@article{243, abstract = {Let P(t) ∈ ℚ[t] be an irreducible quadratic polynomial and suppose that K is a quartic extension of ℚ containing the roots of P(t). Let N K/ℚ(X) be a full norm form for the extension K/ℚ. We show that the variety P(t) =N K/ℚ(X)≠ 0 satisfies the Hasse principle and weak approximation. The proof uses analytic methods.}, author = {Timothy Browning and Heath-Brown, Roger}, journal = {Geometric and Functional Analysis}, number = {5}, pages = {1124 -- 1190}, publisher = {Springer Basel}, title = {{Quadratic polynomials represented by norm forms}}, doi = {10.1007/s00039-012-0168-5}, volume = {22}, year = {2012}, } @inproceedings{2440, abstract = {We present an algorithm for computing [X, Y], i.e., all homotopy classes of continuous maps X → Y, where X, Y are topological spaces given as finite simplicial complexes, Y is (d - 1)-connected for some d ≥ 2 (for example, Y can be the d-dimensional sphere S d), and dim X ≤ 2d - 2. These conditions on X, Y guarantee that [X, Y] has a natural structure of a finitely generated Abelian group, and the algorithm finds generators and relations for it. We combine several tools and ideas from homotopy theory (such as Postnikov systems, simplicial sets, and obstruction theory) with algorithmic tools from effective algebraic topology (objects with effective homology). We hope that a further extension of the methods developed here will yield an algorithm for computing, in some cases of interest, the ℤ 2-index, which is a quantity playing a prominent role in Borsuk-Ulam style applications of topology in combinatorics and geometry, e.g., in topological lower bounds for the chromatic number of a graph. In a certain range of dimensions, deciding the embeddability of a simplicial complex into ℝ d also amounts to a ℤ 2-index computation. This is the main motivation of our work. We believe that investigating the computational complexity of questions in homotopy theory and similar areas presents a fascinating research area, and we hope that our work may help bridge the cultural gap between algebraic topology and theoretical computer science.}, author = {Čadek, Martin and Marek Krcál and Matoušek, Jiří and Sergeraert, Francis and Vokřínek, Lukáš and Uli Wagner}, pages = {1 -- 10}, publisher = {SIAM}, title = {{Computing all maps into a sphere}}, year = {2012}, } @article{2438, abstract = {The colored Tverberg theorem asserts that for eve;ry d and r there exists t=t(d,r) such that for every set C ⊂ ℝ d of cardinality (d + 1)t, partitioned into t-point subsets C 1, C 2,...,C d+1 (which we think of as color classes; e. g., the points of C 1 are red, the points of C 2 blue, etc.), there exist r disjoint sets R 1, R 2,...,R r⊆C that are rainbow, meaning that {pipe}R i∩C j{pipe}≤1 for every i,j, and whose convex hulls all have a common point. All known proofs of this theorem are topological. We present a geometric version of a recent beautiful proof by Blagojević, Matschke, and Ziegler, avoiding a direct use of topological methods. The purpose of this de-topologization is to make the proof more concrete and intuitive, and accessible to a wider audience.}, author = {Matoušek, Jiří and Martin Tancer and Uli Wagner}, journal = {Discrete & Computational Geometry}, number = {2}, pages = {245 -- 265}, publisher = {Springer}, title = {{A geometric proof of the colored Tverberg theorem}}, doi = {10.1007/s00454-011-9368-2}, volume = {47}, year = {2012}, } @article{244, abstract = {We investigate the solubility of the congruence xy ≡ 1 (mod p), where p is a prime and x, y are restricted to lie in suitable short intervals. Our work relies on a mean value theorem for incomplete Kloosterman sums.}, author = {Timothy Browning and Haynes, Alan K}, journal = {International Journal of Number Theory}, number = {2}, pages = {481 -- 486}, publisher = {World Scientific Publishing}, title = {{Incomplete kloosterman sums and multiplicative inverses in short intervals}}, doi = { https://doi.org/10.1142/S1793042112501448}, volume = {9}, year = {2012}, } @article{2439, abstract = {A Monte Carlo approximation algorithm for the Tukey depth problem in high dimensions is introduced. The algorithm is a generalization of an algorithm presented by Rousseeuw and Struyf (1998) . The performance of this algorithm is studied both analytically and experimentally.}, author = {Chen, Dan and Morin, Pat and Uli Wagner}, journal = {Computational Geometry: Theory and Applications}, number = {5}, pages = {566 -- 573}, publisher = {Elsevier}, title = {{Absolute approximation of Tukey depth: Theory and experiments}}, doi = {10.1016/j.comgeo.2012.03.001}, volume = {46}, year = {2012}, }