Showing papers in "Forum of Mathematics, Pi in 2019"
TL;DR: In particular, the authors showed that the set of (tensor-product) quantum correlations is not closed, and that it is undecidable to determine if a linear system game can be played perfectly with a finite-dimensional strategy, or a limit of finite dimensional quantum strategies.
Abstract: We construct a linear system nonlocal game which can be played perfectly using a limit of finite-dimensional quantum strategies, but which cannot be played perfectly on any finite-dimensional Hilbert space, or even with any tensor-product strategy. In particular, this shows that the set of (tensor-product) quantum correlations is not closed. The constructed nonlocal game provides another counterexample to the ‘middle’ Tsirelson problem, with a shorter proof than our previous paper (though at the loss of the universal embedding theorem). We also show that it is undecidable to determine if a linear system game can be played perfectly with a finite-dimensional strategy, or a limit of finite-dimensional quantum strategies.
98 citations
TL;DR: In this paper, the authors consider Brownian last passage percolation in scaled coordinates and show that the energy of long energy-maximizing paths may be studied as a function of the paths' pair of endpoint locations.
Abstract: In last passage percolation models lying in the Kardar–Parisi–Zhang (KPZ) universality class, the energy of long energy-maximizing paths may be studied as a function of the paths’ pair of endpoint locations. Scaled coordinates may be introduced, so that these maximizing paths, or polymers, now cross unit distances with unit-order fluctuations, and have scaled energy, or weight, of unit order. In this article, we consider Brownian last passage percolation in these scaled coordinates. In the narrow wedge case, when one endpoint of such polymers is fixed, say at . This result is proved by harnessing an understanding of the uniform coalescence structure in the field of polymers developed in Hammond [‘Exponents governing the rarity of disjoint polymers in Brownian last passage percolation’, Preprint (2017a), arXiv:1709.04110] using techniques from Hammond (2016) and [‘Modulus of continuity of polymer weight profiles in Brownian last passage percolation’, Preprint (2017b), arXiv:1709.04115].
57 citations
TL;DR: In this paper, an upper bound on the log canonical threshold of a hypersurface that satisfies a certain power condition was shown to cover optimally all situations of the conjectures for nonrational singularities by comparing it with a local notion of the motivic oscillation index.
Abstract: We prove an upper bound on the log canonical threshold of a hypersurface that satisfies a certain power condition and use it to prove several generalizations of Igusa’s conjecture on exponential sums, with the log canonical threshold in the exponent of the estimates. We show that this covers optimally all situations of the conjectures for nonrational singularities by comparing the log canonical threshold with a local notion of the motivic oscillation index.
11 citations
TL;DR: In this article, the authors use microlocal sheaf theory to show that knots can only have Legendrian isotopic conormal tori if they themselves are isotopic or mirror images.
Abstract: We use microlocal sheaf theory to show that knots can only have Legendrian isotopic conormal tori if they themselves are isotopic or mirror images.
8 citations
TL;DR: In this article, a graded extension of the usual Hecke algebra is described, which acts in a graded fashion on the cohomology of an arithmetic group and can be seen as an algebraic version of the algebra of arithmetic groups.
Abstract: We describe a graded extension of the usual Hecke algebra: it acts in a graded fashion on the cohomology of an arithmetic group .
6 citations
TL;DR: In this article, it was shown that a universally measurable homomorphism between Polish groups is automatically continuous and used to calibrate the strength of the existence of discontinuous homomorphisms between groups.
Abstract: Answering a longstanding problem originating in Christensen’s seminal work on Haar null sets [Math. Scand. 28 (1971), 124–128; Israel J. Math. 13 (1972), 255–260; Topology and Borel Structure. Descriptive Topology and Set Theory with Applications to Functional Analysis and Measure Theory, North-Holland Mathematics Studies, 10 (Notas de Matematica, No. 51). (North-Holland Publishing Co., Amsterdam–London; American Elsevier Publishing Co., Inc., New York, 1974), iii+133 pp], we show that a universally measurable homomorphism between Polish groups is automatically continuous. Using our general analysis of continuity of group homomorphisms, this result is used to calibrate the strength of the existence of a discontinuous homomorphism between Polish groups. In particular, it is shown that, modulo has finite chromatic number.
5 citations
TL;DR: Coates et al. as discussed by the authors showed that the higher genus equivariant Gromov-Witten theory of the Hilbert scheme of is also equivalent to the theories of the triangle in all genera.
Abstract: We study the higher genus equivariant Gromov–Witten theory of the Hilbert scheme of is also shown to be equivalent to the theories of the triangle in all genera. The result establishes a complete case of the crepant resolution conjecture [Bryan and Graber, Algebraic Geometry–Seattle 2005, Part 1, Proceedings of Symposia in Pure Mathematics, 80 (American Mathematical Society, Providence, RI, 2009), 23–42; Coates et al., Geom. Topol. 13 (2009), 2675–2744; Coates & Ruan, Ann. Inst. Fourier (Grenoble) 63 (2013), 431–478].
5 citations