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We survey the techniques used in our recent resolution of the Kadison-Singer problem and proof of existence of Ramanujan Graphs of every degree: mixed characteristic polynomials and the method of interlacing families of polynomials.
Linear hyperbolic partial differential equations with constant coefficients
L. Gårding · 1951
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Extensions of pure states
R. V. Kadison and I. M. Singer · 1959
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Theory of monomer-dimer systems
O. J. Heilmann and E. H. Lieb · 1972
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Explicit constructions of concentrators
G. A. Margulis · 1973
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Extensions, restrictions, and representations of states on C ∗ C^{*} -algebras
J. Anderson · 1979
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Extreme points in sets of positive linear maps on ℬ ( ℋ ) \mathcal{B}(\mathcal{H})
J. Anderson · 1979
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Zeros of convex combinations of polynomials
H. J. Fell · 1980
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A conjecture concerning the pure states of ℬ ( ℋ ) \mathcal{B}(\mathcal{H}) and related theorem
J. Anderson · 1981
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Matchings and walks in graphs
C. D. Godsil · 1981
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On the matching polynomial of a graph
C. D. Godsil and I. Gutman · 1981
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A general Lee-Yang theorem for one-component and multicomponent ferromagnets
E. H. Lieb and A. D. Sokal · 1981
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Eigenvalues and expanders
N. Alon · 1986
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Invertibility of large submatrices with applications to the geometry of Banach, spaces and harmonic analysis
J. Bourgain and L. Tzafriri · 1987
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Ramanujan graphs
A. Lubotzky, R. Phillips, and P. Sarnak · 1988
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Explicit group theoretical constructions of combinatorial schemes and their application to the design of expanders and concentrators
G. A. Margulis · 1988
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Lyapunov theorems for operator algebras
C. A. Akemann and J. Anderson · 1991
Cited alongside, same era.
On a problem of Kadison and Singer
J. Bourgain and L. Tzafriri · 1991
Cited alongside, same era.
On the second eigenvalue of a graph
A. Nilli · 1991
Cited alongside, same era.
Obreschkoff’s theorem revisited: what convex sets are contained in the set of hyperbolic polynomials?
J. P. Dedieu · 1992
Cited alongside, same era.
Hyperbolic polynomials and convex analysis
H. H. Bauschke, O. Güler, A. S. Lewis, and H. S. Sendov · 2001
Cited alongside, same era.
John’s decompositions: Selecting a large part
R. Vershynin · 2001
Cited alongside, same era.
Strong converse for identification via quantum channels
Applications of stable polynomials to mixed determinants: Johnson’s conjectures, unimodality, and symmetrized fischer products
J. Borcea and P. Brändén · 2008
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Van der Waerden/Schrijver-Valiant like conjectures and stable (aka hyperbolic) homogeneous polynomials: one theorem for all
L. Gurvits · 2008
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The Lee–Yang and Pólya–Schur programs, I. Linear operators preserving stability
J. Borcea and P. Brändén · 2009
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Multivariate Polya-Schur classification problems in the Weyl algebra
J. Borcea and P. Brändén · 2010
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Sparse quadratic forms and their geometric applications (after batson, spielman and srivastava)
A. Naor · 2011
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R. Ahlswede and A. Winter · 2002
Cited alongside, same era.
The Kadison–Singer problem in discrepancy theory
N. Weaver · 2004
Cited alongside, same era.
Lifts, discrepancy and nearly optimal spectral gap
Y. Bilu and N. Linial · 2006
Cited alongside, same era.
Expander graphs and their applications
S. Hoory, N. Linial, and A. Wigderson · 2006
Cited alongside, same era.
On extreme zeros of classical orthogonal polynomials
I. Krasikov · 2006
Cited alongside, same era.
Hyperbolic programs, and their derivative relaxations
J. Renegar · 2006
Cited alongside, same era.
Graph sparsification by effective resistances
D. Spielman and N. Srivastava · 2011
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Spectral sparsification of graphs
D. A. Spielman and S.-H. Teng · 2011
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Twice-Ramanujan sparsifiers
J. Batson, D. A. Spielman, and N. Srivastava · 2012
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User-friendly tail bounds for sums of random matrices
J. A. Tropp · 2012
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Restricted invertibility and the Banach–Mazur distance to the cube
P. Youssef · 2012
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Interlacing families I: Bipartite Ramanujan graphs of all degrees
A. W. Marcus, D. A. Spielman, and N. Srivastava · 2013
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Interlacing families II: Mixed characteristic polynomials and the Kadison–Singer problem
A. W. Marcus, D. A. Spielman, and N. Srivastava · 2013
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Windows on theory: Discrepancy, graphs, and the kadison-singer problem, 2013
N. Srivastava · 2013
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Covariance estimation for distributions with 2 + ε 2+\varepsilon moments
N. Srivastava and R. Vershynin · 2013
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Pipage rounding, pessimistic estimators and matrix concentration
N. J. A. Harvey and N. Olver · 2014
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