The geometry of 3-manifolds

If you want to get an idea of the famous Poincaré conjecture for which Grigoriy Perelman was recently awarded a Millenium prize, I highly recommend watching this very entertaining lecture by Curtis McMullen at Harvard in 2006.

I had never realised that tori of genus greater than one are hyperbolic! It makes sense, if you think that a sphere (genus 0) has positive curvature, a torus (genus 1) is flat, that tori of genus 2 and more have negative curvature. These higher tori can be constructed by gluing together the edges of polygons.

I wonder whether anti-de Sitter space can also be viewed as a high genus space. Are the many black holes in the universe implying that it has a high "genus"? But the notion of genus isn't clear in higher dimensions...

So I started (very naively) to think about black holes on a latex surface, like the ones shown in the lecture. Consider a sphere and put a stone at the north pole. The sphere will bend, just like spacetime bends around the sun. Now imagine that you had a way to increase the mass of the stone, it would create a well at the north pole which would eventually touch the south pole from the inside. What happens then? The well could continue to become deeper and deeper but now it would create a spike out of the south pole. This is certainly misleading. More likely, once the stone approaches the south pole from the inside, its mass sucks it in, creating a well (as well).

Ultimately, what you'll end up with is a torus. The Planckian mass is at the centre of this torus, where there is really no spacetime: that's the black hole!

I had an unexpected confirmation of this picture in my kitchen. There was a frying pan full of a layer of greasy water and I noticed that it tended to leave discs where there was no water. By slowly pouring more water into the pan, the discs would shrink until there reach zero size, at which point a circular wave was emitted. I had just witnessed a topology change;) I could also reverse the process by taking some water out of the pan and stirring up the water. When the water had stabilised enough, discs were suddenly appearing and expanding quickly. That's my version of frying pan black holes: try it at home!

PS: the key is the circular wave.

"Mirror symmetry, Langlands duality, and the Hitchin system"

Today, at the Geometry and Analysis Seminar organized by Nigel Hitchin, Tamas Hausel talked about his paper with Michael Thaddeus Mirror symmetry, Langlands duality, and the Hitchin system. The room was packed. He started by giving some background informations about the three concepts in his title.

Mirror symmetry

The basic idea is that the symplectic geometry of a d-dimensional Calabi-Yau manifold $X$ can be related to the complex geometry of another CY manifold $Y$. There is a topological test of this relation, referred to as topological mirror symmetry, which equates (mirror pairs of) Hodge numbers of the two CYs:
\[ h^{p,q} (X) = h^{d-p,q}(Y)~. \]

Note that any hyperkaehler manifold satisfies $h^{p,q} (X) = h^{d-p,q}(X)$, so in a certain sense TMS is already built-in. Tamas mentioned two important developments in the history of mirror symmetry: homological mirror symmetry proposed by Kontsevich in 1994, which reads
\[
\mathcal{D}^b (\text{Fuk}(X,\omega)) \cong \mathcal{D}^b (\text{Coh}(Y,I))~,
\] where $\omega$ is the symplectic form and $I$ the complex structure. Another breakthrough was the geometric construction of $Y$ from $X$ elaborated by Strominger, Yau and Zaslow in 1996.

Langlands duality

The aim of the Langlands program is to describe $\text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ via representation theory.
To each reductive group $G$ is associated a Langlands dual $^LG$. The Langlands conjecture leads for instance to class field theory, in the case $G=GL_1$; in the case $G=GL_2$, it leads to the Taniyama-Shimura conjecture (which is famous because it implies Fermat's last theorem). An important progress towards the proof of the conjecture was made by Ngo in 2008 with his proof of the fundamental lemma for the function field $\mathbb{F}_q(X)$.
There is a geometric version of the conjecture, obtained by replacing $\mathbb{F}_q(X)$ by $\mathbb{C}(X)$ for $X/\mathbb{C}$ (Laumon 1987, Beilinson & Drinfeld 1995):
\[
\{ G\text{-local systems on $X$} \}
\]\[
\leftrightarrow \{ \text{Hecke eigensheaves on Bun$_{^LG}(X)$}\}~.\]
Hitchin systems

Recall that a Hamiltonian system $(X^{2d}, \omega)$ has an energy functional $H: X\to \mathbb{R}$ and an Hamiltonian vector field $X_H$ such that $\text{d}H=\omega(X_H,\cdot)$. A function $f:X\to \mathbb{R}$ is a first integral if $X_H f = \omega(X_H, X_f) =0$ (involution). The system is completely integrable if there is $d$ first integrals. The generic fibre is then a torus (examples: Euler and Kovalevskaya tops, spherical pendulum).
An algebraic version is obtaiend by replacinf $\mathbb{R}$ by $\mathbb{C}$, and many examples can be formulated as Hitchin systems (1987).

Now I cannot say I completely followed the rest of the talk in all its glory, but I'll try to restate what I understood. Tamas was considering different moduli spaces, which are all smooth non-compact varieties: $\mathcal{M}_{\text{Dol}}^d (G)$ is the moduli space of rank $n$ and degree $d$ Higgs bundles $(E,\phi)$, $\mathcal{M}_{\text{DR}}^d (G)$ is the moduli space of flat $G$-connections on a genus $g$ curve, $\mathcal{M}_{\text{B}}^d (G)$ is yet another thing -- but they're all equivalent by a non-Abelian Hodge theorem. The Hitchin map $\chi(\phi)$ is completely integrable and its fibre $\chi^{-1}(a)$ is a torsor.

Inspired by the SYZ conjecture, Hausel and Thaddeus noticed in 2003 that $\chi^{-1}_{SL_n}(a)$ and $\chi^{-1}_{PGL_n}(a)$ are torsors for dual Abelian varieties. (I think this means they are related by T-duality on the toroidal Hitchin fibres, but he said "fibrewise Fourier-Mukai tranform" instead :)

A confirmation that their conjecture are more or less sane came from the 2006 work of Kapustin and Witten on S-duality (electric-magnetic duality) in $\mathcal{N}=4$ super-Yang-Mills in four dimensions, which Tamas qualified as "a major work" with many fertile ideas. (In fact T-duality in the Hitchin moduli space corresponds to S-duality in the gauge theory, see Witten's Strings on the Beach! talk in 2005.)

Using stringy Hodge numbers (also known as "orbifold cohomology") Tamas made a conjecture with a TMS test, but he also made another one using mixed Hodge numbers.
His final questions were: "Why two conjectures? Why same Hodge numbers instead of mirrored ones? Why Geometric Langlands and not classical Langlands?

He ended up by mentioning a curious hard Lefschetz conjecture for weight and perverse filtrations that left the crowd speechless.

Alien mathematics

Unlike what you might be expecting from the title, this post is not about Großendieck's fundamental reploughing of algebraic geometry, but about the following hypothetical question:

Imagine we discover another civilisation living in some corner of some galaxy, where we know that the physics is essentially the same than on earth. Would they have the same mathematics than us? And if yes, would they have the same mathematics history than us?

David Gross was proposing this thought exercise at the end of a public talk by Robbert Dijkgraaf at KITP, and I figured the best way to attack it was by taking a hot bath. In fact, the bath was so hot that I could feel my spirits evaporate and I felt that were I to think about this question in such circumstances, I would develop biased conceptions that would definitely screw all my chances to reach any interesting conclusions... Nevertheless I did it, and it's a mess.

(Note that Dijkgraaf wasn't too inspired by the question -- I guess this is the difference between a Nobelised and a non-Nobelised physicist: the former becomes philosophically oriented (remember Josephson))

Probably it is sensible to start by the second part of the question, assuming the answer to the first is positive. Then it is obvious that it would be quite challenging to defend the opinion that their math history is exactly the same as ours. They would have needed to have e.g. an Evariste Galois killed at 20 in an obscure duel, etc. But might they have had the same structure in the development of their mathematics?

What would Kant say about this question? I believe that it would be something along the line of "Their mathematics would resemble ours inasmuch as we are able to perceive them, and them us. (After all they could be made of "dark matter", in which case it would be strange if they had the same concepts as ours...)