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12
On the unique games conjecture
 In FOCS
, 2005
"... This article surveys recently discovered connections between the Unique Games Conjecture and computational complexity, algorithms, discrete Fourier analysis, and geometry. 1 ..."
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This article surveys recently discovered connections between the Unique Games Conjecture and computational complexity, algorithms, discrete Fourier analysis, and geometry. 1
NONLINEAR SPECTRAL CALCULUS AND SUPEREXPANDERS
"... Nonlinear spectral gaps with respect to uniformly convex normed spaces are shown to satisfy a spectral calculus inequality that establishes their decay along Cesàro averages. Nonlinear spectral gaps of graphs are also shown to behave submultiplicatively under zigzag products. These results yield a ..."
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Cited by 15 (4 self)
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Nonlinear spectral gaps with respect to uniformly convex normed spaces are shown to satisfy a spectral calculus inequality that establishes their decay along Cesàro averages. Nonlinear spectral gaps of graphs are also shown to behave submultiplicatively under zigzag products. These results yield a combinatorial construction of superexpanders, i.e., a sequence of 3regular graphs that does not admit a coarse embedding into any uniformly convex normed space.
SPARSE QUADRATIC FORMS AND THEIR GEOMETRIC APPLICATIONS
"... In what follows all matrices are assumed to have real entries, and square matrices are always assumed to be symmetric unless stated otherwise. The support of a k × n matrix A = (aij) will be denoted below by supp(A) = { (i, j) ∈ {1,..., k} × {1,..., n} : aij ̸ = 0}. ..."
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Cited by 11 (0 self)
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In what follows all matrices are assumed to have real entries, and square matrices are always assumed to be symmetric unless stated otherwise. The support of a k × n matrix A = (aij) will be denoted below by supp(A) = { (i, j) ∈ {1,..., k} × {1,..., n} : aij ̸ = 0}.
Sharp quantitative nonembeddability of the Heisenberg group into superreflexive Banach spaces
, 2010
"... Let H denote the discrete Heisenberg group, equipped with a word metric dW associated to some finite symmetric generating set. We show that if (X, ‖ · ‖) is a pconvex Banach space then for any Lipschitz function f: H → X there exist x, y ∈ H with dW (x, y) arbitrarily large and ‖f(x) − f(y)‖ dW ..."
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Cited by 8 (4 self)
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Let H denote the discrete Heisenberg group, equipped with a word metric dW associated to some finite symmetric generating set. We show that if (X, ‖ · ‖) is a pconvex Banach space then for any Lipschitz function f: H → X there exist x, y ∈ H with dW (x, y) arbitrarily large and ‖f(x) − f(y)‖ dW (x, y) log log dW (x, y) log dW (x, y)
Inapproximability of NPcomplete problems, discrete fourier analysis, and geometry
 In Proc. the International Congress of Mathematicians
, 2004
"... Abstract. This article gives a survey of recent results that connect three areas in computer science and mathematics: (1) (Hardness of) computing approximate solutions to NPcomplete problems. (2) Fourier analysis of boolean functions on boolean hypercube. (3) Certain problems in geometry, especiall ..."
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Abstract. This article gives a survey of recent results that connect three areas in computer science and mathematics: (1) (Hardness of) computing approximate solutions to NPcomplete problems. (2) Fourier analysis of boolean functions on boolean hypercube. (3) Certain problems in geometry, especially related to isoperimetry and embeddings between metric spaces.
Expanders with respect to Hadamard spaces and random graphs
, 2012
"... Abstract. It is shown that there exists a sequence of 3regular graphs {Gn} ∞ n=1 and a Hadamard space X such that {Gn} ∞ n=1 forms an expander sequence with respect to X, yet random regular graphs are not expanders with respect to X. This answers a question of [NS11]. {Gn} ∞ n=1 are also shown t ..."
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Cited by 6 (2 self)
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Abstract. It is shown that there exists a sequence of 3regular graphs {Gn} ∞ n=1 and a Hadamard space X such that {Gn} ∞ n=1 forms an expander sequence with respect to X, yet random regular graphs are not expanders with respect to X. This answers a question of [NS11]. {Gn} ∞ n=1 are also shown to be expanders with respect to random regular graphs, yielding a deterministic sublinear time constant factor approximation algorithm for computing the average squared distance in subsets of a random graph. The proof uses the Euclidean cone over a random graph, an auxiliary continuous geometric object that allows for the implementation of
VERTICAL VERSUS HORIZONTAL POINCARÉ INEQUALITIES ON THE HEISENBERG GROUP
"... Let H = 〈a, b  a[a, b] = [a, b]a ∧ b[a, b] = [a, b]b 〉 be the discrete Heisenberg group, equipped with the leftinvariant word metric dW (·, ·) associated to the generating set {a, b, a −1, b −1}. Letting Bn = {x ∈ H: dW (x, eH) � n} denote the corresponding closed ball of radius n ∈ N, and wri ..."
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Cited by 4 (1 self)
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Let H = 〈a, b  a[a, b] = [a, b]a ∧ b[a, b] = [a, b]b 〉 be the discrete Heisenberg group, equipped with the leftinvariant word metric dW (·, ·) associated to the generating set {a, b, a −1, b −1}. Letting Bn = {x ∈ H: dW (x, eH) � n} denote the corresponding closed ball of radius n ∈ N, and writing c = [a, b] = aba −1 b −1, we prove that if (X, ‖·‖X) is a Banach space whose modulus of uniform convexity has power type q ∈ [2, ∞) then there exists K ∈ (0, ∞) such that every f: H → X satisfies n 2 k=1 x∈Bn ‖f(xc k) − f(x) ‖ q X k1+q/2 � K x∈B21n ‖f(xa) − f(x) ‖ q
Hardness of Approximation
"... Abstract. This article accompanies the talk given by the author at the International Congress of Mathematicians, 2014. The article sketches some connections between approximability of NPcomplete problems, analysis and geometry, and the role played by the Unique Games Conjecture in facilitating the ..."
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Abstract. This article accompanies the talk given by the author at the International Congress of Mathematicians, 2014. The article sketches some connections between approximability of NPcomplete problems, analysis and geometry, and the role played by the Unique Games Conjecture in facilitating these connections. For a more extensive introduction to the topic, the reader is referred to survey articles [39, 40, 64].