Also around 1995, Kontsevich analyzed the results of Candelas, which gave a general formula for the problem of counting rational curves on a quintic threefold, and he reformulated these results as a precise mathematical conjecture. In 1996, Alexander Givental posted a paper that claimed to prove this conjecture of Kontsevich. Initially, many mathematicians found this paper hard to understand, so there were doubts about its correctness. Subsequently, Bong Lian, Kefeng Liu, and Shing-Tung Yau published an independent proof in a series of papers. Despite controversy over who had published the first proof, these papers are now collectively seen as providing a mathematical proof of the results originally obtained by physicists using mirror symmetry. In 2000, Kentaro Hori and Cumrun Vafa gave another physical proof of mirror symmetry based on T-duality.