The topological Tverberg conjecture was considered a central unsolved problem of topological combinatorics. The conjecture asserts that for any integers $r,d>1$ and any continuous map $f:\Delta\to\mathbb R^d$ of the $(d+1)(r-1)$-dimensional simplex there are pairwise disjoint faces $\sigma_1,\ldots,\sigma_r\subset\Delta$ such that $f(\sigma_1)\cap \ldots \cap f(\sigma_r)\ne\emptyset$. The conjecture was proved for a prime power $r$. Recently counterexamples for other $r$ were found. Analogously, the $r$-fold van Kampen-Flores conjecture holds for a prime power $r$ but does not hold for other $r$. The arguments form a beautiful and fruitful interplay between combinatorics, algebra and topology. We present a simplified exposition accessible to non-specialists in the area. We also mention some recent developments and open problems.
翻译：顶端的 Tverberg 参数被认为是一个尚未解决的表层组合学中心问题。 猜测称, 对于任何整数 $r, d>$ 和任何连续的地图 $f:\ Delta\ to\mathbRd$( d+1)(r-1)(r-1)- 立方简单x 美元( d+1)(r-1) 美元) 中的任何美元, 上面有双向脱节面容 $sgma_ 1,\\ sigma_r\ subset\ Delta$, 以至于 $f( gma_ 1)\ cap f (\ cap f (\ sigma_r)\ nne\ suntyset $) 。 预测称, 任何整数 美元和任何连续的地图 $( $) $: dr$( d) $( d+1)( d+1) ( mall) (mathbbbbbbb) 简单 $( ) $) $( ) $( ) $( ) $ ( mark) 。 $( $ 1) $ ( $ 1) 和 $( gap (n\ $) $) $) discap (nence of acap (nationt) $) lab) coutus cotor) sub) sub) sub) cotoric cotoric ( sub) sub) sub) sub) subles sub sub sub sub sub sub sub subs subly subly subly subly subly subly.
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