Mathematics Colloquia and Seminars

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Let $Delta$ be a $(d-1)$-dimensional simplicial manifold with $n$ vertices. One of the problems in combinatorics is: What is the maximum possible number of $i$-dimensional faces of $Delta$? What is the maximum possible Euler characteristic of $Delta$? The Upper Bound Conjecture (UBC) asserts that if $Delta$ is a Eulerian simplicial manifold than for any $i$ the number of $i$-dimensional faces of $Delta$ is not greater than the number of $i$-faces of the cyclic $d$-polytope with $n$ vertices. In this talk we will outline the proof of the UBC for all Eulerian manifolds. We also will sketch the proof of the analog of the UBC for arbitrary (non-Eulerian) simplicial manifolds and the (partial) proof of conjecture by K\"{u}hnel concerning the maximum Euler characteristic of even-dimensional simplicial manifolds.