### analysis

Let ${\phi: \Omega_{SO} \rightarrow \Lambda}$ be a genus. We might ask when ${\phi}$ satisfies the following multiplicative property:

Property: For any appropriate fiber bundle ${F \rightarrow E \rightarrow B}$ of manifolds, we have

$\displaystyle \phi(E) = \phi(B) \phi(F). \ \ \ \ \ (1)$

When ${B}$ is simply connected, this is true for the signature by an old theorem of Chern, Hirzebruch, and Serre.

A special case of the property (1) is that whenever ${E \rightarrow B}$ is an even-dimensional complex vector bundle, then we have

$\displaystyle \phi(\mathbb{P}(E)) = 0,$

for ${\mathbb{P}(E)}$ the projectivization: this is because ${\mathbb{P}(E) \rightarrow B}$ is a fiber bundle whose fibers are odd-dimensional complex projective spaces, which vanish in the cobordism ring.

Ochanine has given a complete characterization of the genera which satisfy this property.

Theorem 1 (Ochanine) A genus ${\phi}$ annihilates the projectivizations ${\mathbb{P}(E)}$ of even-dimensional complex vector bundles if and only if the associated log series ${g(x) = \sum \frac{\phi(\mathbb{CP}^{2i})}{2i+1} x^{2i+1}}$ is given by an elliptic integral

$\displaystyle g(x) = \int_0^x Q(u)^{-1/2} du,$

for ${Q(u) = 1 - 2\delta u^2 + \epsilon u^4}$ for constants ${\delta, \epsilon}$.

Such genera are called elliptic genera. Observe for instance that in the case ${\epsilon = 1, \delta = 1}$, then

$\displaystyle g(x) = \int_0^x \frac{du}{1 - u^2} = \tanh^{-1}(u),$

so that we get the signature as an example of an elliptic genus (the signature has ${\tanh^{-1}}$ as logarithm, as we saw in the previous post).

I’d like to try to understand the proof of Ochanine’s theorem in the next couple of posts. In this one, I’ll describe the proof that an elliptic genus in fact annihilates projectivizations ${\mathbb{P}(E)}$ of even-dimensional bundles ${E}$. (more…)

Apologies for the lack of posts lately; it’s been a busy semester. This post is essentially my notes for a talk I gave in my analytic number theory class.

Our goal is to obtain bounds on the distribution of prime numbers, that is, on functions of the form ${\pi(x)}$. The closely related function

$\displaystyle \psi(x) = \sum_{n \leq x} \Lambda(n)$

turns out to be amenable to study by analytic means; here ${\Lambda(n)}$ is the von Mangolt function,

$\displaystyle \Lambda(n) = \begin{cases} \log p & \text{if } n = p^m, p \ \text{prime} \\ 0 & \text{otherwise} \end{cases}.$

Bounds on ${\psi(x)}$ will imply corresponding bounds on ${\pi(x)}$ by fairly straightforward arguments. For instance, the prime number theorem is equivalent to ${\psi(x) = x + o(x)}$.

The function ${\psi(x)}$ is naturally connected to the ${\zeta}$-function in view of the formula

$\displaystyle - \frac{\zeta'(s)}{\zeta(s)} = \sum_{n=1}^\infty \Lambda(n) n^{-s}.$

In other words, ${- \frac{\zeta'}{\zeta}}$ is the Dirichlet series associated to the function ${\Lambda}$. Using the theory of Mellin inversion, we can recover partial sums ${\psi(x) = \sum_{n \leq x} \Lambda(x)}$ by integration of ${-\frac{\zeta'}{\zeta}}$ along a vertical line. That is, we have

$\displaystyle \psi(x) = \frac{1}{2\pi i} \int_{\sigma - i \infty}^{\sigma + i \infty} -\frac{\zeta'(s)}{\zeta(s)} \frac{x^s}{s} ds ,$

at least for ${\sigma > 1}$, in which case the integral converges. Under hypotheses on the poles of ${-\frac{\zeta'}{\zeta}}$ (equivalently, on the zeros of ${\zeta}$), we can shift the contour appropriately, and estimate the integral to derive the prime number theorem. (more…)

Let ${M}$ be a compact manifold, ${E, F}$ vector bundles over ${M}$. Last time, I sketched the definition of what it means for a differential operator

$\displaystyle D: \Gamma(E) \rightarrow \Gamma(F)$

to be elliptic: the associated symbol

$\displaystyle \sigma(D): \pi^* E \rightarrow \pi^* F, \quad \pi: T^* X \rightarrow X$

was required to be an isomorphism outside the zero section. The goal of the index theorem is to use this symbol ${\sigma(D)}$ to compute the index of ${D}$, which we saw last time was a well-defined number

$\displaystyle \mathrm{index} D = \dim \ker D - \dim \mathrm{coker} D \in \mathbb{Z}$

invariant under continuous perturbations of ${D}$ through elliptic operators (by general facts about Fredholm operators).

The main observation is that ${D}$, in virtue of its symbol, determines an element of ${K(TX)}$. (Henceforth, we shall identify the tangent bundle ${TX}$ with the cotangent bundle ${T^*X}$, by choice of a Riemannian metric; the specific metric is not really important since ${K}$-theory is a homotopy invariant.) In fact, we have that ${K(TX)}$ is the (reduced) ${K}$-theory of the Thom space, so it is equivalently ${K(BX, SX)}$ for ${BX}$ the unit ball bundle and ${SX}$ the unit sphere bundle. But we have seen that to give an element of ${K(BX, SX)}$ is the same as giving a pair of vector bundles on ${BX}$ together with an isomorphism on ${SX}$, modulo certain relations.

Observation: The symbol of an elliptic operator determines an element in ${K(TX)}$. (more…)

This is the first in a series of posts about the Atiyah-Singer index theorem.

Let ${V, W}$ be finite-dimensional vector spaces (over ${\mathbb{C}}$, say), and consider the space ${\hom_{\mathbb{C}}(V, W)}$ of linear maps ${T: V \rightarrow W}$. To each ${T \in \hom_{\mathbb{C}}(V, W)}$, we can assign two numbers: the dimension of the kernel ${\ker T}$ and the dimension of the cokernel ${\mathrm{coker} T}$. These are obviously nonconstant, and not even locally constant. However, the difference ${\dim \ker T - \dim \mathrm{coker} T = \dim V - \dim W}$ is constant in ${T}$.

This was a trivial observation, but it leads to something deeper. More generally, let’s consider an operator (such as, eventually, a differential operator), on an infinite-dimensional Hilbert space. Choose separable, infinite-dimensional Hilbert spaces ${V, W}$; while they are abstractly isomorphic, we don’t necessarily want to choose an isomorphism between them. Consider a bounded linear operator ${T: V \rightarrow W}$.

Definition 1 ${T}$ is Fredholm if ${T}$ is “invertible up to compact operators,” i.e. there is a bounded operator ${U: W \rightarrow U}$ such that ${TU - I}$ and ${UT - I}$ are compact.

In other words, if one forms the category of Hilbert spaces and bounded operators, and quotients by the ideal (in this category) of compact operators, then ${T}$ is invertible in the quotient category. It thus follows that adding a compact operator does not change Fredholmness: in particular, ${I + K}$ is Fredholm if ${V = W}$ and ${K: V \rightarrow V}$ is compact.

Fredholm operators are the appropriate setting for generalizing the small bit of linear algebra I mentioned earlier. In fact,

Proposition 2 A Fredholm operator ${T: V \rightarrow W}$ has a finite-dimensional kernel and cokernel.

Proof: In fact, let ${V' \subset V }$ be the kernel. Then if ${v' \in V'}$, we have

$\displaystyle v' = UT v' + (I - UT) v' = (I - UT) v'$

where ${U}$ is a “pseudoinverse” to ${T}$ as above. If we let ${v'}$ range over the elements of ${v'}$ of norm one, then the right-hand-side ranges over a compact set by assumption. But a locally compact Banach space is finite-dimensional, so ${V'}$ is finite-dimensional. Taking adjoints, we can similarly see that the cokernel is finite-dimensional (because the adjoint is also Fredholm). $\Box$

The space of Fredholm operators between a pair of separable, infinite-dimensional Hilbert spaces is interesting. For instance, it has the homotopy type of ${BU \times \mathbb{Z}}$, so it is a representing space for K-theory. In particular, the space of its connected components is just ${\mathbb{Z}}$. The stratification of the space of Fredholm operators is given by the index.

Definition 3 Given a Fredholm operator ${T: V \rightarrow W}$, we define the index of ${T}$ to be ${\dim \ker T - \dim \mathrm{coker} T}$. (more…)

I’ve been trying to understand some complex analytic geometry as of late; here is an overview of Oka’s theorem.

Consider the space ${\mathbb{C}^n}$ and the sheaf ${\mathcal{O}}$ of holomorphic functions on it. One should think of this as the analog of complex affine space ${\mathbb{C}^n}$, with the Zariski topology, and with the sheaf ${\mathcal{O}_{reg}}$ of regular functions.

In algebraic geometry, if ${I \subset \mathbb{C}[x_1, \dots, x_n]}$ is an ideal, or if ${\mathcal{I} \subset \mathcal{O}_{reg}}$ is a coherent sheaf of ideals, then we can define a closed subset of ${\mathbb{C}[x_1,\dots, x_n]}$ corresponding to the roots of the polynomials in ${I}$. This construction gives the notion of an affine variety, and by gluing these one gets general varieties.

More precisely, here is what an affine variety is. If ${\mathcal{I} \subset \mathcal{O}_{reg}}$ is a coherent sheaf of ideals, then we define a ringed space ${(\mathrm{supp} \mathcal{O}_{reg}/\mathcal{I}, \mathcal{O}_{reg}/\mathcal{I})}$; this gives the associated affine variety. Here the “support” corresponds to taking the common zero locus of the functions in ${\mathcal{I}}$. In this way an affine variety is not just a subset of ${\mathbb{C}^n}$, but a locally ringed space.

Now we want to repeat this construction in the holomorphic category. If ${\mathcal{I} \subset \mathcal{O}}$ is a finitely generated ideal—that is, an ideal which is locally finitely generated—in the sheaf of holomorphic functions on ${\mathbb{C}^n}$, then we define the space cut out by ${\mathcal{I}}$ to be ${(\mathrm{supp} \mathcal{O}/\mathcal{I}, \mathcal{O}_{reg}/\mathcal{I})}$. We can think of these as “affine analytic spaces.”

Definition 1 An analytic space is a locally ringed space which is locally isomorphic to an “affine analytic space.” (more…)

First, thanks to all who kindly contributed advice on the previous post.

A long on-and-off project of mine has been to learn several complex variables. My latest attempt started a few days back, though the commencement of the fall semester may derail it. As is now customary for myself, I have started writing a set of notes that I intend to make grow reasonably large, though right now it is less than thirty pages (even with parts of my AG notes on coherence copied and pasted in). I shall post here an excerpt on the Weierstrass preparation theorem, which will assume only the definition of a holomorphic function in several variables). (more…)

So.  First off, surely the five remaining readers of this increasingly erratic blog have noticed the change of theme.

I want to next discuss the second inequality in class field theory, which is an upper bound on the norm index of the idele group.  There are two ways I know of to prove this: one analytic, one algebraic.  I will first sketch the analytic one. I say sketch because to do a full proof would get into the details of Dirichlet series, lattice points in homogeneously expanding domains, the construction of a certain fundamental domain for the action of the units, etc., etc., and I’d rather outline those ideas rather than do all the details because this is a series on class field theory. What I do plan on doing properly, however, is the algebraic (due to Chevalley in 1940) proof of the second inequality, which heavily uses results of field theory (e.g. Kummer theory) and local fields (e.g. power index computations).  I still thought it worthwhile to sketch the analytic approach, though. Rather than jumping right into it (I have to first say something about how the ideal and idele groups are connected), I decided to give an expository post on L-functions and Dirichlet’s theorem—in the case of the rational numbers.

1. Ramblings on the Riemann-zeta function

Recall that the Riemann-zeta function is defined by ${\zeta(s) = \sum n^{-s}}$, and that it is intimately connected with the distribution of the prime numbers because of the product formula

$\displaystyle \zeta(s) = \prod_p (1 - p^{-s})^{-1}$

valid for ${Re(s)>1}$, and which is a simple example of unique factorization. In particular, we have

$\displaystyle \log \zeta(s) = \sum_p p^{-s} + O(1) , \ s \rightarrow 1^+.$

It is known that ${\zeta(s)}$ has an analytic continuation to the whole plane with a simple pole with residue one at ${1}$. The easiest way to see this is to construct the analytic continuation for ${Re (s)>0}$. For instance, ${\zeta(s) - \frac{1}{s-1}}$ can be represented as a certain integral for ${Re(s)>1}$ that actually converges for ${Re(s)>0}$ though. (The functional equation is then used for the rest of the analytic continuation.) The details are here for instance. As a corollary, it follows that

$\displaystyle \sum_p p^{-s} = \log \frac{1}{s-1} + O(1) , \ s \rightarrow 1^+.$

This fact can be used in deducing properties about the prime numbers. (Maybe sometime I’ll discuss the proof of the prime number theorem on this blog.) Much simpler than that, however, is the proof of Dirichlet’s theorem on the infinitude of primes in arithmetic progressions. I will briefly outline the proof of this theorem, since it will motivate the idea of L-functions.

Theorem 1 (Dirichlet) Let ${\{an+b\}_{n \in \mathbb{Z}}}$ be an arithmetic progression with ${a,b}$ relatively prime. Then it contains infinitely many primes.

The idea of this proof is to note that the elements of the arithmetic progression ${\{an+b\}}$ can be characterized by so-called “Dirichlet characters.” This is actually a general and very useful (though technically trivial) fact about abelian groups, which I will describe now.

I fell a bit behind on the continuation of the class field theory series because I was setting up a new laptop. Before I resume that, I want to talk about something very weird that I learned today.

Let ${U \subset \mathbb{C}}$ be a set that omits at least two points. If ${f: U \rightarrow U}$ holomorphic and is such that $f(w)=w, {f'(w)=1}$ at one ${w \in U}$, then ${f}$ is the identity.

This is a striking rigidity phenomenon!

But how do we prove it? The idea is to consider the sequence of iterates ${f, f \circ f, \dots}$. Suppose for simplicity ${P=0}$. Then in a neighborhood of ${0}$, we can write ${f = z + cz^m + \dots }$, where the ${\dots}$ are omitted higher terms. If ${f}$ is not identically the identity, then ${c \neq 0}$.

So, similarly, by direct computation, in some neighborhood of ${P}$, we have ${f \circ f = z + 2c z^m + \dots}$. Similarly, if we define ${g_1 = f, g_2 = f \circ f, }$ for notational convenience, we have

$\displaystyle g_k = z + kc z^m + \dots.$

But the ${g_k}$ are all holomorphic maps into ${U}$. Since ${U}$ omits at least two points, the family ${g_k}$ is normal by Montel’s theorem and consequently has a subsequence ${g_{k_i}}$ that converges uniformly on compact sets.

Thus the derivatives ${g^{(m)}_{k_i}(0) = m! k_i c}$ converge, which is impossible unless ${c=0}$.

Huh? I didn’t exactly see that coming. If ${U}$ is the unit disk, then at least it looks familiar. A holomorphic map ${f}$ of the unit disk into itself sending zero to zero must satisfy ${|f'(0)| \leq 1}$, and if equality holds ${f}$ is a rotation. So perhaps this result should be thought of as a generalization of Schwarz’s lemma? (Nevertheless, the use of Montel’s theorem is quite a sledgehammer to prove something as elementary as Schwarz.)

I should say where I got this from: Krant’z Complex Analysis: The Geometric Viewpoint. Krantz didn’t prove exactly this, but the argument is the same.  Either this is standard fare that I missed when learning basic complex analysis, or I’m turning Climbing Mount Bourbaki into a comedy routine.

We will now apply the machinery already developed to a few concrete problems.

Proposition 1 Let ${G}$ be a compact abelian group and ${T}$ the rotation by ${a \in G}$. Then ${T}$ is uniquely ergodic (with the Haar measure invariant) if ${a^{\mathbb{Z}}}$ is dense in ${G}$.

The proof is straightforward. Suppose ${\mu}$ is invariant with respect to rotations by ${a}$. Then for ${f \in C(G)}$, we have

$\displaystyle \int f(a^m x ) d \mu = \int f(x) d \mu, \quad \forall m \in \mathbb{Z}$

and hence

$\displaystyle \int f(bx ) d \mu = \int f(x) d \mu, \quad \forall m \in \mathbb{Z},$

for any ${b \in G}$, which means that ${\mu}$ must be Haar measure (which is unique).

Corollary 2 An irrational rotation of the unit circle ${S^1}$ is uniquely ergodic.

Application: Equidistribution

Theorem 3 Let ${\xi \in \mathbb{R}}$ be irrational and let ${f: \mathbb{R} \rightarrow \mathbb{C}}$ be continuous and ${2 \pi }$-periodic. Then$\displaystyle \boxed{ \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{i=0}^{N-1} f( n \xi) = \int_0^1 f(x) dx .}$ (more…)

So, let’s fix a compact metric space ${X}$ and a transformation ${T: X \rightarrow X}$ which is continuous. We defined the space ${M(X,T)}$ of probability Borel measures which are ${T}$-invariant, showed it was nonempty, and proved that the extreme points correspond to ergodic measures (i.e. measures with respect to which ${T}$ is ergodic). We are interested in knowing what ${M(X,T)}$ looks like, based solely on the topological properties of ${T}$. Here are some techniques we can use:

1) If ${T}$ has no fixed points, then ${\mu \in M(X,T)}$ cannot have any atoms (i.e. ${\mu(\{x\})=0, x \in X}$). Otherwise ${\{x, Tx , T^2x, \dots \}}$ would have infinite measure.

2) The set of recurrent points in ${X}$ (i.e. ${x \in X}$ such that there exists a sequence ${n_i \rightarrow \infty}$ with ${T^{n_i}x \rightarrow x}$) has ${\mu}$-measure one. We proved this earlier.

3) The set of non-wandering points has measure one. We define this notion now. Say that ${x \in X}$ is wandering if there is a neighborhood ${U}$ of ${X}$ such that ${T^{-n}(U) \cap U = \emptyset, \forall n \in \mathbb{N}}$. In other words, the family of sets ${T^{i}(U), i \in \mathbb{Z}_{\geq 0}}$ is disjoint. If not, say that ${x}$ is non-wandering. Any recurrent point, for instance, is non-wandering, which implies that the set of non-wandering points has measure one.

Here is an example. (more…)

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