This is a list of inventions and discoveries by Israeli scientists and researchers, working locally or overseas. There are over 6,000 startups currently in Israel. There are currently more than 30 technology companies valued over US$1 billion (unicorn startups) in Israel, more than all of Europe combined. Mathematics Johnson–Lindenstrauss lemma, a mathematical result concerning low-distortion embeddings of points from high-dimensional into low-dimensional Euclidean space contributed by Joram Lindenstrauss. Development of the measurement of rigidity by Elon Lindenstrauss in ergodic theory, and their applications to number theory. Proof of Szemerédi's theorem solved by mathematician Hill Furstenberg. Expansion of axiomatic set theory and the ZF set theory by Abraham Fraenkel. Development of the area of automorphic forms and L-functions by Ilya Piatetski-Shapiro.Tel Aviv University obituary Development of Sauer–Shelah lemma and Shelah cardinal. Development of the first proof of the alternating sign matrix conjecture. Development of Zig-zag product of graphs, a method of combining smaller graphs to produce larger ones used in the construction of expander graphs by Avi Wigderson. Development of Bernstein–Sato polynomial and proof of the Kazhdan–Lusztig conjectures by Joseph Bernstein Generalization of the marriage theorem by obtaining the right transfinite conditions for infinite bipartite graphs. He subsequently proved the appropriate versions of the Kőnig theorem and the Menger theorem for infinite graphs by Ron Aharoni. Development of the Amitsur–Levitzki theorem by Shimshon Amitsur. False discovery rate, a statistical method for regulating Type I errors. Science Chemistry Discovery of quasicrystals by Dan Shechtman of the Technion. The discovery led him to receive the Nobel Prize in Chemistry. Discovery of the role of protein Ubiquitin by Avram Hershko and Aaron Ciechanover of the Technion Institute (together with the American biologist Irwin Rose). The discovery led them to receive the Nobel Prize in Chemistry. Ada Yonath - 2009 Nobel Prize in Chemistry for discovery of the structure and function of ribosomes, the universal cellular factory for the translation of the genetic code to proteins. Development of multiscale models for complex chemical systems by Arieh Warshel and Michael Levitt of the Weizmann Institute of Science (presently at University of Southern California and Stanford University, respectively), together with the Austrian-born American chemist Martin Karplus. The discovery led them to receive the Nobel Prize in Chemistry. Physics Prediction of Quarks by Yuval Ne'eman of Tel Aviv University (together with the American physicist Murray Gell-Mann). Discovery of the Aharonov–Bohm effect by Yakir Aharonov and David Bohm. Formulation