News

Lecture of Károly Simon

Projection of random Cantor sets


The Department of Geometry is pleased to divulge that

Simon, Károly Károly Simon
(BME, Budapest, Hungary)

gives a lecture at the Kerékjártó Seminar with title

Véletlen Cantor halmazok vetületei
(Projection of random Cantor sets)

Date and place of the lecture is:

Thursday February 14, 2019, at 12:30,
room Riesz (BO-107)

Abstract of the lecture:
As a mathematical model for turbulence, Mandelbrot introduced the so-called fractal percolation Cantor sets (also called Mandelbrot percolation Cantor sets) in the 1970s. These are random subsets of the unit cube $K=[0,1]^d$ which are generated as follows: We partition $K$ into $M^d$ cubes $K_{i_1,\dots,i_d}$ of side length $1/M$ and we assign to each of them a retention-probability $p_{i_1,\dots,i_d}$. That is each cubes $K_{i_1,\dots,i_d}$ are retained with probability $p_{i_1,\dots,i_d}$ and discarded with probability $1-p_{i_1,\dots,i_d}$ independently. The union of these level-1 cubes forms the first approximation $E_1$ of the fractal percolation Cantor set $E$. Then we repeat this process in every cube retained in the previous step. The union of the cubes (of side length $1/M^2$) retained in the second step $E_2$ is the second approximation of the fractal percolation Cantor set. We construct the $n$-th approximation $E_n$ in the same way. It happens with a positive probability that $E_n$ is empty. Then $\{E_n\}$ is a nested sequence of compact sets. Their intersection is the fractal percolation Cantor set $E$. If all the probabilities $p_{i_1,\dots,i_d}$ are the same then $E$ is called homogeneous otherwise $E$ is inhomogeneous.
We study the projections of $E$ to lower dimensional planes and the intersections of $E$ with lower dimensional planes. Due to the many interesting results of the recent years on this field, we have a good understanding of the projections and slices of the fractal percolation Cantor sets in the homogeneous case but our understanding is far from complete in the inhomogeneous case.

 

Here are some snapshots of the event:

images/math-site/meetings/Seminar/20190214-SimonK/web/big/IMG_20190214_123302.jpg

Tájékoztatás:
az eseményeken rendszerint kép- és hangfelvétel is készül tömegfelvételek formájában, egyben az esemény sajtónyilvános rendezvény is.
A Polgári Törvénykönyvről szóló 2013. évi V. törvény 2:48. § (2) bekezdése alapján a tömegfelvételek, valamint a nyilvános közéleti szereplés esetén nincs szükség a résztvevők hozzájárulására sem a felvétel elkészítéséhez, sem annak felhasználásához, de az érintetteket erről előzetesen tájékoztatni kell.
Kötelezettségünknek jelen szöveg megjelenítésével teszünk eleget azzal megtoldva, hogy jelezzük: a felvételeket az esemény népszerűsítésére, marketing céllal, online és nyomtatott csatornáinkon keresztül használjuk fel.


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