Singularities, Expanders and Topology of Maps. Part 2: from Combinatorics to Topology Via Algebraic Isoperimetry

By Misha Gromov. 2010

266 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3861447e9eed766feef113e34d1a0c67ec240b91

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Citations in the corpus

The eigenvalue method for cross t-intersecting families (2013, background): “Gromov [10] showed that such inequalities concerning cross intersecting structures can be equivalently reformulated in terms of monomial subsets in the N-torus, and for example he obtained a homological separation inequality for pairs of disjoint subsets in N-torus from an inequality obtained in [11”
An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves (2013, background): “how that valid (p;q) and [[E A(p;q)]] exist given that we have xed valid (p0;q0) and [[E A(p0;q0)]] for all 19That this should be a contractible choice is suggested by the Dold{Thom{Almgren theorem [Gro10, p430]. 18 pp0˚q0q(other than (p 0;q) = (p;q)). The inductive nature of the existence proof is ”
Sharp vanishing thresholds for cohomology of random flag complexes (2012, unknown): “es random complexes are known to have very large torsion groups on average [19]. This work can also be viewed in the context of higher-dimensional expanders; see for example the recent work of Gromov [13]. Acknowledgements I thank NogaAlon, Eric Babson, Chris Hoﬀman, RoyMeshulam, Elliot Paquette, an”
Ramanujan complexes and high dimensional expanders (2012, methodology): “ensional complete complexes have the overlap property. This was proved for the geometric overlap in [BF84] for dim 2 and in [Bár82] for all dimensions. For the topological overlap, this was proved in [Gro10] (see also [MW11, Kar12]). 25 The main result of [FGL+10] asserts that there even exist simpl”
Generalizations of the Kolmogorov–Barzdin embedding estimates (2011, methodology): “ Fill(j) is ﬁnite, but for an inﬁnite complex, it could be +∞.) The combinatorial width of a complex can be controlled in terms of its ﬁlling inequalities by the following theorem. Width and ﬁlling. ([Gr2]) Suppose that X is a k-dimensional simplicial complex or cubical complex with N vertices. Supp”
Spectral Gaps of Random Graphs and Applications (2012, background): “See, for example, recent work by Gromov and collaborators in [26], [27], and [20]).”