novel types of short and long range order, and to understand their physical properties.

And they're sometimes quite surprising.

And an especially nice example I want to present to you today are the photonic band gaps.

But before I come to that, let me just give you a very brief introduction to the types

of structures that I'm working with.

So my research is at the interplay between mathematics and physics, particularly between

the geometry and physics of complex spatial structures.

You can see here a variety of structures that I've worked on from astronomical scales, the

supernova remnant, down to exotic states of nuclear matter.

And here in Erlangen, we are particularly interested in quantification, a very universal,

comprehensive way of quantifying these complex structures using integral geometry, and in

particular the so-called Minkowski functionals or intrinsic volumes.

And of course, I won't go here into any detail, but just the key idea is that these functionals

in 3D, they are four functionals, are all additive, meaning the functional of the union

of two sets is just the functional of the sum of the functions of the single sets minus

the intersection.

And what's now so special about these Minkowski functionals and why we're so interested in

that they comprise actually all, so to speak, additive shape information, which is the content

of the celebrated Hartvigga theorem stating that any additive, continuous, and motion

invariant functional on the set of convex bodies is actually a linear combination of

these, for example, four functionals in 3D.

So we use them a lot and also provide software for them, but that's not the theme of this

talk.

Another, as I said, central theme of my research is that of novel types of disorder and order.

And an especially fascinating example is that of hyperuniformity.

It's an emerging field in physics, and what you can think of it as a kind of a hidden

type of long-range order.

Let me visualize this.

So what you see on the left-hand side is the snapshot of what you could call a garden variety

type of disorder.

It's actually just a Poincare point process, snapshot of the ideal gas, just complete spatial

randomness.

What you see on the right-hand side might look quite similar.

And in fact, on such a local scale, I can make it as indistinguishable as you like.

Now, what's the difference?

The difference you only see as we zoom out.

As we zoom out, you will see that on the left-hand side, there will be density fluctuations on

all scales.

But on the right-hand side, the large-scale density fluctuations suddenly cancel out,

and the system becomes homogeneous, homogeneous like crystal.

And in fact, that can be made mathematically precise.

How?

Well, it is actually pretty simple.

We throw in a ball, B, let's say, of some certain radius inside our system and count

simply the number of points of particles inside that ball.

And then we see how does this is a random number, so it has a variance.

How does this variance scale with the size of the ball?

Now, in a typically disordered system, any typical type of liquid or disorder that in

physics you usually encounter, this variance will scale like the volume of the ball, the

density fluctuations in the bulk all throughout the ball.