The Dawn of an Algebraic Era in Discrete Geometry?

To me, 2010 looks as annus mirabilis, a miraculous year, in several areas of my mathematical interests. Below I list seven highlights and breakthroughs, mostly in discrete geometry, hoping to share some of my wonder and pleasure with the readers. Of course, hardly any of these great results have come out of the blue: usually the paper I refer to adds the last step to earlier ideas. Since this is an extended abstract (of a nonexistent paper), I will be rather brief, or sometimes completely silent, about the history, with apologies to the unmentioned giants on whose shoulders the authors I do mention have been standing. A careful reader may notice that together with these great results, I will also advertise some smaller results of mine.

[1]  Endre Szemerédi,et al.  Extremal problems in discrete geometry , 1983, Comb..

[2]  László Lovász,et al.  Discrepancy of Set-systems and Matrices , 1986, Eur. J. Comb..

[3]  Leonidas J. Guibas,et al.  Combinatorial complexity bounds for arrangements of curves and spheres , 1990, Discret. Comput. Geom..

[4]  G. Kalai,et al.  A quasi-polynomial bound for the diameter of graphs of polyhedra , 1992, math/9204233.

[5]  L. A S Z L,et al.  Crossing Numbers and Hard Erdős Problems in Discrete Geometry , 1997 .

[6]  L. A. Oa,et al.  Crossing Numbers and Hard Erd} os Problems in Discrete Geometry , 1997 .

[7]  Lajos Rónyai,et al.  A Combinatorial Problem on Polynomials and Rational Functions , 2000, J. Comb. Theory, Ser. A.

[8]  Terence Tao,et al.  A sum-product estimate in finite fields, and applications , 2003, math/0301343.

[9]  J. Pach.  Towards a Theory of Geometric Graphs , 2004 .

[10]  Larry Guth,et al.  Algebraic methods in discrete analogs of the Kakeya problem , 2008, 0812.1043.

[11]  Zeev Dvir,et al.  On the size of Kakeya sets in finite fields , 2008, 0803.2336.

[12]  Miklós Simonovits,et al.  A Combinatorial Distinction Between Unit Circles and Straight Lines: How Many Coincidences Can they Have? , 2009, Comb. Probab. Comput..

[13]  Francisco Santos,et al.  A counterexample to the Hirsch conjecture , 2010, ArXiv.

[14]  M. Gromov.  Singularities, Expanders and Topology of Maps. Part 2: from Combinatorics to Topology Via Algebraic Isoperimetry , 2010 .

[15]  June Huh,et al.  Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs , 2010, 1008.4749.

[16]  Micha Sharir,et al.  Incidences in three dimensions and distinct distances in the plane , 2010, Combinatorics, Probability and Computing.

[17]  Jivr'i Matouvsek.  The number of unit distances is almost linear for most norms , 2010 .

[18]  Nikhil Bansal,et al.  Constructive Algorithms for Discrepancy Minimization , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[19]  Noga Alon,et al.  A Non-linear Lower Bound for Planar Epsilon-nets , 2010, Discrete & Computational Geometry.

[20]  Uri Zwick,et al.  Subexponential lower bounds for randomized pivoting rules for the simplex algorithm , 2011, STOC '11.

[21]  János Pach,et al.  Tight lower bounds for the size of epsilon-nets , 2010, SoCG '11.

[22]  J. Matousek,et al.  The determinant bound for discrepancy is almost tight , 2011, 1101.0767.

[23]  Aleksandar Nikolov,et al.  Tight hardness results for minimizing discrepancy , 2011, SODA '11.

[24]  Haim Kaplan,et al.  Simple Proofs of Classical Theorems in Discrete Geometry via the Guth–Katz Polynomial Partitioning Technique , 2011, Discret. Comput. Geom..

[25]  Roman N. Karasev,et al.  A Simpler Proof of the Boros–Füredi–Bárány–Pach–Gromov Theorem , 2010, Discret. Comput. Geom..