Give comments and the talks will be around 25 minutes and then we will have time to discuss

and the schedule is not completely fixed so it might take a bit, hopefully it will take

a bit longer than 12 to be finished for the four speakers that we have.

And we'll start with Ilias, I'm sorry your last name is a bit hard for me to pronounce

yet, and he will give a talk on the estimation of the fundamental

frequency.

Okay, thank you Tobias for the invitation and for the introduction, and thank you everybody

that are attending the seminary for today that allows me to present a little bit of

what I've been working on during my PhD thesis with the Jimmy Lombolet in Paris.

So today I picked up one really I find it interesting subject that I worked on during

my PhD, it's about the estimation of the fundamental frequency and I will explain the details during

the talk.

So before I will just give the plan of the talk, I will give a brief introduction where

I will talk about shape optimization which is the topic that I've been working on for

my PhD.

And I will give an introduction on Blaschko-Santalo diagrams that I will explain the use and I

will give examples of people working on these kind of diagrams.

Then I will be interested in the study of a special Blaschko-Santalo diagram relating

three involved functionals, I will give the details about these functional after.

So I will discuss as I said the class of open sets and the class of convex sets and give

some sketches of proofs and numerical results obtained in a WIDGIM.

So the last part will be about some perspective and current research projects that I'm willing

to perform here in the FAO.

So first of all let us begin by the introduction, what is shape optimization?

Okay, so because we are mathematicians I will go to the formulas.

So if J is a functional that means a function that takes an argument as a set and gives

a real number.

So this functional can be for example the perimeter, can be the volume, so the volume

or the measure you give it set omega and it gives you a real value and this is what we

call a shape functional.

So we are interested in studying problems of the type, so we are talking about optimization,

so I am interested in studying problems of the following type, infimum of omega in a

certain class of subsets of omega, it can be for example the class of simply connected

domains or the class of doubly connected domains or for example the class of convex sets.

This is a constraint that you add on your set omega in order to have a problem which

is non-trivial to resolve as every optimization problem.

So many questions can be asked in the framework of shape optimization.

The first one is does this problem admits a solution?

So is this infimum, a minimum or just an infimum?

So there are several techniques to prove the existence of solutions and there are examples

where the solution does not exist.

This is related to some, for example, homogenization methods that prove that you cannot attend

sometimes this infimum.

So if the infimum exists, which means that there is a set omega star which is satisfying

the infimum here, so can we prove some qualitative results about it?

Because the graph will be to find this optimal set, but as you can imagine, it is quite complicated

and quite difficult to solve every shape optimization problem in a theoretical point of view.

So sometimes we try to prove regularity or symmetries of the set and have more knowledge

about these optimal sets.

Another interesting question too is that can we prove that a given shape is local solution