Yeah, okay, great. I consent. This is me, I'm Alden, and I'm at the Bernoulli Institute

in the Netherlands. This is joint work with Professor Alexander Stromae, who is at the

University of Leeds. Alexander Stromae is also my husband, so anyhow. And now my slides don't

work. There we go. So let me first start with an outline of the talk. So I'm going to introduce

those of you to what the idea of reachability is, what exactly are we talking about, what

does it mean to be reachable. I'll talk a little bit about background on analytic functions,

some notation which should be a review to some of you and for others is always sort

of part of the lore of analytic functions. I'll give a forward direction theorem, and

how this is proved is using what we call thermal layer potential theory. And this is an idea

that actually came from another paper that I was writing, and this has been known to

the numeric community for a while. Usually people do layer potential theory for the Helmholtz

equation, and so I was using that technique and I wondered if people had done it for the

heat equation, and then I figured out I could apply to it here. I'll give an example, which

is a cute little example, that shows you that this forward direction theorem is sharp. So

I don't expect too much better in the case of the forward direction for Lipschitz domains.

I'll talk about analyticity and convergence of heat kernels. So one of the parts of the

title is the analytic properties of heat equation solutions. So we did just a little bit more

than prove facts about reachable sets. We also proved some things about how you can

extend heat kernels into the complex plane. This is an interesting topic of separate value

to harmonic analysts. And then I'll give the converse direction theorem and talk about

where this fits in. Some other people sitting in the audience have actually made the significant

contributions that started this field, Professor Zuzura being one of them. And then I'll show

you some room for improvement because I don't think that this converse direction theorem

is quite sharp in higher dimensions. Okay, so it's a good place to start with the problem.

So we're looking at the heat equation, so there's nothing super fancy. We have the heat

equation here. I don't know if you can see my pointer or not, but let's pretend you can

and you're looking at the first line. So this is on a bounded Lipschitz domain. So this

is the least amount of regularity that you could expect for such a domain. Otherwise,

this problem is not necessarily well posed. We have some initial data in the bounded Lipschitz

domain, and we have some data which lives on the boundary cylinder. Okay, and this data

is his non-zero, and this makes the problem actually quite hard. So, all right, if we

have initial data in H0, omega on a finite time interval for positive times t, then the

solution u is in this mixed Sobolev space. This little comma, this kind of funny Sobolev

space is the intersection of two Sobolev spaces. We'll get to that definition a bit later.

So with some regularity of H, this problem is well posed, and we're asking ourselves

the question of, if I know that I have a certain class of functions and the analytic, do they

come from a heat equation problem like this? So let us look at the formal definition of

the reachable set. Okay, so here I define the reachable set. So this is the set of states

v equals u capital T of x, where u at all times t beforehand solves this boundary value

problem with Dirichlet boundary controls, and this is referred to as the reachable set.

Okay, so this is r here. I'm using some notation which is, I think, more formalized by Sylvan

and Jeremy Dardet in a paper that they have in Siam. Okay, so null controllability of

the heat equation with boundary controls gives us that, so if this was just zero, we know

that this equation is null controllable, so you can force it to the state zero and some

finite time t. So you can subtract off all initial states by linearity, and so what happens

is that it's instead sufficient to consider the problem with u of zero equals to zero,

and so we're looking at ht of x and the set of reachable sets with these types of boundary

controls in the boundary cylinder. So for lack of a board, I keep pretending like I'm

a pantomime, but that's okay. So this doesn't depend on the event time horizon. All right,

so we have omega as a bounded Lipschitz domain in rd. This is the reachable set, and the