Семинары: O. D. Frolkina,

СЕМИНАРЫ


Узлы и теория представлений 17 ноября 2025 г. 18:30, г. Москва, Online, Zoom

[Properties of a compact set in $R^n$ and its projections] О. Д. Фролкина
Аннотация: Properties of projections of zero-dimensional sets were considered already at the end of 19th century. In 1884 G.Cantor described a surjection of the middle-thirds Cantor set onto the unit segment. Cantor sets in plane all of whose projections are segments were constructed by L.Antoine (1924), H.Otto (1933), A.Flores (1933), G.Noebeling (1933). In 1947, K.Borsuk described a Cantor set in $R^n$ such that its projection into any $(n-1)$-plane contains an $(n-1)$-ball. As a corollary, Borsuk obtained a knot in $R^n$ such that its projection into any $(n-1)$-plane contains an $(n-1)$-ball. There are many later results in this field. The author remarked that for any Cantor set $K\subset R^n$ there exists an arbitrarily small isotopy $\{ f_t \} :R^n\cong R^n$ such that the projection of $f_1(K)$ into any $(n−1)$-plane has dimension $(n−1)$; and there exists an arbitrarily small isotopy $\{ g_t \} : R^n \cong R^n$ such that the projection of $ g_1(K)$ into any $(n−1)$-plane has dimension $(n−2)$. In the talk, we will discuss these and other similar results using the Baire category approach. The questions on typical behaviour (in the sense of Baire category) are classic. A typical continuous function is nowhere differentiable (S.Banach-S.Mazurkiewicz 1931). A typical knot is wild (J.Milnor 1964) and moreover wild at any of its points (H.G.Bothe 1966). A typical compactum in Rn is a Cantor set (K.Kuratowski 1973). We will discuss the behavior of projections of a compactum $X\subset R^n$ under a typical isotopy of $R^n$, and as a corollary we will strengthen a theorem of J.Vaisala (1979). Язык доклада: английский Website: https://us02web.zoom.us/j/81866745751?pwd=bEFqUUlZM1hVV0tvN0xWdXRsV2pnQT09