Can anybody sneak into a math seminar?
Apparently one can! I tried two days ago, and it was fun! Living in Bonn (as I do) has many advantages: There's the traditional Rheinische Friedrich Wilhelm University
, relating to many mathematical celebreties such as Hausdorff
, and Tits
, to name a few; famous physicists, among them names like Pauli
, and Hertz
; as well as illustrous figures like Karl Marx
, and Nietzsche
And besides other mathematical institutes there's the renowed Max Planck Intitute
(MPI), attracting nerds of the field from all over the world. They have a nice location right at the heart of the city, and when I happened to come close to +Urs Schreiber
's seminar at the right time, I couldn't resist - and rang the bell.
When you get through the entrance on the second floor, the view's a bit surprising. There's another stairway in a glass box that, together with the windows, evokes a transparent and varied atmosphere. I remember thinking that in case I see people walking by, who are intensely staring at their shoes, that would be because of vertigo...
Much of the lecture was far beyond me, even though I had looked at the notes (versioned about 12 hours before, way out of date), and the notation strongly reminded me of stuff I ought to know reasonably well. I felt like an impostor, or like a child, albeit an ugly one - because I'm old :)
In the following days I managed to make sense of a larger part of it, by tediously looking stuff up, staring at the slides, and puzzling over my notes. I'll give you the maths in a bit, just let me tell you first why it took me a while to get home.
My fierce plan was this: Meet someone I only know from google+ and... shake his hand. That's me aiming low because missing a high target makes me tumble, a trait that had confused me and others before. So after the lecture, and some innocent looking chasing, happy to meet in person a guy I've had nice online chats with: "Professor, ..."
There was no need to be afraid to begin with. We had a nice quick exchange of random bits of personality, and I learned that some of the audience was gathering to jointly hunt for lunch outdoors. I happily followed to make a group of five, pertained myself an ascetic glass of liquid, but really nourshing from the topics that came up.
That has been even more fun than letting the snake in my head swallow an elephant! Should there be physical effects in software development? Did you know that you can have nice chats, similar to the occasional comment parties here, but in real life - if you manage to get enough mathematicians around a table with food! And I also got an answer to: What is an AKSZ manifold? Which leads us back to the topic of the seminar lecture:"Prequantum field theories from Shifted symplectic structures"
A field in the physical sense is a map that assigns an object to every point in space (-time). If the object is a number (e.g. temperature), it's called a scalar field. If it's a vector (wind direction), it's a vector field. People also study tensor fields, spinor fields, and other stuff.
Quantum field theories study "excitations" in those fields, which are similar to water waves, but ones that don't dissipate and exist only at specific energy levels (heights). Most unfortunately, these excitations are called "particles". Then what are prequantum field theories
> Prequantization produces a natural Hilbert space together with a quantization procedure for observables that exactly [...] transforms Poisson brackets on the classical side into commutators on the quantum sidehttps://en.wikipedia.org/wiki/Geometric_quantization
A little further on that page we read:
> Prequantization of a symplectic manifold (M, Omega) provides a representation of elements f in C-infinity(M) of the Poisson algebra of smooth real functions on M by first order differential operators f-hat on sections of a complex line bundle L -> M. In accordance with the Kostant-Souriau prequantization formula, these operators are expressed via a connection on M whose curvature form R obeys the prequantization condition R = i·Omega.
Wait, you say? That's too fast? You may have heard about phase spaces. Well, symplectic manifolds
are just that. Every point, or phase
, is equipped with two vectors, a position and a direction (momentum), the latter is called covector
and written as a differential form
. That's a derivative arranged such that it doesn't change under coordinate transformations. The position vector is M
, the differential form Omega
Feeling lost? Warm yourself up here:https://en.wikipedia.org/wiki/Phase_space
More on track, a longer, gentle take on symplectic manifolds
by Ben Webster
at the secret blogging seminar
You often see a symplectic space
handled using the corresponding Poisson algebra
, an equivalent formalism. f
is an infinitely often differentiable function returning complex numbers defined on the manifold M
gives us f-hat
on sections of a family of complex lines
, by finding the connection
(that's another derivative/direction along the manifold), where the curvature R
is orthogonal to the phase.
If all that reminds you of Hamiltonian mechanics
in high-brow language, I suppose that's because it is, carefully arranging things to generalize it all.
I haven't quite gotten to get what exactly shifted
means, it's getting late here, and I'm too lazy to connect it all to AKSZ manifolds
now. But you can try reading the lecture notes on your own here:http://ncatlab.org/schreiber/show/Prequantum+field+theories+from+Shifted+symplectic+structures
I spent much of the remaining afternoon watching things getting wrecked at a large demolition site with my son. It's been going on for months now, right next to his kindergarden, groundshaking excavators