Well, this is my title, Mathematics of Topological Materials. You see I'm from Erlangen, there

is a little bit of an abstract, but the first question which I would like to answer is why

I'm here to speak about this because I'm certainly an odd outlier of the topics of the conferences.

I don't work in control, optimization, nonlinear PDEs, networks. The true reason is that Enrique,

who's not sitting here right now, has his office one floor above me, but there are also

a number of other questions. I have a lot of collaborators in Latin America. Currently

I have one PhD student in UNAM and we are battling to have him have a Cotutel PhD thesis.

It's something which is very, very difficult to attain, but we are getting very, very close

to it now. I have collaborators in Cuanabaca, you see, and then also in Chile, and I have

many others had before in other Latin American countries. So that makes sense. My trend,

which I want to talk about, is rather a trend in physics than in mathematics about topological

materials and I suppose that almost nobody has ever heard about something like that.

So mainly I try to give you a little bit of an appetizer of what we are doing in that

field and why it's interesting, both from a physics and from a math point of view. So

the whole story actually starts a long time ago in 1980 with the quantum Hall effect.

What is that? The quantum Hall effect was a two-dimensional electron gas which people

could actually produce at the interface of two semiconductors, which has very particular

properties due to a topological invariant. Topological invariant is something which is

known from topological, well, from differential topology. It's churn numbers in two dimensions.

I'll tell you a little bit what that is in a second. And these topological invariants,

they are due to somehow a global twist in the wave packets, the quantum mechanical wave

packets inside of the models or inside of the electron system. And this topological

quantity is really in this case an integer number and all the situations which I talk

about, they're integer numbers, so they're very, very robust invariants. They're important

because there is something called the bulk boundary correspondence, which tells you that

in such materials where you have topologically, topological non-trivial invariants, there

is an effect that you can see on the boundaries of these systems. And I want to illustrate

you what that means, but it means that this topology really is responsible for physical

phenomena which also are mathematical phenomena, of course, that have many applications. In

particular, they are behind this quantum Hall effect. So the quantum Hall effect was from

the 80s and there was not much happening on the physics side. There are lots of math

papers about that, but on the physics side there was a crucial event in 2005. People

started realizing that what was crucial in quantum Hall effect that you have next strong

magnetic fields was not really necessary. So people started having topological insulators

which were time reversal invariant, so without magnetic fields. They could realize also that

these effects are not restricted to dimension two. You can have similar things in dimension

one, three, four. Four makes sense because you could have time-driven systems. And the

bulk boundary correspondence is something which is valid in all of these systems, okay?

Something extremely robust. And I would like to give you sort of a flavor of what is behind

that. So since 2010 then there's also a bunch of new things which happened, so there were

topological materials. These are topological photonics, topological mechanical systems.

Actually people realized that whatever kind of a system that you describe by a sort of

wave type equation is susceptible to have this phenomenon of having topological invariance

and associated bulk boundary correspondence. So what is common to all these systems is

that they have periodic structure behind them and this periodic structure leads to

the fact that if you go to the Bloch analysis of this system, the associated Bloch wave

packets, they have twists that lead to these topological invariants. And then there is

this bulk boundary correspondence which is generally robust principle, okay? So there's

a bunch of Nobel prizes associated to this phenomenon. I mean, from Plitzing for the

quantum Hall effect, but then for fractional quantum Hall effect there was Laflin and well