, Riemann zeta function). Indeed infinite series arise naturally in both arithmetic (as Dirichlet series, etc.) and complex geometry (meromorphic functions).   The most important Lie groups are analytic...

We will discuss later the necessity of generalizing analytic functions to be suitable for physics and applied mathematics (such as Sato's theory of hyperfunctions), to be able to furnish an alternative to smooth functions or the standard classes of Lebesgue measurable functions and distributions.

String theory is not science and not physics (and it does not even address what we feel to be the main philosophical and mathematical problems of quantum theory) although it uses abstract versions of some physical concepts. Rather it is an alternative (experimental) form of pure mathematics, perhaps somewhat in the spirit of Ramanujan.  String theory is actually a beautiful form of alternative pure mathematics which uses some transposed concepts of physics, to obtain surprising results or insights.

How to visualize the microsupport of a sheaf ?

https://mathoverflow.net/questions/314465/how-to-visualize-the-microsupport-of-a-sheaf

Coherent sheaves

The concept of coherent sheaf is of great interest.  The definition is subtle but it rests on very basic ideas. Consider a stalk $F_x$ of a sheaf $F$. Then each element $s \in F_x$ has a representative $s_U$ for some $U$ containing $x$.  But we cannot in general find a $U$ which does for all such $s$. Consider a countable set of points having the origin of the complex plane as an accumulation point. And consider the sheaf of holomorphic functions having zeros at the origin and on the points of the set. Then the stalk at the origin is isomorphic to 0 but locally around 0 our sheaf is not isomorphic to the null sheaf. Thus a morphism of sheaves which is an isomorphism on the stalk at at a point $x$ is not necessarily locally an isomorphism of sheaves around $x$. Note the similarity to uniform continuity. Coherence expresses a coincidence between locality and infinitesimal locality which "coheres" - one of the characteristics of holomorphic functions.

An excellent discussion and motivation is given in Greuel and Pfister's Local Analytic Geometry.

Additive categories

In an additive category show that we have products iff we have coproducts. Show the equivalence between the definition of (co)product in terms of representable functors and the usual one in terms of universal properties. An additive category is a special case of a category enriched in pointed sets. A remarkable property is that every two objects $X$ and $Y$ are connected by some morphism $f: X \rightarrow Y$. If we have a terminal $1$ and initial object $0$ then there is a morphism $1 \rightarrow 0$.  Note that $hom(0,0)$ and $hom(1,1)$ are singletons. Thus the mere existence of $1 \rightarrow 0$ entails that $0\backsimeq 1$.

Let $\mathcal{C}$ be a triangulated category and $\mathcal{N}$ a null system and $Q: \mathcal{C} \rightarrow \mathcal{C}/\mathcal{N}$ the canonical functor. Suppose that $X \in \mathcal{N}$. Then $Q(X) \backsimeq 0$. To see this take $X \rightarrow 0$. By the triangulated category axioms this can be extended to a triangle $X \rightarrow 0 \rightarrow Z \rightarrow TX$. By the definition of null system we have that $Z \in \mathcal{N}$ since by definition also $0 \in \mathcal{N}$. Hence by definition we have that $X \rightarrow 0$ is in $S(\mathcal{N})$ so that it is an isomorphism in $\mathcal{C}/\mathcal{N}$.

Sheaf Theory

Note that a locally constant sheaf is nothing more than a covering space. The definitions of constant and locally constant sheaf are sometimes given in a confusing and misleading way.  Correct category theoretic notions should be given up to isomorphism. Given a space $U$ and an object $A$ there is a presheaf $\mathcal{F'}^A_U$ which yields $A$ on every open set and whose restriction map is $id_A$.  This generates a sheaf $\mathcal{F}^A_U$.  We say that a sheaf $F$ on a space $X$ is a locally constant sheaf if $X$ has a cover such that on each component $U$ of the cover it is isomorphic to a sheaf of the form $\mathcal{F}^A_U$. Monodromy is an abstract way at looking at the liftings of a closed path to a covering space.

Exercise: A locally constant sheaf on $[0,1]$ is a constant sheaf, that is, is isomorphic to a sheaf of the form $\mathcal{F}^A_{[0,1]}$. More generally: a locally constant sheaf on a simply connected space is a constant sheaf.

For $F$ an abelian sheaf,  we define $sup \, F = \{ x \in X: F_x \neq 0 \}$. Show that $sup\,F$ need not be closed. Consider the sheaf $F$ and fix a point $p$ in a Hausdorff space $X$. Then define $F(U) = 0$ for $p \in U$ and $F(U) = \mathbb{Z}_2$ otherwise for some abelian group $A \neq 0$.  Then clearly $sup\, F$ is not closed.   Do not confuse the support of an abelian sheaf with the support of a section $s$ over $U$, the points of $U$ in which $s_x \neq 0$. $sup\, s$ is indeed closed and this follows essentially from the algebraic constraint on restriction map homomorphisms where the image of $0$ must be $0$. Thus the stalk of a zero section must be zero.

Exercise: Show that sections of locally constant abelian sheaf $F$ have closed support (hence unique extensions).  Hint: suppose that $s_x \neq 0$. Then there is an open set $V(x)$ and a section $s_V$ which represents $s_x$. But since $F$ is locally constant we can assume, by taking intersections, that $F_{V(x)}$ is isomorphic to a constant sheaf for a given group $A$. But then clearly we must have $s_y  = s_x$ for all $y \in V(x)$ and hence the complement of $sup\, s$ is open. 

Thus (locally) constant abelian sheaves are abstract versions of many important properties of holomorphic functions.

A locally constant sheaf on a simply connected topological space is a constant sheaf. This follows from homotopy invariance of unique extensions. This is perhaps the most fundamental result about locally constant sheaves; more generally this would be the result concerning the representation of a the fundamental group based at $x$ by automorphisms of the stalk at $x$.