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作者：Calmes, Baptiste; Dotto, Emanuele; Harpaz, Yonatan; Hebestreit, Fabian; Land, Markus; Moi, Kristian; Nardin, Denis; Nikolaus, Thomas; Steimle, Wolfgang
作者单位：Universite d'Artois; University of Warwick; Sorbonne Universite; Universite Paris Cite; University of Bielefeld; Johannes Gutenberg University of Mainz; Royal Institute of Technology; University of Regensburg; University of Munster; University of Augsburg
摘要：We establish a fibre sequence relating the classical Grothendieck-Witt theory of a ring R to the homotopy C-2-orbits of its K-theory and Ranicki's original (non-periodic) symmetric L-theory. We use this fibre sequence to remove the assumption that 2 is a unit in R from various results about Grothendieck-Witt groups. For instance, we solve the homotopy limit problem for Dedekind rings whose fraction field is a number field, calculate the various flavours of Grothendieck-Witt groups of Z, show t...
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作者：Guth, Larry; Maynard, James
作者单位：Massachusetts Institute of Technology (MIT); University of Oxford; University of Oxford
摘要：We prove new bounds for how often Dirichlet polynomials can take large values. This gives improved estimates for a Dirichlet polynomial of length N taking values of size close to N3/4, which is the critical situation for several estimates in analytic number theory connected to prime numbers and the Riemann zeta function. As a consequence, we deduce a zero density estimate N(sigma, T) <= T30(1-sigma )/13+o(1) and asymptotics for primes in short intervals of length x17/30+o(1).
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作者：Campos, Marcelo; Griffiths, Simon; Morris, Robert; Sahasrabudhe, Julian
摘要：The Ramsey number R(k) is the minimum n is an element of N such that every red-blue colouring of the edges of the complete graph Kn on n vertices contains a monochromatic copy of K-k. We prove that R(k) <= (4-epsilon)(k) for some constant epsilon > 0. This is the first exponential improvement over the upper bound of Erdos and Szekeres, proved in 1935.
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作者：Chodosh, Otis; Li, Chao; Minter, Paul; Stryker, Douglas
作者单位：Stanford University; New York University; Princeton University; Institute for Advanced Study - USA; University of Cambridge
摘要：We show that a complete, two-sided stable minimal hypersurface in R5 is flat.
